# ltc3

***

## maximum\_consecutive\_floors\_without\_special\_floors

**Problem Statement:**

You are given an array of integers representing the floor numbers of a building. The building has special floors where you can only go up or down one floor. You can only go up to the highest floor and down to the lowest floor.

You want to find the maximum number of consecutive floors you can traverse without encountering any special floors.

**Example:**

```python
floors = [0, 2, 3, 4, 5, 6, 7, 8, 9, 10]
max_consecutive_floors = 5  # 5 consecutive floors without special floors (0, 2, 3, 4, 5)
```

**Solution:**

The idea is to keep track of the current maximum consecutive floor count and reset it to 0 whenever a special floor is encountered.

1. **Initialize** the maximum consecutive floor count to 0.
2. **Iterate** over the floors in the array.
3. **If** the current floor is a special floor, reset the maximum consecutive floor count to 0.
4. **Otherwise**, increment the maximum consecutive floor count by 1 and update the maximum consecutive floor count if it exceeds the current maximum.
5. **Return** the maximum consecutive floor count.

**Python Code:**

```python
def maximum_consecutive_floors_without_special_floors(floors):
    max_consecutive_floor_count = 0
    current_consecutive_floor_count = 0

    for floor in floors:
        if floor % 2 == 1:  # Special floors are odd-numbered
            current_consecutive_floor_count = 0
        else:
            current_consecutive_floor_count += 1
            max_consecutive_floor_count = max(max_consecutive_floor_count, current_consecutive_floor_count)

    return max_consecutive_floor_count
```

**Explanation:**

The `maximum_consecutive_floors_without_special_floors()` function takes an array of floor numbers as input and returns the maximum number of consecutive floors you can traverse without encountering any special floors.

The function initializes the maximum consecutive floor count to 0 and the current consecutive floor count to 0. Then, it iterates over the floors in the array. If the current floor is a special floor (odd-numbered), it resets the current consecutive floor count to 0. Otherwise, it increments the current consecutive floor count by 1 and updates the maximum consecutive floor count if it exceeds the current maximum.

Finally, the function returns the maximum consecutive floor count.

**Applications in Real World:**

This problem can be applied in real-world scenarios such as:

* **Building design:** Determining the maximum number of consecutive floors in a building without special floors can help architects design buildings that are accessible and easy to navigate.
* **Crowd management:** In crowded areas such as shopping malls or airports, this algorithm can be used to identify the areas with the highest foot traffic and plan for crowd control measures.
* **Disaster planning:** In the event of a disaster, this algorithm can be used to find the areas in a building that are most accessible and can be used as evacuation points.

***

## find\_latest\_group\_of\_size\_m

**Find Latest Group of Size M**

**Problem Statement:** Given an array of integers representing the arrival time of customers, find the latest group of customers whose size is exactly `m`.

**Example:**

```
Input: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], m = 3
Output: [7, 8, 9]
```

**Approach:**

1. **Initialize a queue:** Use a queue to keep track of the arrival times of customers.
2. **Maintain a window of size m:** Slide a window of size `m` through the queue.
3. **Update the latest group:** As you slide the window, update the latest group of size `m` if the window contains all distinct customers.
4. **Return the latest group:** Once you have processed the entire queue, return the latest group of size `m`.

**Python Implementation:**

```python
def find_latest_group_of_size_m(arrivals, m):
    """
    Finds the latest group of customers of size m.

    Args:
        arrivals (list): The arrival times of customers.
        m (int): The size of the group.

    Returns:
        list: The latest group of customers of size m.
    """

    # Initialize the queue and the latest group.
    queue = []
    latest_group = []

    # Iterate over the arrival times.
    for arrival in arrivals:

        # Add the arrival to the queue.
        queue.append(arrival)

        # If the queue has size m, check if the window contains all distinct customers.
        if len(queue) == m:

            # Get the current window.
            window = queue.copy()

            # Sort the window.
            window.sort()

            # Check if the window contains all distinct customers.
            if len(window) == len(set(window)):

                # Update the latest group.
                latest_group = queue.copy()

        # Remove the oldest arrival from the queue.
        queue.pop(0)

    # Return the latest group.
    return latest_group
```

**Time Complexity:**

The time complexity of this algorithm is `O(n)`, where `n` is the number of customers.

**Space Complexity:**

The space complexity of this algorithm is `O(m)`, where `m` is the size of the group.

**Applications:**

This algorithm can be used in various real-world applications, such as:

* Predicting customer behavior
* Optimizing resource allocation
* Managing customer queues

***

## average\_height\_of\_buildings\_in\_each\_segment

**Problem Statement:**

You are given an array of heights representing the heights of buildings in a city skyline. Each index in the array corresponds to a segment of the skyline, and the height at that index represents the average height of the buildings in that segment.

Your task is to find the average height of the buildings in each segment.

**Example:**

Given an array of heights `[2, 3, 4, 5, 6, 7]`

The output should be:

```
[2, 3.5, 4.5, 5.5, 6.5, 7]
```

**Breakdown and Explanation:**

1. **Input:** The problem provides an array of heights. Each element in the array represents the average height of buildings in a specific segment of the skyline.
2. **Output:** The goal is to compute and return an array where each element represents the average height of buildings in each segment of the skyline.
3. **Algorithm:**
   * **Naive Approach:** Iterate over the input array and compute the average height for each segment. This approach has a time complexity of O(n), where n is the number of elements in the array.
   * **Optimized Approach:** We can slightly optimize the algorithm by using a cumulative sum to compute the average height. Here's how it works:
     * Initialize an array `cumsum` of the same length as the input array.
     * Iterate over the input array and populate `cumsum` by adding the current height to the previous cumulative sum.
     * Compute the average height for each segment by dividing the cumulative sum at the end of each segment by the number of elements in that segment.
4. **Real World Applications:**
   * Urban planning: Determining the average building heights in different areas of a city can help urban planners make decisions about zoning and land use.
   * Real estate analysis: Data on building heights can inform real estate investors about the potential return on investment in different areas.
   * Environmental studies: Measuring the average height of buildings can provide insights into the urban heat island effect, which is the phenomenon of increased temperature in urban areas due to the presence of buildings and other structures.

**Code Implementation:**

```python
def average_height_of_buildings_in_each_segment(heights):
  """
  Computes the average height of buildings in each segment.

  Args:
    heights: An array of average building heights in each segment.

  Returns:
    An array of average building heights for each segment.
  """

  # Initialize cumulative sum array
  cumsum = [0] * len(heights)

  # Compute the cumulative sum
  for i in range(len(heights)):
    cumsum[i] = heights[i] if i == 0 else cumsum[i - 1] + heights[i]

  # Compute the average height for each segment
  average_heights = []
  for i in range(len(heights)):
    start = i
    end = i + 1
    segment_length = end - start
    average_height = cumsum[end - 1] / segment_length if start > 0 else cumsum[end - 1]
    average_heights.append(average_height)

  return average_heights
```

**Example Usage:**

```python
heights = [2, 3, 4, 5, 6, 7]
average_heights = average_height_of_buildings_in_each_segment(heights)
print(average_heights)  # [2, 3.5, 4.5, 5.5, 6.5, 7]
```

***

## minimum\_consecutive\_cards\_to\_pick\_up

**Problem:**

You have a deck of playing cards with numbers from 1 to 9. You can pick up cards from the deck in consecutive order. What is the minimum number of cards you need to pick up to guarantee having at least one of each number?

**Solution:**

To solve this problem, we need to consider the maximum number of cards we can pick up without getting all the numbers. This maximum number is 4, because we can pick up 1, 2, 3, and 4. Picking up any more cards will result in picking up a duplicate card.

Therefore, we need to pick up at least 5 cards to guarantee having at least one of each number.

**Python Implementation:**

```python
def minimum_consecutive_cards_to_pick_up(deck):
  """
  Finds the minimum number of cards to pick up to guarantee having at least one of each number.

  Args:
    deck: A list of integers representing the cards in the deck.

  Returns:
    The minimum number of cards to pick up.
  """

  # Initialize the maximum number of cards we can pick up without getting all the numbers.
  max_cards = 4

  # Check if the deck has fewer than max_cards + 1 cards. If so, we cannot guarantee getting all the numbers.
  if len(deck) < max_cards + 1:
    return -1

  # Sort the deck in ascending order.
  deck.sort()

  # Pick up the first max_cards + 1 cards.
  picked_up_cards = deck[:max_cards + 1]

  # Check if the picked up cards include all the numbers. If not, we need to pick up more cards.
  if len(set(picked_up_cards)) == len(deck):
    return max_cards + 1
  else:
    # Pick up the next card in the deck and check if it includes all the numbers.
    while len(set(picked_up_cards)) < len(deck):
      picked_up_cards.append(deck[len(picked_up_cards)])
    return len(picked_up_cards)
```

**Example:**

```python
deck = [1, 2, 3, 4, 5, 6, 7, 8, 9]
result = minimum_consecutive_cards_to_pick_up(deck)
print(result)  # Output: 5
```

**Applications in the Real World:**

This problem has applications in inventory management and resource allocation. For example, a company may have a warehouse with different types of products. The company needs to decide how many of each product to order to guarantee having at least one of each product in stock.

***

## put\_boxes\_into\_the\_warehouse\_i

**Problem Statement (Simplified)**

There's a warehouse with shelves of different heights. You have a bunch of boxes with heights, and you need to put them on the shelves. You can only put a box on a shelf if the box's height is less than or equal to the shelf's height.

Find the minimum number of shelves you need to use to store all the boxes.

**Optimal Solution**

We can sort the boxes from smallest to largest. Then, we can start putting them on shelves. We always put the smallest boxes on the lowest shelves first. This way, we'll use the minimum number of shelves possible.

**Implementation**

```python
def put_boxes_into_warehouse(boxes, shelf_heights):
  """Returns the minimum number of shelves needed to store all the boxes.

  Args:
    boxes: A list of box heights.
    shelf_heights: A list of shelf heights.

  Returns:
    The minimum number of shelves needed.
  """

  # Sort the boxes and shelves from smallest to largest.
  boxes.sort()
  shelf_heights.sort()

  # Initialize the current shelf index to 0.
  current_shelf = 0

  # Loop through the boxes.
  for box in boxes:
    # Find the smallest shelf that can fit the box.
    while current_shelf < len(shelf_heights) and box > shelf_heights[current_shelf]:
      current_shelf += 1

    # If there's no shelf that can fit the box, return -1.
    if current_shelf == len(shelf_heights):
      return -1

    # Put the box on the shelf.
    current_shelf += 1

  # Return the number of shelves used.
  return current_shelf
```

**Explanation**

The `put_boxes_into_warehouse` function takes two lists as input: `boxes` and `shelf_heights`. It sorts both lists from smallest to largest. Then, it initializes the current shelf index to 0.

The function then loops through the boxes. For each box, it finds the smallest shelf that can fit the box. If there's no shelf that can fit the box, the function returns -1. Otherwise, it puts the box on the shelf and increments the current shelf index.

After looping through all the boxes, the function returns the number of shelves used.

**Real-World Applications**

This problem can be applied to a variety of real-world scenarios, such as:

* Packing items into boxes or suitcases
* Loading items onto a truck or ship
* Assigning employees to tasks based on their skills and experience

***

## maximum\_product\_of\_the\_length\_of\_two\_palindromic\_subsequences

**Problem Statement:**

Given a string `s`, find the maximum product of the lengths of two non-overlapping palindromic subsequences of `s`.

**Best & Performant Solution (Dynamic Programming):**

We can use Dynamic Programming to solve this problem efficiently.

**Breakdown:**

1. **Create a 2D array `dp` to store the longest palindromic subsequence ending at each index:**
   * `dp[i][j] = True` if the substring from index `i` to `j` is a palindrome.
2. **Initialize `dp` for base cases:**
   * `dp[i][i] = True` for all `i`.
   * `dp[i][i+1] = True` if `s[i] == s[i+1]`.
3. **Fill in the remaining entries of `dp` using the following recursive relation:**
   * `dp[i][j] = dp[i+1][j-1] and s[i] == s[j]`
4. **Find the maximum product of the lengths of two non-overlapping palindromic subsequences:**
   * Iterate through `dp` to find pairs of palindromic subsequences that do not overlap.
   * Compute the product of their lengths and keep track of the maximum product.

**Simplified Explanation:**

Imagine a string as a row of letters. We create a grid (`dp` array) where each box represents a pair of letters in the string. We want to find the largest rectangle (subsequence) in this grid that reads the same forwards and backwards (palindromic).

We start with boxes that contain single letters (base cases). Then, for each pair of letters, we check if they're the same and if the box between them is also a palindrome. If so, we mark the box as a palindrome.

Finally, we go through all the boxes and find the largest pair of non-overlapping palindromic rectangles. We multiply their lengths to get the maximum product.

**Example Code:**

```python
def maxProduct(s):
  """
  :type s: str
  :rtype: int
  """

  n = len(s)
  dp = [[False] * n for _ in range(n)]

  # Base cases
  for i in range(n):
    dp[i][i] = True
  for i in range(n-1):
    if s[i] == s[i+1]:
      dp[i][i+1] = True

  # Fill in the remaining entries of dp
  for length in range(3, n+1):
    for i in range(n-length+1):
      j = i + length - 1
      dp[i][j] = dp[i+1][j-1] and s[i] == s[j]

  max_product = 0

  # Find the maximum product of two non-overlapping palindromic subsequences
  for i in range(n):
    for j in range(n):
      if dp[i][j] and i != j:
        max_product = max(max_product, (j-i+1) * (i-dp[0][i]+1) * (n-dp[j][n-1]+1))

  return max_product
```

**Potential Applications:**

* **Sequence Alignment:** Comparing the genomes of different species.
* **Pattern Recognition:** Identifying repeated patterns in text or DNA sequences.
* **Speech Recognition:** Identifying palindromic words in spoken language.

***

## widest\_vertical\_area\_between\_two\_points\_containing\_no\_points

**Problem Statement:**

You have a set of points on a plane. You want to find the widest vertical area between two points that does not contain any other points.

**Brute Force Solution:**

The brute force solution is to consider all pairs of points and find the widest vertical area between them. This solution has a time complexity of O(N^2), where N is the number of points.

```python
def widest_vertical_area_brute_force(points):
  """Finds the widest vertical area between two points that does not contain any other points.

  Args:
    points: A list of points on a plane.

  Returns:
    The width of the widest vertical area.
  """

  max_width = 0
  for point1 in points:
    for point2 in points:
      if point1[0] != point2[0]:
        width = abs(point1[0] - point2[0])
        if width > max_width:
          max_width = width

  return max_width
```

**Optimal Solution using Sorting:**

The optimal solution is to sort the points by their x-coordinates. This allows us to find the widest vertical area by considering only the adjacent pairs of points. The time complexity of this solution is O(N log N).

```python
def widest_vertical_area_optimal(points):
  """Finds the widest vertical area between two points that does not contain any other points.

  Args:
    points: A list of points on a plane.

  Returns:
    The width of the widest vertical area.
  """

  # Sort the points by their x-coordinates.
  points.sort(key=lambda point: point[0])

  max_width = 0
  for i in range(1, len(points)):
    width = abs(points[i][0] - points[i-1][0])
    if width > max_width:
      max_width = width

  return max_width
```

**Real-World Applications:**

This problem can be applied to a variety of real-world problems, including:

* **Computer graphics:** Finding the widest vertical area between two points can be used to create shadows or other effects in computer graphics.
* **Physics:** Finding the widest vertical area between two points can be used to calculate the area of a fluid flow.
* **Robotics:** Finding the widest vertical area between two points can be used to plan the path of a robot.

***

## find\_the\_winner\_of\_an\_array\_game

**Problem Statement** Given an array of integers, return the winner of the game where players take turns taking at least 1 and at most 3 elements from the front of the array and add their values to their score.

**Example**

```
Input: nums = [1, 2, 3, 4]
Output: 1
Explanation: Player 1 takes 1, Player 2 takes 2, Player 1 takes 3, Player 2 takes 4. Both players have 4 points, so Player 1 wins because they started first.
```

**Optimal Solution (Dynamic Programming)**

**Step 1: Initialize a DP array**

* Create a 2D array dp of size (n+1) x (4), where n is the length of the input array nums.
* dp\[i]\[j] will store the maximum score player 1 can achieve starting from index i with j elements taken.

**Step 2: Populate the DP array**

* Start from the bottom-right cell (dp\[n]\[3]) and fill the array diagonally upwards.
* For each cell dp\[i]\[j], calculate the maximum score player 1 can achieve by taking 1, 2, or 3 elements from the array.

```python
def findTheWinner(nums):
    n = len(nums)
    dp = [[0 for _ in range(4)] for _ in range(n+1)]
    
    for i in range(n-1,-1,-1):
        for j in range(3,0,-1):
            dp[i][j] = max(nums[i] + dp[i+j][3], dp[i+1][j])

    if dp[0][1] > dp[0][2]:
        return 1
    return 2
```

**Explanation:**

* We start from the bottom-right corner of the DP array and work our way up diagonally.
* At each cell dp\[i]\[j], we consider taking 1, 2, or 3 elements from the array.
* We calculate the maximum score player 1 can achieve by taking each of these options and store it in dp\[i]\[j].
* For the base case (i = n), player 1 cannot take any elements, so dp\[n]\[j] = 0.
* Finally, we compare the scores player 1 can achieve by taking 1 or 2 elements from the beginning and return the player with the higher score.

**Real-World Applications**

* Can be used in competitive programming games like CodeChef and Codeforces.
* Can be used to analyze optimal strategies in sports, board games, or card games.

***

## maximum\_split\_of\_positive\_even\_integers

**Problem Statement:**

You are given an array of positive even integers. You can split the array into as many subsequences as you want, where each subsequence consists of at least one element. The score of a subsequence is the sum of its elements. The total score of a split is the sum of the scores of each subsequence.

**Goal:**

Find the maximum possible total score that you can achieve by splitting the array.

**Example:**

Input: \[2, 4, 6, 8, 10] Output: 40 Explanation: Split into subsequences \[2, 4], \[6], \[8], \[10]. Total score is 2+4+6+8+10 = 40.

**Approach:**

1. **Sort the array in ascending order:** This ensures that the elements are in order of their smallest to largest values.
2. **Initialize a variable `score` to 0:** This variable will store the maximum total score.
3. **Loop through the sorted array using two pointers, `i` and `j`:**
   * Start `i` and `j` both at the beginning of the array (i.e., `i = 0` and `j = 0`).
   * While `j` is less than the length of the array:
     * If the sum of the elements between `i` and `j` (inclusive) is even, add it to the `score`.
     * Increment `j` by 1.
   * Update `i` to the index of the first even element after `j`.
4. **Return the `score`:** This will be the maximum possible total score.

**Python Implementation:**

```python
def maximum_split_of_positive_even_integers(nums):
  # Sort the array in ascending order
  nums.sort()

  # Initialize the maximum score to 0
  score = 0

  # Loop through the sorted array using two pointers
  i = 0
  j = 0
  while j < len(nums):
    # Check if the sum of the elements between i and j is even
    if sum(nums[i:j+1]) % 2 == 0:
      # Add the sum to the maximum score
      score += sum(nums[i:j+1])
    # Increment j
    j += 1
    # Update i to the index of the first even element after j
    while i < j and nums[i] % 2 != 0:
      i += 1

  # Return the maximum score
  return score
```

**Real-World Application:**

This problem has applications in resource allocation scenarios, such as:

* **Scheduling:** Dividing available resources (e.g., time slots, servers) into subsequences of even workloads to maximize overall efficiency.
* **Inventory Management:** Splitting inventory into subsequences to optimize storage space and minimize waste due to uneven distribution.
* **Financial Planning:** Dividing financial resources into even-sized investments or payments to achieve the best possible return or minimize risk.

***

## check\_if\_two\_expression\_trees\_are\_equivalent

**Problem Statement**

Given the roots of two binary expression trees `root1` and `root2`, return `true` if they are equivalent, otherwise return `false`.

A binary expression tree is a binary tree where each node contains an integer or an operator.

**Solution**

The key to solving this problem is to recursively compare the two trees. We can do this by comparing the values of the root nodes. If they are equal, then we recursively compare the left and right subtrees. If any of the comparisons fail, then the trees are not equivalent.

Here is the Python code for the solution:

```python
def check_if_two_expression_trees_are_equivalent(root1, root2):
    """
    :type root1: TreeNode
    :type root2: TreeNode
    :rtype: bool
    """
    if root1 is None and root2 is None:
        return True
    if root1 is None or root2 is None:
        return False
    if root1.val != root2.val:
        return False
    return check_if_two_expression_trees_are_equivalent(root1.left, root2.left) and check_if_two_expression_trees_are_equivalent(root1.right, root2.right)
```

**Explanation**

The `check_if_two_expression_trees_are_equivalent` function takes two arguments, `root1` and `root2`, which are the roots of two binary expression trees.

The function first checks if both `root1` and `root2` are `None`. If they are, then the trees are equivalent.

If either `root1` or `root2` is `None`, then the trees are not equivalent.

If the values of `root1` and `root2` are not equal, then the trees are not equivalent.

Otherwise, the function recursively compares the left and right subtrees of `root1` and `root2`. If any of the comparisons fail, then the trees are not equivalent.

**Real World Applications**

Binary expression trees can be used to represent mathematical expressions. This can be useful in a variety of applications, such as:

* Compiling expressions
* Evaluating expressions
* Simplifying expressions

**Example**

Consider the following two binary expression trees:

```
     +
    / \
   2   *
      / \
     3   4

     +
    / \
   2   *
      / \
     3   5
```

These two trees are not equivalent because the value of the right child of the root node is different.

***

## paths\_in\_maze\_that\_lead\_to\_same\_room

**Problem Statement:**

The given problem asks us to find all the paths in a maze that lead to the same room. A maze is represented as a 2D array where 0 represents an open path and 1 represents a blocked path.

**Approach:**

We will use a Depth-First Search (DFS) to traverse the maze and keep track of the paths that lead to each room. We will use a stack to keep track of the path we have traversed so far.

**Implementation:**

```python
def find_paths(maze, start_room, end_room):
  """Finds all the paths from start_room to end_room in the maze.

  Args:
    maze: A 2D array representing the maze.
    start_room: The starting room.
    end_room: The ending room.

  Returns:
    A list of all the paths from start_room to end_room.
  """

  # Create a stack to keep track of the path we have traversed so far.
  stack = [start_room]

  # Create a set to keep track of the rooms we have visited.
  visited = set()

  # While the stack is not empty, pop the current room from the stack and check if it is the end room.
  while stack:
    current_room = stack.pop()

    # If the current room is the end room, return the stack, which contains the path from start_room to end_room.
    if current_room == end_room:
      return stack

    # If the current room has not been visited, add it to the visited set and push all of its neighbors onto the stack.
    if current_room not in visited:
      visited.add(current_room)
      for neighbor in get_neighbors(current_room, maze):
        stack.append(neighbor)

  # If the stack is empty, there is no path from start_room to end_room.
  return []

  # Helper function to get the neighbors of a room in the maze.
  def get_neighbors(room, maze):
    """Gets the neighbors of a room in the maze.

    Args:
      room: The room to get the neighbors of.
      maze: A 2D array representing the maze.

    Returns:
      A list of the neighbors of the room.
    """

    neighbors = []

    # Check the room to the north.
    if room[0] > 0 and maze[room[0] - 1][room[1]] == 0:
      neighbors.append((room[0] - 1, room[1]))

    # Check the room to the east.
    if room[1] < len(maze[0]) - 1 and maze[room[0]][room[1] + 1] == 0:
      neighbors.append((room[0], room[1] + 1))

    # Check the room to the south.
    if room[0] < len(maze) - 1 and maze[room[0] + 1][room[1]] == 0:
      neighbors.append((room[0] + 1, room[1]))

    # Check the room to the west.
    if room[1] > 0 and maze[room[0]][room[1] - 1] == 0:
      neighbors.append((room[0], room[1] - 1))

    return neighbors
```

**Example:**

Consider the following maze:

```
0 0 0 0 1
0 1 1 0 1
0 1 0 0 0
1 1 0 0 0
```

The path from (0, 0) to (4, 4) is:

```
(0, 0) -> (1, 0) -> (1, 1) -> (1, 2) -> (1, 3) -> (1, 4) -> (2, 4) -> (3, 4) -> (4, 4)
```

**Real-World Applications:**

This problem has applications in many areas, such as:

* **Robotics:** Robots can use path planning algorithms to navigate through mazes and other complex environments.
* **Networking:** Routing algorithms can use path planning algorithms to find the best path for data to travel through a network.
* **Computer games:** Path planning algorithms can be used to create AI characters that can navigate through game worlds.

***

## design\_bitset

**Problem Statement:** You are given an array of integers, each representing a different word. Find the longest word that can be formed by concatenating any two or more words in the array.

**Example:** Input: \["cat", "dog", "fish", "food", "catfish"] Output: "catfish"

**Solution:** We can use a bitset to solve this problem. A bitset is a data structure that stores a set of bits. We can use a bitset to represent a set of words. If the i-th bit of the bitset is set, then it means that the i-th word is in the set.

We can start by creating a bitset for each word in the array. Then, we can iterate over all the pairs of words in the array. For each pair of words, we can create a new bitset that represents the union of the two bitsets for the individual words. If the length of the new bitset is greater than the length of the longest word we have seen so far, then we update the longest word.

Here is the Python code for the solution:

```python
def longest_word(words):
  """Finds the longest word that can be formed by concatenating any two or more words in the array.

  Args:
    words: A list of integers, each representing a different word.

  Returns:
    The longest word that can be formed by concatenating any two or more words in the array.
  """

  # Create a bitset for each word in the array.
  bitsets = [bitset() for _ in range(len(words))]
  for i, word in enumerate(words):
    bitsets[i].set(word)

  # Iterate over all the pairs of words in the array.
  longest_word = ""
  for i in range(len(words)):
    for j in range(i + 1, len(words)):
      # Create a new bitset that represents the union of the two bitsets for the individual words.
      new_bitset = bitsets[i] | bitsets[j]

      # If the length of the new bitset is greater than the length of the longest word we have seen so far, then we update the longest word.
      if new_bitset.size() > len(longest_word):
        longest_word = ""
        for word in bitsets[i].keys():
          longest_word += word
        for word in bitsets[j].keys():
          longest_word += word

  return longest_word
```

**Time Complexity:** O(n^2), where n is the number of words in the array.

**Space Complexity:** O(n), where n is the number of words in the array.

**Applications:** This problem can be applied in real-world scenarios where we need to find the longest word that can be formed by combining multiple words. For example, this problem can be used to find the longest word that can be formed by anagramming a set of words.

***

## minimum\_deletions\_to\_make\_character\_frequencies\_unique

**Problem Statement:**

Given a string, determine the minimum number of characters that must be deleted to make the frequency of each character unique.

**Solution:**

1. **Character Frequency Count:**

   Count the frequency of each character in the string. Store it in a dictionary `freq_count`.
2. **Unique Frequency Count:**

   Find the unique count of the characters in the string. This is the maximum count allowed for each character. Store it in a variable `unique_count`.
3. **Minimum Deletions:**

   Iterate over the characters in the string. For each character, subtract its frequency from the unique count. If the result is negative, add the absolute value to the `minimum_deletions` variable.
4. **Return the Minimum Deletions:**

   Return the `minimum_deletions` variable.

**Python Implementation:**

```python
def minimum_deletions_to_make_character_frequencies_unique(string):
  # Count character frequencies
  freq_count = {}
  for char in string:
    freq_count[char] = freq_count.get(char, 0) + 1

  # Find unique frequency count
  unique_count = len(freq_count)

  # Calculate minimum deletions
  minimum_deletions = 0
  for char in freq_count:
    deletions = freq_count[char] - unique_count
    if deletions > 0:
      minimum_deletions += deletions

  return minimum_deletions
```

**Example:**

```python
string = "aaabb"
result = minimum_deletions_to_make_character_frequencies_unique(string)
print(result)  # Output: 2
```

**Explanation:**

In the example, the character 'a' appears twice, and the character 'b' appears three times. The maximum unique count is 2 (since there are only two different characters). Therefore, we need to delete one 'a' and one 'b' to make the frequencies unique.

**Real-World Applications:**

* **Data Analysis:** Analyzing the distribution of data and identifying anomalies or duplicate values.
* **Data Cleaning:** Removing duplicate or unnecessary data before further analysis or processing.
* **Text Processing:** Ensuring that character frequencies are unique to improve search results, language models, and other NLP tasks.

***

## minimum\_health\_to\_beat\_game

**Problem:**

In a video game, you have to beat a boss to win. The boss has a certain amount of health, and you have a certain amount of health. You can attack the boss to damage it, but the boss can also attack you. If your health reaches 0 before the boss's health reaches 0, you lose.

You want to find the minimum amount of health you need to have in order to beat the boss.

**Example:**

* Boss health: 100
* Your health: 50
* Attack damage: 10

In this example, you would need to have at least (100 / 10) + 1 = 11 health to beat the boss. This is because you would need to attack the boss 10 times to kill it, and you would lose 10 health each time you are attacked. Therefore, you need to have at least 11 health to survive the 10 attacks and still be able to defeat the boss.

**Solution:**

The best and most performant solution for this problem is to use the following formula:

```
minimum_health = (boss_health / attack_damage) + 1
```

This formula calculates the minimum amount of health you need to have in order to beat the boss. It takes into account the boss's health, your attack damage, and the fact that you will lose 1 health each time you are attacked.

**Code:**

```python
def minimum_health_to_beat_game(boss_health, attack_damage):
  """
  Calculates the minimum amount of health needed to beat a boss in a video game.

  Args:
    boss_health: The health of the boss.
    attack_damage: The damage of your attacks.

  Returns:
    The minimum amount of health needed to beat the boss.
  """

  minimum_health = (boss_health / attack_damage) + 1

  return minimum_health


# Example usage
boss_health = 100
attack_damage = 10
minimum_health = minimum_health_to_beat_game(boss_health, attack_damage)
print(minimum_health)  # Output: 11
```

**Explanation:**

The `minimum_health_to_beat_game()` function takes two arguments: the boss's health and your attack damage. It then calculates the minimum amount of health you need to have in order to beat the boss using the formula described above. The function returns the minimum amount of health as an integer.

**Real-World Applications:**

This problem is a simplified version of a real-world problem that arises in game development. In order to balance a game, developers need to determine how much health players should have in order to have a fair chance of winning. The formula provided in this solution can be used to calculate the minimum amount of health that players need to have in order to beat a boss.

***

## minimum\_moves\_to\_make\_array\_complementary

**Problem:**

Given an array of integers, the task is to find the minimum number of moves required to make the array complementary, where a pair of elements is called complementary if their sum is equal to the length of the array.

**Solution:**

We can use a greedy approach to solve this problem. The idea is to sort the array in ascending order and then iterate through it. For each element, we check if its complement (i.e., the sum of the element and the required length) is present in the array. If it is, we remove both elements from the array and continue. If it is not, we add the element to a list of uncomplemented elements.

**Python Implementation:**

```python
def minimum_moves_to_make_array_complementary(arr, n):
    """
    Returns the minimum number of moves required to make the array complementary.

    :param arr: The input array.
    :type arr: List[int]
    :param n: The length of the array.
    :type n: int
    :return: The minimum number of moves required to make the array complementary.
    :rtype: int
    """

    # Sort the array in ascending order.
    arr.sort()

    # Initialize the number of moves to 0.
    moves = 0

    # Initialize the list of uncomplemented elements.
    uncomplemented = []

    # Iterate through the array.
    for i in range(len(arr)):
        # Check if the current element's complement is present in the array.
        if arr[i] + n in arr:
            # If the complement is present, remove both elements from the array.
            arr.remove(arr[i])
            arr.remove(arr[i] + n)
            moves += 1
        else:
            # If the complement is not present, add the current element to the list of uncomplemented elements.
            uncomplemented.append(arr[i])

    # Return the number of moves plus the number of uncomplemented elements.
    return moves + len(uncomplemented)
```

**Example:**

```python
arr = [1, 2, 3, 4, 5]
n = 5

result = minimum_moves_to_make_array_complementary(arr, n)
print(result)  # Output: 2
```

**Real-World Applications:**

This problem has applications in various real-world scenarios, such as:

* **Inventory Management:** A company can use this approach to determine the minimum number of additional items it needs to add to its inventory to ensure that it has complementary products available.
* **Scheduling:** A scheduler can use this approach to determine the minimum number of shifts that need to be added to a schedule to ensure that employees are working complementary shifts.
* **Resource Allocation:** A resource manager can use this approach to determine the minimum number of additional resources that need to be allocated to a project to ensure that all required resources are available.

***

## maximum\_erasure\_value

**Problem Description**

Given an array of integers nums and an integer k, you can erase up to k elements from the array. Return the maximum sum you can achieve after erasing elements.

**Example**

```
Input: nums = [1,2,3,4], k = 2
Output: 7
Explanation: Erase the two smallest numbers 1 and 2.
```

**Solution**

**Idea:**

The optimal solution is to erase the k smallest elements from the array.

**Algorithm:**

1. Sort the array in ascending order.
2. Remove the k smallest elements from the array.
3. Calculate the sum of the remaining elements.

**Implementation:**

```python
def maximum_erasure_value(nums, k):
    """
    :type nums: List[int]
    :type k: int
    :rtype: int
    """
    # Sort the array in ascending order
    nums.sort()

    # Remove the k smallest elements from the array
    nums = nums[k:]

    # Calculate the sum of the remaining elements
    return sum(nums)
```

**Complexity Analysis:**

* Time complexity: O(n log n), where n is the length of the array. Sorting the array takes O(n log n) time. The rest of the operations take O(n) time.
* Space complexity: O(1), since we do not allocate any additional space.

**Real World Applications**

This problem can be applied in a variety of real-world scenarios, such as:

* **Data cleaning:** Removing outliers or noisy data points from a dataset.
* **Feature selection:** Selecting the most important features from a dataset for a machine learning model.
* **Resource optimization:** Allocating resources to the most important tasks.

***

## count\_lattice\_points\_inside\_a\_circle

**LeetCode Problem:** <https://leetcode.com/problems/count-lattice-points-inside-a-circle/>

**Problem Statement:**

Given a circle with the center at `(0, 0)` and the radius `r`, and a rectangle with the lower left corner at `(x1, y1)` and the upper right corner at `(x2, y2`), return the number of lattice points (i.e., points with integer coordinates) that are inside the circle.

**Best & Performant Solution in Python:**

```python
def countLatticePoints(circles: int, rectangles: int) -> int:
    # Initialize a set to store the lattice points inside the circle
    lattice_points = set()

    # Iterate over the circles
    for circle in range(circles):
        # Get the center and radius of the circle
        x_c, y_c, r = map(int, input().split())

        # Iterate over the rectangles
        for rectangle in range(rectangles):
            # Get the coordinates of the rectangle
            x1, y1, x2, y2 = map(int, input().split())

            # Check if the rectangle intersects the circle
            if is_intersect(x_c, y_c, r, x1, y1, x2, y2):
                # Get the lattice points inside the rectangle that intersect the circle
                points = get_lattice_points(x_c, y_c, r, x1, y1, x2, y2)
                
                # Add the lattice points to the set
                lattice_points.update(points)

    # Return the number of unique lattice points in the set
    return len(lattice_points)


def is_intersect(x_c, y_c, r, x1, y1, x2, y2):
    # Check if the center of the rectangle is inside the circle
    if (x_c - x1) ** 2 + (y_c - y1) ** 2 <= r ** 2:
        return True

    # Check if the center of the rectangle is above the circle
    if y_c > y2:
        return False

    # Check if the center of the rectangle is to the right of the circle
    if x_c > x2:
        return False

    # There must be an intersection
    return True


def get_lattice_points(x_c, y_c, r, x1, y1, x2, y2):
    # Get the center of the rectangle
    x_r = (x1 + x2) // 2
    y_r = (y1 + y2) // 2

    # Initialize a set to store the lattice points
    lattice_points = set()

    # Iterate over the lattice points within the rectangle
    for x in range(x1, x2 + 1):
        for y in range(y1, y2 + 1):
            # Check if the lattice point is inside the circle
            if (x - x_c) ** 2 + (y - y_c) ** 2 <= r ** 2:
                # Add the lattice point to the set
                lattice_points.add((x, y))

    # Return the set of lattice points
    return lattice_points
```

**Breakdown and Explanation:**

**Step 1: Handle User Input:**

The program starts by reading the number of circles and rectangles from the standard input.

**Step 2: Iterate Over Circles:**

For each circle, the program reads its center and radius from the input.

**Step 3: Iterate Over Rectangles:**

For each rectangle, the program reads its coordinates from the input.

**Step 4: Check for Intersection:**

The `is_intersect` function determines if the rectangle and the circle intersect. It calculates the distance between the rectangle's center and the circle's center and checks if it is less than or equal to the circle's radius.

**Step 5: Get Lattice Points:**

If there is an intersection, the `get_lattice_points` function returns a set of lattice points (integer coordinate points) that lie inside both the rectangle and the circle.

**Step 6: Update Lattice Points Set:**

For each intersection, the program updates the `lattice_points` set with the lattice points obtained from `get_lattice_points`.

**Step 7: Count Lattice Points:**

Finally, the program returns the number of unique lattice points in the `lattice_points` set.

**Applications:**

This algorithm can be used in various real-world applications, such as:

* Calculating the area of overlap between a circle and a rectangle in geometry.
* Finding the number of customers within a certain radius of a store location in marketing.
* Simulating particle collisions in physics.

***

## find\_all\_lonely\_numbers\_in\_the\_array

### Problem Statement

Given an array of integers `nums`, return a list of all the **lonely numbers** in the array. A **lonely number** is a number that appears **exactly once** in the array.

### Breakdown and Explanation

Here's a breakdown of the problem statement:

1. **Input**: An array of integers `nums`.
2. **Output**: A list of all the lonely numbers in `nums`.
3. **Lonely Number**: A number that appears exactly once in `nums`.

### Simplified Explanation

Imagine you have a group of friends, and each friend has a unique number assigned to them. You want to find out which friends are lonely, meaning they don't have any other friends with the same number.

### Performant Solution in Python

The most performant solution in Python is to use a `hash table` to store the numbers and their frequencies. Here's how it works:

1. Create a hash table called `num_freq` to store the frequency of each number.
2. Iterate over the array `nums` and update the frequency of each number in `num_freq`.
3. Iterate over the keys in `num_freq` and add the numbers with a frequency of 1 to the output list.

```python
def find_all_lonely_numbers_in_the_array(nums):
  # Create a hash table to store number frequencies
  num_freq = {}

  # Iterate over the array and update frequencies
  for num in nums:
    if num not in num_freq:
      num_freq[num] = 0
    num_freq[num] += 1

  # Find lonely numbers
  lonely_numbers = []
  for num, freq in num_freq.items():
    if freq == 1:
      lonely_numbers.append(num)

  return lonely_numbers
```

### Real-World Application

This algorithm can be used in various real-world applications, such as:

* **Market research**: Identifying unique preferences among customers.
* **Financial analysis**: Detecting outliers in financial data.
* **Fraud detection**: Identifying suspicious transactions that appear only once.

***

## number\_of\_smooth\_descent\_periods\_of\_a\_stock

**Problem Statement:**

You are given a stock's price history over several days. Each day's price is represented by a positive integer. You want to find out the number of smooth descent periods within the given price history.

A smooth descent period is defined as a sequence of two or more consecutive days where the stock price strictly decreases from one day to the next.

**Solution:**

**Breakdown:**

To solve this problem, we can iterate through the given price history and keep track of the current smooth descent period. If the price on the next day is lower than the current day's price, we increment the current smooth descent period by 1. Otherwise, we reset the current smooth descent period to 0.

**Implementation:**

```python
def number_of_smooth_descent_periods(prices):
    smooth_descent_period = 0
    count = 0

    for i in range(1, len(prices)):
        if prices[i] < prices[i - 1]:
            smooth_descent_period += 1
        else:
            count += (smooth_descent_period > 1)
            smooth_descent_period = 0

    count += (smooth_descent_period > 1)

    return count
```

**Example:**

```python
prices = [3, 2, 1, 4, 3, 2, 6, 5]
result = number_of_smooth_descent_periods(prices)
print(result)  # Output: 2
```

**Explanation:**

In the given price history, there are two smooth descent periods:

1. \[3, 2, 1]
2. \[6, 5]

Therefore, the output is 2.

**Potential Applications:**

This problem can be applied to various real-world scenarios, such as:

* **Stock market analysis:** Identifying smooth descent periods can help investors identify potential selling opportunities or market downtrends.
* **Financial forecasting:** By analyzing historical price patterns, analysts can predict future stock movements and make informed investment decisions.
* **Risk management:** Understanding the frequency and duration of smooth descent periods can help financial institutions assess and mitigate their risk exposure.

***

## first\_day\_where\_you\_have\_been\_in\_all\_the\_rooms

**Problem Statement:**

You have n rooms numbered from 0 to n - 1. Initially, all rooms are locked. You start at room 0.

You are given an array keys where keys\[i] is the key that unlocks room i.

Return the first day where you have been in all the rooms. If no such day exists, return -1.

**Example 1:**

```
Input: keys = [1, 2, 3]
Output: 3
Explanation:
On day 1, you go to room 0 and open it with the key.
On day 2, you go to room 1 and open it with the key.
On day 3, you go to room 2 and open it with the key.
Therefore, you have visited all rooms by day 3.
```

**Example 2:**

```
Input: keys = [1]
Output: -1
Explanation:
You cannot open any rooms since you only have one key for room 1.
```

**Solution:**

The key to solving this problem is to realize that you can only visit a room if you have the key. Therefore, we can use a queue to keep track of the rooms we can visit and a set to keep track of the rooms we have already visited.

Here is a step-by-step explanation of the solution:

1. Initialize a queue with room 0.
2. Initialize a set with room 0.
3. While the queue is not empty: a. Pop the first room from the queue. b. Iterate through the keys of the current room. c. For each key, if it corresponds to a room that has not been visited, add it to the queue and the set.
4. If the number of rooms in the set is equal to the total number of rooms, return the current day.
5. Otherwise, return -1.

**Code:**

```python
def first_day_where_you_have_been_in_all_the_rooms(keys):
    n = len(keys)
    queue = [0]
    visited = set([0])
    day = 1

    while queue:
        room = queue.pop(0)

        for key in keys[room]:
            if key not in visited:
                queue.append(key)
                visited.add(key)

        if len(visited) == n:
            return day

    return -1
```

**Applications:**

This problem can be applied to real-world scenarios such as:

* Managing access to rooms in a building.
* Planning a route through a maze.
* Solving puzzles involving locked doors.

***

## rearrange\_array\_elements\_by\_sign

Q: **Rearrange Array Elements by Sign** Given an integer array nums, rearrange the elements to make all the positive elements appear before the negative elements. Return any valid rearrangement of the elements.

Example 1: Input: nums = \[3,4,-1,1] Output: \[3,1,4,-1]

Example 2: Input: nums = \[7,-6,5,-3] Output: \[7,5,-6,-3]

**Solution Breakdown:**

We will use two pointers, one for the positive elements and one for the negative elements. We will iterate through the array and move the positive elements to the left and the negative elements to the right.

**Simplified Explanation:**

We have an array of numbers, some are positive and some are negative. We want to rearrange the numbers so that all the positive numbers come first, followed by all the negative numbers.

We can use two pointers to do this. One pointer will start at the beginning of the array and the other pointer will start at the end of the array.

We will then iterate through the array, moving the positive numbers to the left and the negative numbers to the right.

**Python Implementation:**

```python
def rearrange_array_elements_by_sign(nums):
  """
  Rearranges the elements of an array to make all the positive elements appear before the negative elements.

  Args:
    nums: A list of integers.

  Returns:
    A list of integers.
  """

  # Initialize two pointers, one for the positive elements and one for the negative elements.
  positive_pointer = 0
  negative_pointer = len(nums) - 1

  # Iterate through the array.
  while positive_pointer < negative_pointer:
    # If the current element is positive, move it to the left.
    if nums[positive_pointer] >= 0:
      positive_pointer += 1

    # If the current element is negative, move it to the right.
    else:
      nums[positive_pointer], nums[negative_pointer] = nums[negative_pointer], nums[positive_pointer]
      negative_pointer -= 1

  # Return the rearranged array.
  return nums
```

**Real World Applications:**

This problem can be applied to any situation where you need to sort a list of numbers by their sign. For example, you could use this problem to:

* Sort a list of financial transactions by their type (e.g., income or expenses).
* Sort a list of customer reviews by their sentiment (e.g., positive or negative).
* Sort a list of votes by their direction (e.g., for or against).

***

## maximum\_binary\_string\_after\_change

### Problem Statement

Given a binary string `s`, you can perform the following operation any number of times:

* Select any two adjacent digits of `s` and flip them (i.e., "0" becomes "1" and "1" becomes "0").

Return the maximum binary string you can obtain after any number of operations.

#### Solution

1. **Convert the string to a list:** We convert the input binary string `s` to a list of characters `s_list`. This allows us to easily manipulate and flip the digits in the string.
2. **Iterate over the list:** We iterate through the list of characters, checking each pair of adjacent digits.
3. **Flip "01" pairs:** If we encounter a pair of adjacent digits "01", we flip them by replacing "0" with "1" and "1" with "0". We keep doing this until there are no more "01" pairs in the list.
4. **Check for all "0"s:** After flipping all "01" pairs, we check if the list contains only "0"s. If it does, it means we cannot flip any more pairs, so we return the updated list as a string.
5. **Flip the last "1" to "0" (if possible):** If the list contains a "1" at the end, we flip it to "0" because "10" can be flipped to "01" and we want to maximize the binary string.
6. **Convert the list back to a string:** Finally, we convert the updated list back to a string and return it as the maximum binary string.

#### Python Implementation

```python
def maximum_binary_string_after_change(s):
    
    # Convert the string to a list
    s_list = list(s)
    
    # Iterate over the list and flip "01" pairs
    i = 0
    while i < len(s_list) - 1:
        if s_list[i] == '0' and s_list[i + 1] == '1':
            s_list[i] = '1'
            s_list[i + 1] = '0'
            i += 1  # Skip the next pair since it's already flipped
        i += 1
    
    # Check for all "0"s
    if all(c == '0' for c in s_list):
        return ''.join(s_list)
    
    # Flip the last "1" to "0" (if possible)
    if s_list[-1] == '1':
        s_list[-1] = '0'
    
    # Convert the list back to a string
    return ''.join(s_list)
```

#### Example

**Input:** `s = "00110"`

**Output:** `"11111"`

**Explanation:** We can flip the "01" pairs at indices (1, 2) and (3, 4) to get "11111".

#### Applications

The `maximum_binary_string_after_change` function can be applied in various real-world scenarios:

* **Data Optimization:** In data mining, optimizing digital signals or data transmissions often involves maximizing binary strings to represent information efficiently.
* **Error Correction:** In communication systems, binary strings are used to transmit data. This function can be used to correct errors in transmitted data by flipping incorrect bits.
* **Binary Search Optimization:** In computer science, binary search algorithms can be made more efficient by maximizing the binary representation of the search range.
* **Secret Codes and Encryption:** Binary strings are commonly used in cryptography to represent secret codes. The given function can be utilized to create stronger and more secure codes.

***

## diameter\_of\_n\_ary\_tree

***

ERROR OCCURED diameter\_of\_n\_ary\_tree

Can you please implement the best & performant solution for the given leetcode problem in python, then simplify and explain the given content for competitive coding?

* breakdown and explain each topic or step in detail and simplified manner (simplify in very plain english like explaining to a child).
* give real world complete code implementations and examples for each. provide potential applications in real world.

  ```
    500 Internal error encountered.
  ```

***

***

## subrectangle\_queries

**Problem:**

Given a rectangular matrix and a list of queries, where each query is either:

* `update(row, col, val)`: Update the value of the cell at `(row, col)` to `val`.
* `getSum(row1, col1, row2, col2)`: Return the sum of all cells within the rectangle `[(row1, col1), (row2, col2)]`.

**Solution:**

We can use a prefix sum matrix to efficiently process both types of queries.

**Prefix Sum Matrix:**

A prefix sum matrix is a matrix where each element `(i, j)` stores the sum of all elements in the submatrix `[(0, 0), (i, j)]`. It can be constructed in O(n^2) time.

**Queries:**

* **`update(row, col, val)`:** To update a cell, we only need to update the prefix sum for the cell and its descendants (i.e., cells to its right and below).
* **`getSum(row1, col1, row2, col2)`:** To get the sum of a rectangle, we can simply subtract the prefix sum at `(row1, col1)` from the prefix sum at `(row2, col2)`.

**Implementation:**

```python
class SubrectangleQueries:
    def __init__(self, matrix):
        self.matrix = matrix
        self.prefix_sum = [[0 for _ in row] for row in matrix]

        # Build the prefix sum matrix
        for i in range(len(matrix)):
            for j in range(len(matrix[0])):
                self.prefix_sum[i][j] = matrix[i][j] + (self.prefix_sum[i - 1][j] if i > 0 else 0) + (self.prefix_sum[i][j - 1] if j > 0 else 0) - (self.prefix_sum[i - 1][j - 1] if i > 0 and j > 0 else 0)

    def update(self, row, col, val):
        diff = val - self.matrix[row][col]
        self.matrix[row][col] = val

        # Update the prefix sum for the cell and its descendants
        for i in range(row, len(self.matrix)):
            for j in range(col, len(self.matrix[0])):
                self.prefix_sum[i][j] += diff

    def getSum(self, row1, col1, row2, col2):
        # Subtract the prefix sum at (row1, col1) from the prefix sum at (row2, col2)
        return self.prefix_sum[row2][col2] - (self.prefix_sum[row1 - 1][col2] if row1 > 0 else 0) - (self.prefix_sum[row2][col1 - 1] if col1 > 0 else 0) + (self.prefix_sum[row1 - 1][col1 - 1] if row1 > 0 and col1 > 0 else 0)
```

**Potential Applications:**

* Image processing
* Heat transfer modeling
* Fluid dynamics simulations

***

## smallest\_greater\_multiple\_made\_of\_two\_digits

**Problem Statement:**

Given a two-digit number, find the smallest multiple of that number that is also a two-digit number.

**Example:**

* Input: 12
* Output: 24

**Optimal Python Solution with Explanation:**

```python
def smallest_greater_multiple(number):
  """
  :param number: The two-digit number.
  :return: The smallest multiple of that number that is also a two-digit number.
  """

  # Check if the number is already a two-digit number.
  if number >= 10 and number <= 99:
    return number

  # Find the smallest multiple of the number that is also a two-digit number.
  result = number
  while result < 100:
    result += number

  # Return the smallest multiple.
  return result
```

**Breakdown of the Solution:**

1. **Check if the number is already a two-digit number.** If the number is already a two-digit number, then it is already the smallest multiple of itself that is also a two-digit number. So, we can simply return the number.
2. **Find the smallest multiple of the number that is also a two-digit number.** We can do this by starting with the number and adding the number to itself until we get a two-digit number.
3. **Return the smallest multiple.** Once we have found the smallest multiple, we can return it.

**Real-World Applications:**

This problem can be used in various real-world applications, such as:

* **Financial calculations:** In finance, we often need to calculate multiples of numbers. For example, we may need to calculate the interest on a loan or the value of a stock.
* **Inventory management:** In inventory management, we may need to calculate the number of items we need to order to meet a certain demand.
* **Scheduling:** In scheduling, we may need to calculate the time it will take to complete a task or the number of people we need to assign to a project.

By understanding the problem and implementing the optimal solution, we can efficiently solve real-world problems that require finding the smallest multiple of a number that is also a two-digit number.

***

## make\_sum\_divisible\_by\_p

**Problem Statement:** Given an array of positive integers `nums` and a positive integer `p`, find the smallest number of integers that need to be removed from `nums` such that the sum of the remaining integers is divisible by `p`.

**Solution:** We can use the concept of modular arithmetic to solve this problem. Let's represent the sum of all integers in `nums` modulo `p` as `s`. By removing an integer, we reduce `s` by that integer modulo `p`. Since `p` is a positive integer, `s - p` is equivalent to `s - (s % p)` modulo `p`.

Therefore, to make the sum of the remaining integers divisible by `p`, we need to find the smallest subset of integers that sum up to `s - (s % p)` modulo `p`. We can use a greedy approach to find such a subset.

**Implementation:**

```python
def make_sum_divisible_by_p(nums, p):
    """
    Finds the smallest number of integers that need to be removed from nums such that the sum of the remaining integers is divisible by p.

    Args:
        nums (list): A list of positive integers.
        p (int): A positive integer.

    Returns:
        int: The smallest number of integers that need to be removed.
    """
    # Calculate the sum of all integers in nums modulo p.
    s = sum(nums) % p
    
    # If the sum is already divisible by p, no integers need to be removed.
    if s == 0:
        return 0
    
    # Sort the integers in nums in ascending order.
    nums.sort()
    
    # Initialize a variable to keep track of the number of integers that need to be removed.
    count = 0
    
    # Iterate over the integers in nums.
    for num in nums:
        # If the current sum plus the current integer is divisible by p, remove the current integer.
        if (s + num) % p == 0:
            count += 1
            s = (s + num) % p
        # Otherwise, continue to the next integer.
        else:
            continue
    
    # Return the number of integers that need to be removed.
    return count
```

**Example:**

```python
nums = [3, 1, 4, 2]
p = 6
res = make_sum_divisible_by_p(nums, p)
print(res)  # Output: 1
```

**Explanation:** The sum of the integers in `nums` modulo `p` is `3 + 1 + 4 + 2 = 10`, which is not divisible by `p = 6`. We can remove the integer `1` to make the sum divisible by `p`, which is the smallest possible number of integers that can be removed. Therefore, the function returns `1`.

**Applications:** This problem has applications in various fields, including:

* **Computer science:** In designing algorithms to solve problems that involve modular arithmetic, such as finding the remainder of a division operation or finding the inverse of an element in a modular ring.
* **Cryptography:** In designing cryptographic protocols that rely on modular arithmetic, such as the RSA encryption algorithm.
* **Number theory:** In studying the properties of numbers and their relationships to each other, including topics such as prime numbers, divisibility, and modular arithmetic.

***

## stone\_game\_vi

**Problem Statement:**

You are playing a game with your friend. There are **n** piles of stones, and each pile has a certain number of stones.

On each turn, you and your friend can choose **any pile** and remove **any number** of stones from it. The one who takes the **last stone** wins the game.

You want to determine if you can win the game, regardless of your friend's strategy.

**Example:**

```
piles = [2, 7, 9]
Output: True
```

**Breakdown:**

* **Game Theory:** This problem involves game theory, where two players make optimal moves to achieve a desired outcome.
* **Dynamic Programming:** We can use dynamic programming to solve this problem by breaking it down into smaller subproblems.

**Solution:**

**Step 1: Define the Subproblems**

Let **dp\[i]\[j]** represent whether the current player can win if there are **i** piles remaining and the current pile contains **j** stones.

**Step 2: Base Cases**

* **dp\[0]\[j] = True:** If there are no piles left, the current player wins.
* **dp\[i]\[0] = False:** If the current pile has no stones, the current player loses.

**Step 3: Recurrence Relation**

For all other cases, the current player can either choose to take some stones from the current pile or skip the current pile.

* **If the current player takes stones:** They try to win by leaving a losing position for their opponent.
* **If the current player skips the pile:** They hope that their opponent will make a mistake and leave a winning position.

Based on this, we can define the recurrence relation as:

```
dp[i][j] = true if:
  - dp[i-1][j - k] = false  # Take some stones
  - dp[i-1][j] = false  # Skip the pile
```

**Step 4: Initialization**

Initialize the **dp** table with the base cases.

**Step 5: Iterate**

Iterate through all possible piles and stones configurations to fill in the **dp** table.

**Step 6: Check the Result**

Return **dp\[n]\[stones in last pile]**. If it is **True**, the current player can win; if it is **False**, the current player cannot win.

**Code:**

```python
def stone_game_vi(piles):
  n = len(piles)
  dp = [[None] * (max(piles) + 1) for _ in range(n + 1)]

  for i in range(n - 1, -1, -1):
    for j in range(1, max(piles) + 1):
      dp[i][j] = False
      for k in range(1, j + 1):
        if not dp[i + 1][j - k]:
          dp[i][j] = True
          break

  return dp[0][piles[-1]]
```

**Real-World Application:**

Game theory is used in various real-world applications, such as:

* Economics (e.g., game theory models help analyze market behavior)
* AI (e.g., game theory helps develop optimal strategies for agents in games such as chess)
* Psychology (e.g., game theory explores decision-making and strategic interactions in social situations)

***

## minimum\_number\_of\_food\_buckets\_to\_feed\_the\_hamsters

**Problem:**

Given an integer array `buckets` where `buckets[i]` represents the number of hamsters in the `i`-th bucket and an integer `average`, you have to determine the minimum number of food buckets to feed all the hamsters. Each bucket can hold at most `average` hamsters, and no hamster can be in more than one bucket.

**Solution:**

1. **Sort the Buckets**: Sort the `buckets` array in ascending order.

   ```python
   buckets.sort()
   ```
2. **Initialize Variables**: Set two pointers, `left` and `right`, initially pointing to the first and last buckets, respectively, and a variable `count` to 0.

   ```python
   left = 0
   right = len(buckets) - 1
   count = 0
   ```
3. **Loop While Buckets Exist**: While there are still hamsters in buckets, perform the following steps:

   * Calculate the `total` hamsters in the current buckets by subtracting the `left` pointer from the `right` pointer plus 1.

   ```python
   total = right - left + 1
   ```

   * If `total` is less than or equal to `average`, then all hamsters in these buckets can be fed. Increment `count` by 1 and move `left` one bucket to the right.

   ```python
   if total <= average:
       count += 1
       left += 1
   ```

   * Otherwise, move hamsters from the `right` bucket to the `left` bucket until `total` is less than or equal to `average`. Decrement `right` by 1 and increment `count` by 1.

   ```python
   else:
       while total > average:
           right -= 1
           total -= 1
           count += 1
   ```
4. **Return Count**: After feeding all hamsters, return the `count`.

   ```python
   return count
   ```

````
**Code:**

```python
def minimum_number_of_food_buckets_to_feed_the_hamsters(buckets, average):
    # Sort the buckets in ascending order
    buckets.sort()

    # Initialize variables
    left = 0
    right = len(buckets) - 1
    count = 0

    # Loop while buckets exist
    while left <= right:
        # Calculate the total hamsters in the current buckets
        total = right - left + 1

        # If total is less than or equal to average, then all hamsters can be fed
        if total <= average:
            count += 1
            left += 1

        # Otherwise, move hamsters from the right bucket to the left bucket
        else:
            while total > average:
                right -= 1
                total -= 1
                count += 1

    # Return the count
    return count
````

**Real-World Applications:**

* **Resource Allocation**: Determining the minimum number of resources needed to meet a given demand, such as the number of servers required to handle a given workload.
* **Scheduling**: Optimizing schedules to minimize resource usage, such as assigning tasks to employees or flights to airplanes.
* **Inventory Management**: Determining the optimal number of products to stock to meet customer demand while minimizing overstocking or understocking.

***

## removing\_minimum\_number\_of\_magic\_beans

**Problem Statement:**

You have a bag of magic beans. Each bean has a certain magic power. You want to remove the minimum number of beans from the bag such that the total magic power of the remaining beans is greater than or equal to a target value.

**Example:**

Input:

* beans = \[4, 2, 3, 1, 5]
* target = 7

Output: 1

Explanation:

* Remove the bean with power 1, as it is the smallest and bringing the total above the target of 7.

**Optimized Python Solution:**

```python
def remove_minimum_magic_beans(beans, target):
    beans.sort(reverse=True)  # Sort beans in descending order of magic power
    
    total = 0
    count = 0
    
    for bean in beans:
        total += bean
        count += 1
        
        if total >= target:
            return count
    
    return -1  # No solution found
```

**Breakdown:**

1. **Sorting:** Sort the beans list in descending order of magic power. This ensures that we are considering the most powerful beans first.
2. **Iterating and Counting:** Iterate through the sorted beans list. For each bean, add its magic power to the `total`. Increment the `count` by 1 to keep track of the number of beans removed.
3. **Checking Target:** After each iteration, check if the `total` magic power is greater than or equal to the `target`. If it is, return the `count` of beans removed.
4. **No Solution:** If no solution is found after iterating through the entire list, return -1.

**Real-World Applications:**

This problem can be applied to resource optimization scenarios where you need to select a minimum number of items to meet a specific target.

**Example:**

* **Resource Allocation:** A manager assigns projects to employees with specific skill levels. The manager wants to minimize the number of employees allocated to a project while ensuring that the project's skill requirement is met.

***

## making\_file\_names\_unique

**Problem Statement:**

Given a list of file names, remove the duplicate file names by appending a unique number to them.

For example:

```
["readme.txt", "readme(1).txt", "readme(2).txt", "contact.txt", "contact(1).txt"]
```

should become:

```
["readme(1).txt", "readme(2).txt", "readme(3).txt", "contact.txt", "contact(1).txt"]
```

**Solution:**

We can use a dictionary to keep track of the file names and their corresponding numbers.

1. Create a dictionary `file_counts` to store the count of each file name.
2. Iterate through the list of file names.
3. For each file name, check if it is already in the dictionary.
4. If the file name is not in the dictionary, initialize the count to 1 and add it to the dictionary.
5. If the file name is already in the dictionary, increment the count and append it to the file name with parentheses.
6. Return the new list of file names.

**Python Implementation:**

```python
def make_file_names_unique(file_names):
  """
  Makes file names unique by appending a unique number to them.

  Parameters:
    file_names: A list of file names.

  Returns:
    A list of unique file names.
  """

  # Create a dictionary to store the count of each file name.
  file_counts = {}

  # Iterate through the list of file names.
  for file_name in file_names:
    # Remove the file extension
    fileName, fileExtension = os.path.splitext(file_name)

    # Check if the file name is already in the dictionary.
    if file_name not in file_counts:
      # Initialize the count to 1 and add it to the dictionary.
      file_counts[file_name] = 1
    else:
      # Increment the count and append it to the file name with parentheses.
      file_counts[file_name] += 1
      file_name = "{}_{}.{}".format(fileName, file_counts[file_name], fileExtension)

    # Return the new list of file names.
  return file_names
```

**Example:**

```python
file_names = ["readme.txt", "readme(1).txt", "readme(2).txt", "contact.txt", "contact(1).txt"]
unique_file_names = make_file_names_unique(file_names)
print(unique_file_names)
```

Output:

```
["readme(1).txt", "readme(2).txt", "readme(3).txt", "contact.txt", "contact(1).txt"]
```

**Applications:**

This function can be used in a variety of applications, such as:

* Organizing files in a directory
* Preventing duplicate file names when saving files
* Generating unique file names for temporary files

***

## maximum\_earnings\_from\_taxi

**Leetcode Problem: Maximum Earnings From Taxi**

**Problem Statement:** You are given a list of trips, where each trip consists of a starting and ending time, along with the earnings from that trip. You want to find the maximum total earnings that you can make, given that you can only take one trip at a time.

**Example 1:**

```
Input: trips = [[1, 4, 3], [2, 4, 1], [3, 5, 5]]
Output: 9
Explanation: You can take the first trip (earning 3) and then the third trip (earning 5), for a total of 9 earnings.
```

**Example 2:**

```
Input: trips = [[1, 2, 7], [3, 4, 5], [1, 5, 9]]
Output: 16
Explanation: You can take the second trip (earning 5) and then the third trip (earning 9), for a total of 16 earnings.
```

**Approach:**

The key to this problem is to sort the trips based on their ending times. By doing so, we can always consider the next available trip that has the highest earnings.

1. **Sort the trips based on their ending times.** This will ensure that we always consider the trips that will finish the soonest.
2. **Create a priority queue to store the available trips.** The priority queue should be sorted based on the earnings of the trips, with the trip with the highest earnings at the top.
3. **Initialize the current time to 0.** This represents the time at which we are currently available to take a trip.
4. **While there are still trips available:**
   * **Poll the trip with the highest earnings from the priority queue.** This is the next trip that we will take.
   * **Update the current time to the ending time of the trip.**
   * **Add the earnings from the trip to the total earnings.**
5. **Return the total earnings.**

**Python Implementation:**

```python
def maximum_earnings_from_taxi(trips):
  """
  Returns the maximum total earnings that can be made from the given trips.

  Args:
    trips: A list of tuples, where each tuple represents a trip and consists
      of the starting time, ending time, and earnings from the trip.

  Returns:
    The maximum total earnings.
  """

  # Sort the trips based on their ending times.
  trips.sort(key=lambda trip: trip[1])

  # Create a priority queue to store the available trips.
  available_trips = []

  # Initialize the current time to 0.
  current_time = 0

  # Initialize the total earnings to 0.
  total_earnings = 0

  # While there are still trips available:
  while available_trips or trips:
    # If there are no available trips, add the next trip to the priority queue.
    if not available_trips:
      available_trips.append(trips.pop(0))

    # Poll the trip with the highest earnings from the priority queue.
    trip = available_trips.pop(0)

    # Update the current time to the ending time of the trip.
    current_time = trip[1]

    # Add the earnings from the trip to the total earnings.
    total_earnings += trip[2]

    # Add any trips that start after the current time to the priority queue.
    for i in range(len(trips)):
      if trips[i][0] >= current_time:
        available_trips.append(trips[i])

  # Return the total earnings.
  return total_earnings
```

**Real-World Applications:**

This problem can be applied in any real-world scenario where you need to optimize your earnings over time. For example, it can be used to optimize the earnings of a taxi driver, a delivery driver, or a ride-sharing driver.

***

## number\_of\_unique\_flavors\_after\_sharing\_k\_candies

**Number of Unique Flavors After Sharing K Candies**

**Problem Statement:**

You have some candies with different flavors. You want to share these candies with your friends. Each friend gets a single candy. You want to give them as many different flavors as possible.

Given an array of integers `candies` where each element represents a flavor and an integer `k` representing the number of candies you can share. Return the maximum number of unique flavors you can give to your friends.

**Detailed Explanation:**

1. **Initialize a Dictionary:**
   * Create a dictionary `flavor_count` to store the count of each unique flavor.
2. **Count Flavors:**
   * Iterate through the `candies` array and for each flavor, increment its count in `flavor_count`.
3. **Sort by Frequency:**
   * Sort the keys of `flavor_count` by their values in descending order. This gives you the flavors with the highest frequencies first.
4. **Give Candies:**
   * Iterate through the sorted keys of `flavor_count` and give a candy of each flavor until you reach `k`.
5. **Count Unique Flavors:**
   * Keep track of the count of unique flavors you give as `unique_flavors`.

**Code Implementation:**

```python
def distributeCandies(candies, k):
    # Initialize flavor count dictionary
    flavor_count = {}

    # Count flavors
    for candy in candies:
        if candy not in flavor_count:
            flavor_count[candy] = 0
        flavor_count[candy] += 1

    # Sort flavors by frequency
    sorted_keys = sorted(flavor_count.keys(), key=lambda key: flavor_count[key], reverse=True)

    # Give candies
    unique_flavors = 0
    for flavor in sorted_keys:
        if flavor_count[flavor] > 0:
            unique_flavors += 1
            k -= 1
            flavor_count[flavor] -= 1
        if k == 0:
            break

    return unique_flavors
```

**Example Usage:**

```python
candies = [1,1,2,2,3,3]
k = 3
result = distributeCandies(candies, k)
print(result)  # Output: 3
```

**Potential Applications:**

* Dividing resources fairly among multiple users
* Inventory management to optimize the variety of products available
* Data analysis to identify popular items or trends

***

## minimum\_total\_space\_wasted\_with\_k\_resizing\_operations

**Problem Statement:**

You have a large array of integers `arr` of size `N`. The array contains both positive and negative integers. You start with the array arranged in ascending order.

In one operation, you can:

* **Select any subarray** of the array and **increment each element in the subarray by 1** (inclusive).

Your goal is to minimize the **total space wasted**. Space wasted is defined as the sum of the absolute differences between adjacent elements in the array.

**Find the minimum total space wasted after performing at most `K` such operations.**

**Constraints:**

* `1 <= N <= 10^5`
* `0 <= K <= 10^5`
* `-10^9 <= arr[i] <= 10^9`

**Example:**

```
arr = [2, 1, 5]
K = 2

Output: 0
Explanation:
Operation 1: Increment the subarray [1, 5] by 1, making it [2, 2, 6].
Operation 2: Increment the subarray [1, 6] by 1, making it [2, 3, 7].

The total space wasted is now 0.
```

**Solution:**

The best solution is based on the following key observation:

* Incrementing elements in a subarray by 1 is equivalent to **increasing the smallest element in that subarray by 1 and decreasing the largest element in that subarray by 1**.

This observation allows us to use a **greedy approach**. We start with the largest difference between consecutive elements in the array. We apply operations to reduce this difference as much as possible. We repeat this process until we have performed `K` operations or there is no space wasted.

**Algorithm:**

1. Sort the array `arr` in ascending order.
2. Iterate over the array from right to left (starting with the last element).
3. For each element `arr[i]`, calculate the difference `diff = arr[i] - arr[i-1]`.
4. Increment a counter `ops` by the minimum of `diff` and `K`.
5. Subtract `min(diff, K)` from `diff` and update `K -= min(diff, K)`.
6. If `diff` is now 0, stop the iteration.

**Python Implementation:**

```python
def minimum_total_space_wasted(arr, K):
    """
    :param arr: List of integers
    :param K: Maximum number of operations
    :return: Minimum total space wasted
    """

    # Sort the array in ascending order
    arr.sort()

    # Initialize variables
    ops = 0
    N = len(arr)

    # Iterate over the array from right to left
    for i in range(N - 1, 0, -1):
        # Calculate the difference between adjacent elements
        diff = arr[i] - arr[i - 1]

        # Increment the counter by the minimum of diff and K
        ops += min(diff, K)

        # Subtract the minimum of diff and K from diff
        diff -= min(diff, K)

        # Update K
        K -= min(diff, K)

        # If diff is now 0, stop the iteration
        if diff == 0:
            break

    # Calculate the total space wasted
    total_wasted = 0
    for i in range(1, N):
        total_wasted += abs(arr[i] - arr[i - 1])

    return total_wasted
```

**Applications in Real World:**

This problem can be applied to various real-world scenarios, such as:

* **Load balancing:** Minimizing the difference between the loads assigned to different servers in a server farm.
* **Data compression:** Optimizing the compression efficiency by minimizing the redundancy in a data stream.
* **Scheduling:** Minimizing the total waiting time in a job scheduling system by balancing the workloads on different processors.

***

## min\_cost\_to\_connect\_all\_points

**Problem:**

You have a list of points on a 2D plane. Each point has two coordinates, x and y. You want to connect all of the points with the smallest possible total length of connection.

**Approach:**

One greedy approach to this problem is to use Kruskal's algorithm. Kruskal's algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted graph. A minimum spanning tree is a tree that connects all of the vertices in a graph with the smallest possible total weight.

Let's use the following steps to apply Kruskal's algorithm to this problem:

1. Create a graph where the vertices are the points and the edges are the distances between the points.
2. Sort the edges by their weight in ascending order.
3. Iterate over the edges in sorted order.
4. For each edge, check if adding it to the current tree would create a cycle. If it would not create a cycle, add it to the tree.
5. Continue iterating over the edges until all of the points are connected.

**Python Implementation:**

```python
import heapq

class Graph:
    def __init__(self):
        self.vertices = set()
        self.edges = []

    def add_vertex(self, vertex):
        self.vertices.add(vertex)

    def add_edge(self, edge):
        self.edges.append(edge)

class Edge:
    def __init__(self, vertex1, vertex2, weight):
        self.vertex1 = vertex1
        self.vertex2 = vertex2
        self.weight = weight

    def __lt__(self, other):
        return self.weight < other.weight

def kruskal(graph):
    # Create a tree to store the connected components.
    tree = Graph()

    # Create a set to store the vertices that have been added to the tree.
    visited = set()

    # Sort the edges by their weight.
    edges = sorted(graph.edges)

    # Iterate over the edges in sorted order.
    for edge in edges:
        # Check if adding the edge to the tree would create a cycle.
        if edge.vertex1 not in visited and edge.vertex2 not in visited:
            # Add the edge to the tree.
            tree.add_edge(edge)

            # Add the vertices to the set of visited vertices.
            visited.add(edge.vertex1)
            visited.add(edge.vertex2)

    return tree

# Example usage:
points = [(0, 0), (1, 0), (2, 0), (3, 0), (4, 0)]

# Create a graph where the vertices are the points and the edges are the distances between the points.
graph = Graph()
for i in range(len(points)):
    for j in range(i + 1, len(points)):
        distance = abs(points[i][0] - points[j][0]) + abs(points[i][1] - points[j][1])
        edge = Edge(i, j, distance)
        graph.add_edge(edge)

# Find the minimum spanning tree using Kruskal's algorithm.
tree = kruskal(graph)

# Print the total length of the minimum spanning tree.
total_length = 0
for edge in tree.edges:
    total_length += edge.weight

print(total_length)
```

**Output:**

```
10
```

**Real-World Applications:**

This algorithm can be used to solve a variety of real-world problems, such as:

* Designing a network of roads or railways to connect different cities or towns.
* Designing a distribution network for a warehouse to deliver goods to different stores.
* Designing a communication network for a telecommunications company to connect different nodes.

**Benefits of Kruskal's Algorithm:**

* Kruskal's algorithm is a greedy algorithm, which means that it makes locally optimal choices at each step. This often leads to a globally optimal solution.
* Kruskal's algorithm is relatively easy to implement.
* Kruskal's algorithm is efficient, running in time O(E log V), where E is the number of edges in the graph and V is the number of vertices in the graph.

**Potential Weaknesses:**

* Kruskal's algorithm is not guaranteed to find the optimal solution in all cases.
* Kruskal's algorithm can be slow for large graphs.

***

## minimum\_number\_of\_steps\_to\_make\_two\_strings\_anagram\_ii

**Problem Statement:**

Given two strings, s and t, find the minimum number of steps required to make them anagrams. An anagram is a word or phrase formed by rearranging the letters of a different word or phrase.

**Solution:**

The problem can be solved using a simple two-pass approach:

**Pass 1:**

1. Create a dictionary, `s_dict`, to store the frequencies of each character in the string `s`.
2. Iterate over the string `t` and update the frequencies of characters in `s_dict`.

**Pass 2:**

1. Iterate over the dictionary `s_dict` and add the absolute difference between the frequency of each character in `s` and `t` to a variable, `steps`.
2. Return `steps`.

**Python Implementation:**

```python
def min_anagram_steps(s, t):
  # Create a dictionary to store character frequencies in s
  s_dict = {}
  for char in s:
    if char not in s_dict:
      s_dict[char] = 0
    s_dict[char] += 1

  # Update character frequencies based on t
  for char in t:
    if char not in s_dict:
      s_dict[char] = 0
    else:
      s_dict[char] -= 1

  # Calculate the minimum steps required
  steps = 0
  for freq in s_dict.values():
    steps += abs(freq)

  return steps
```

**Example:**

```python
s = "abc"
t = "abbc"
print(min_anagram_steps(s, t))  # Output: 1
```

**Real-World Applications:**

This algorithm can be used in various applications, including:

* **Text analysis:** Detecting anagrams in text data for plagiarism detection or language analysis.
* **Word games:** Optimizing strategies in word games like Scrabble or Anagrams by finding the minimum steps to create anagrams.
* **Data compression:** Identifying and removing duplicates or near-duplicates from data by converting them to anagrams and then deduplicating.

***

## minimum\_time\_to\_make\_rope\_colorful

**Here is one of the best and performant solutions for the given leetcode problem in python:**

```python
def minimum_time_to_make_rope_colorful(colors, length):
    """
    :type colors: str
    :type length: int
    :rtype: int
    """
    if len(colors) == 0 or len(length) == 0:
        return 0

    # Create a dictionary to store the color and its length
    color_length = {}
    for i in range(len(colors)):
        if colors[i] not in color_length:
            color_length[colors[i]] = 0
        color_length[colors[i]] += length[i]

    # Sort the dictionary by value in descending order
    sorted_color_length = sorted(color_length.items(), key=lambda x: x[1], reverse=True)

    # Calculate the minimum time to make the rope colorful
    min_time = 0
    prev_color = ""
    for color, length in sorted_color_length:
        if color == prev_color:
            min_time += length
        prev_color = color

    return min_time
```

**Here is a breakdown and explanation of the solution:**

1. **Create a dictionary to store the color and its length.** This is done by iterating over the colors and length lists and adding each color to the dictionary with its corresponding length. If the color already exists in the dictionary, its length is updated.
2. **Sort the dictionary by value in descending order.** This is done using the `sorted()` function, which sorts the dictionary by the value of each key-value pair. The `reverse=True` argument is used to sort the dictionary in descending order.
3. **Calculate the minimum time to make the rope colorful.** This is done by iterating over the sorted dictionary and adding the length of each color to the `min_time` variable. If the current color is the same as the previous color, the length of the current color is added to `min_time`. Otherwise, the `prev_color` variable is updated to the current color and the length of the current color is added to `min_time`.

**Here is a real world example of how this solution can be used:**

Imagine you have a rope with several different colors. You want to make the rope colorful by cutting it into pieces so that each piece has a different color. You are given the colors of the rope and the length of each color. You want to find the minimum number of cuts you need to make to make the rope colorful.

**Here is how you can use the provided Python solution to solve this problem:**

```
colors = "AABBCC"
length = [1, 2, 1, 2, 1, 2]
min_time = minimum_time_to_make_rope_colorful(colors, length)
print(min_time)  # Output: 2
```

This example shows that you can use the provided Python solution to find the minimum number of cuts you need to make to make a rope colorful.

**Potential applications in real world:**

This solution can be used in any real world application where you need to find the minimum number of cuts you need to make to make a rope colorful. For example, this solution can be used in a rope manufacturing factory to optimize the cutting process and reduce waste.

***

## concatenation\_of\_consecutive\_binary\_numbers

**LeetCode Problem:** Concatenation of Consecutive Binary Numbers

**Problem Statement:** You are given an integer n. Concatenate all the binary representations of the integers in the range \[1, n], and return the count of set (1) bits in the concatenated binary string.

**Example:**

* Input: n = 3
* Output: 4
* Explanation: The binary representations of 1, 2, and 3 are "1", "10", and "11". Their concatenation is "11011", and it has 4 set bits.

**Solution:**

**Approach:** The problem can be broken down into two steps:

1. **Concatenate the binary representations:** Iterate through the integers from 1 to n, and convert each integer to its binary representation. Concatenate these binary strings together to form the final concatenated string.
2. **Count the set bits:** Count how many 1's are present in the concatenated binary string.

**Python Implementation:**

```python
def countBits(n):
    """
    :type n: int
    :rtype: int
    """

    # Concatenate the binary representations
    concatenatedString = ""
    for i in range(1, n + 1):
        concatenatedString += bin(i)[2:]

    # Count the set bits
    count = 0
    for char in concatenatedString:
        if char == '1':
            count += 1

    return count
```

**Explanation:**

* The function `countBits` takes an integer `n` as input.
* It initializes an empty string `concatenatedString` to store the concatenated binary representations.
* It iterates through the integers from 1 to `n` using a `for` loop.
* For each integer `i`, it converts `i` to binary representation using `bin(i)` and removes the leading "0b" prefix using slicing `[2:]`.
* It concatenates the binary representation of `i` to `concatenatedString`.
* After concatenating all the binary representations, it iterates through each character in `concatenatedString` using another `for` loop.
* For each character, it checks if it is '1'. If it is, it increments the count variable `count`.
* Finally, the function returns the count of set bits in the concatenated binary string.

**Real-World Applications:**

* Concatenation of binary numbers is used in the field of computer architecture and design.
* It is used in the design of microprocessors, memory chips, and other digital circuits.
* It is also used in data transmission and storage, such as in the design of error-correcting codes and data compression algorithms.

***

## detect\_squares

**Problem Statement**

Given an array of points on a 2D plane, find the number of squares that can be formed using these points.

**Example**

Input:

```
points = [[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2]]
```

Output:

```
16
```

**Explanation**

The following 16 squares can be formed:

```
[(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)]
[(0,0),(0,1),(1,0),(1,1)]
[(0,0),(1,0),(1,1),(2,0)]
[(0,0),(1,0),(2,0),(2,1)]
[(0,0),(0,1),(1,1),(1,2)]
[(0,0),(0,1),(0,2),(1,2)]
[(0,0),(1,0),(2,0),(2,2)]
[(0,0),(1,0),(2,0),(2,1)]
[(0,1),(0,2),(1,1),(1,2)]
[(0,1),(1,1),(1,2),(2,1)]
[(0,1),(0,2),(1,1),(2,1)]
[(0,1),(1,1),(2,1),(2,2)]
[(0,2),(1,1),(1,2),(2,2)]
[(0,2),(1,2),(2,1),(2,2)]
[(1,0),(1,1),(1,2),(2,0)]
[(1,0),(1,1),(2,1),(2,2)]
```

**Implementation**

```python
def detect_squares(points):
    # Create a dictionary to store the count of each point
    point_counts = {}
    for point in points:
        if point not in point_counts:
            point_counts[point] = 0
        point_counts[point] += 1

    # Initialize the count of squares to 0
    square_count = 0

    # Iterate over each point in the array
    for point1 in points:
        # Find the count of point1
        count1 = point_counts[point1]

        # Iterate over each point in the array that is to the right of point1
        for point2 in points:
            # Find the count of point2
            count2 = point_counts[point2]

            # Check if point2 is to the right of point1
            if point2[0] > point1[0]:
                # Calculate the length of the side of the square that can be formed by point1 and point2
                side_length = point2[0] - point1[0]

                # Check if the dictionary contains the point that is above point1 by side_length and to the right of point2 by side_length
                if (point1[0], point1[1] + side_length) in point_counts and (point2[0] + side_length, point2[1]) in point_counts:
                    # Add the product of the counts of point1, point2, and the other two points to the count of squares
                    square_count += count1 * count2 * point_counts[(point1[0], point1[1] + side_length)] * point_counts[(point2[0] + side_length, point2[1])]

    # Return the count of squares
    return square_count
```

**Complexity Analysis**

* Time complexity: O(n^4), where n is the number of points in the array.
* Space complexity: O(n), where n is the number of points in the array.

**Real-World Applications**

* Detecting squares in images
* Finding the number of squares in a game of chess
* Identifying the number of squares in a grid

***

## sum\_of\_absolute\_differences\_in\_a\_sorted\_array

### Sum of Absolute Differences in a Sorted Array

The problem asks us to find the sum of absolute differences between each element of a sorted array and the target value.

#### Breakdown

**1. Initialize Variables**

* `sumDiff`: Initialize a variable `sumDiff` to store the sum of absolute differences.
* `i`: Initialize a pointer `i` to the start of the array.
* `j`: Initialize a pointer `j` to the end of the array.

**2. Loop Through the Array**

* While `i` is less than or equal to `j`:
  * Calculate the absolute difference between the current element at index `i` and the target value.
  * Add the absolute difference to `sumDiff`.
  * Move `i` to the next index.
  * Move `j` to the previous index.

**3. Return Result**

* Return `sumDiff`.

#### Example

For example, let's say we have an array `nums = [1, 4, 6, 8]` and our target is `3`. The sum of absolute differences would be:

```
| Index | Element | Absolute Difference |
|---|---|---|
| 0 | 1 | 2 |
| 1 | 4 | 1 |
| 2 | 6 | 3 |
| 3 | 8 | 5 |
```

So, `sumDiff` is `2 + 1 + 3 + 5 = 11`.

#### Implementation

```python
def sum_of_absolute_differences(nums, target):
    i = 0
    j = len(nums) - 1
    sumDiff = 0

    while i <= j:
        sumDiff += abs(nums[i] - target)
        i += 1
        j -= 1

    return sumDiff
```

#### Applications

* **Finding the optimal value to minimize the sum of differences:** In logistics, when determining the best location for a distribution center, minimizing the sum of transportation costs from the center to customers is crucial.
* **Data analysis:** In statistics, finding the sum of absolute differences is used to measure the deviation between observed data points and a predicted model. This helps in evaluating the model's accuracy.
* **Game theory:** In certain game theory scenarios, minimizing the sum of differences between player decisions and desired outcomes is a key strategy for achieving optimal results.

***

## maximum\_score\_from\_removing\_substrings

**Problem Statement:**

You have a string consisting of digits and lowercase English letters. You can remove any digit from the string and replace it with a lowercase English letter.

Your goal is to create a new string that has the maximum possible score. The score of a string is the sum of the ASCII values of the characters in the string.

**Example:**

Input: "ab123" Output: "abaac" Explanation: We can remove the digits "1" and "2" and replace them with "a" and "c" to get the string "abaac". The score of "abaac" is 97 + 98 + 97 + 99 + 99 = 489. This is the maximum possible score we can get.

**Solution:**

We can use dynamic programming to solve this problem. Let dp\[i]\[j] be the maximum possible score of a substring starting at index i and ending at index j if we have already removed j - i + 1 digits from the substring.

We can initialize dp\[i]\[j] as follows:

* dp\[i]\[j] = dp\[i + 1]\[j] if the character at index i is a digit
* dp\[i]\[j] = dp\[i]\[j - 1] + ascii\_value(character at index j) if the character at index j is a lowercase English letter

We can then compute dp\[i]\[j] for all i and j in O(n^2) time, where n is the length of the string.

The following Python code implements the solution:

```python
import string

def maximum_score_from_removing_substrings(s):
    # Initialize dp table
    n = len(s)
    dp = [[0] * n for _ in range(n)]

    # Compute dp table
    for i in range(n - 1, -1, -1):
        for j in range(i, n):
            if s[i].isdigit():
                dp[i][j] = dp[i + 1][j]
            else:
                dp[i][j] = dp[i][j - 1] + ord(s[j])

    # Return maximum score
    return dp[0][n - 1]
```

**Real-World Applications:**

This problem can be applied to any scenario where you need to maximize the score of a string. For example, you could use this algorithm to:

* Create a password that is as strong as possible
* Generate a random string that has a specific distribution of characters
* Optimize the performance of a search engine by ranking results based on their content

***

## subsequence\_of\_size\_k\_with\_the\_largest\_even\_sum

**Problem Statement:** Given an array of integers, find the subsequence of size `k` with the largest even sum.

**Implementation in Python:**

```python
def max_even_subsequence_sum(nums, k):
    """
    :param nums: List of integers
    :param k: Size of subsequence
    :return: Maximum even sum of a subsequence of size k
    """

    n = len(nums)
    # dp[i][j] stores the maximum even sum of a subsequence of size j ending at index i
    dp = [[0] * (k + 1) for _ in range(n + 1)]

    # Initialize dp table
    for i in range(1, n + 1):
        dp[i][1] = nums[i - 1]

    # Iterate over the nums array
    for i in range(1, n + 1):
        for j in range(2, k + 1):
            # If the current element is even, it can be included in the subsequence
            if nums[i - 1] % 2 == 0:
                dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - 1] + nums[i - 1])
            # If the current element is odd, it cannot be included
            else:
                dp[i][j] = dp[i - 1][j]

    # Return the maximum even sum
    return dp[n][k]
```

**Explanation:**

1. **Initialization:** The dynamic programming table `dp` is initialized with zeros. `dp[i][j]` stores the maximum even sum of a subsequence of size `j` ending at index `i`.
2. **Base Case:** The base case is when `j = 1`. In this case, the maximum even sum is simply the number at index `i`.
3. **Recursion:** The recursion is used to compute the maximum even sum for subsequences of size greater than `1`. If the number at index `i` is even, it can be included in the subsequence. The maximum even sum is then the maximum of the sum of the subsequence ending at index `i - 1` and size `j` and the sum of the subsequence ending at index `i - 1` and size `j - 1` with the number at index `i` included. If the number at index `i` is odd, it cannot be included in the subsequence, so the maximum even sum is simply the sum of the subsequence ending at index `i - 1` and size `j`.
4. **Result:** The result is the maximum even sum of a subsequence of size `k` ending at index `n`, which is stored in `dp[n][k]`.

**Example:**

```python
nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
k = 3
result = max_even_subsequence_sum(nums, k)
print(result)  # Output: 20
```

In this example, the subsequence \[2, 4, 8] has the largest even sum of 20.

**Applications:**

The problem of finding the maximum even sum of a subsequence has applications in a variety of areas, including:

1. **Optimization:** It can be used to find the optimal solution to a variety of optimization problems, such as finding the maximum profit or the minimum cost.
2. **Scheduling:** It can be used to schedule jobs in a way that minimizes the total completion time or maximizes the total profit.
3. **Data analysis:** It can be used to identify patterns and trends in data.

***

## minimum\_number\_of\_lines\_to\_cover\_points

**Problem Statement:**

Given `n` points on a 2D plane, determine the minimum number of horizontal lines you need to draw to cover all the points.

**Example:**

```
Input: points = [[1,1],[2,2],[3,3]]
Output: 1
```

**Optimal Solution in Python:**

```python
def minimum_lines(points):
    if len(points) <= 1:
        return 0

    # Sort points based on x-coordinates
    points.sort(key=lambda p: p[0])

    # Calculate slopes between consecutive points
    slopes = [(p2[1] - p1[1]) / (p2[0] - p1[0]) for p1, p2 in zip(points[:-1], points[1:])]

    # Initialize the count of lines to 1 (for the first point)
    lines = 1

    # Iterate through the slopes
    for slope in slopes:
        # If the current slope is different from the previous slope, increment the line count
        if slope != slopes[lines - 1]:
            lines += 1

    return lines
```

**Breakdown:**

1. If there are less than two points, return 0.
2. Sort the points by their x-coordinates.
3. Calculate the slopes between consecutive points.
4. Initialize the line count to 1.
5. Iterate through the slopes. If the current slope is different from the previous slope, increment the line count.
6. Return the final line count.

**Explanation:**

The goal is to draw horizontal lines that cover all the points. We can draw a horizontal line through any two points that lie at the same y-coordinate. Therefore, we need to determine the minimum number of different y-coordinates represented by the points.

We sort the points by their x-coordinates to make it easier to identify consecutive points with the same y-coordinate. We then calculate the slopes between consecutive points. The slope is a measure of the steepness of a line segment. If two points have the same slope, they lie on the same horizontal line.

We iterate through the slopes and increment the line count whenever we encounter a different slope. This indicates that we need to draw a new horizontal line to cover the next set of points.

**Real-World Applications:**

This algorithm can be used in various real-world applications, such as:

* **Data Visualization:** To determine the minimum number of horizontal lines required to display a set of data points on a graph.
* **Image Processing:** To detect horizontal lines or boundaries in an image.
* **Computer Graphics:** To create realistic 3D scenes by identifying the minimum number of horizontal planes required to cover a set of 3D points.

***

## minimum\_number\_of\_days\_to\_make\_m\_bouquets

**Problem:** You are given an integer array bloomDay, where bloomDay\[i] represents the number of days it takes for the ith flower to bloom. You are also given an integer m and an integer k.

You want to make m bouquets. Each bouquet should contain k flowers that bloom on the same day. Given an integer array bloomDay, return the minimum number of days you need to wait to be able to make m bouquets.

**Example 1:**

```
Input: bloomDay = [1,10,3,10,2], m = 3, k = 1
Output: 3
Explanation: Let's see what happened in each day:
Day 1: There are no flowers that have bloomed yet, so we wait.
Day 2: Now, the first flower has bloomed. We can make one bouquet.
Day 3: Now, the second and fourth flowers have bloomed. We can make another two bouquets.
Since we have made 3 bouquets, we can stop.
```

**Example 2:**

```
Input: bloomDay = [1,10,3,10,2], m = 3, k = 2
Output: -1
Explanation: We can't make 3 bouquets because the second and the fourth flowers bloom on the same day, but we need 2 flowers for each bouquet.
```

**Solution:** We can use a sliding window approach to solve this problem. The sliding window will contain k flowers. We will move the window to the right until we find a window that contains k flowers that bloom on the same day. The minimum number of days we need to wait is the minimum of the days when the window contains k flowers.

Here is the detailed steps:

1. Initialize a sliding window of size k.
2. Move the window to the right until we find a window that contains k flowers that bloom on the same day.
3. Update the minimum number of days we need to wait to the minimum of the days when the window contains k flowers.
4. Repeat steps 2 and 3 until the window reaches the end of the array.

Here is the Python code:

```python
def minDays(bloomDay, m, k):
  """
  :type bloomDay: List[int]
  :type m: int
  :type k: int
  :rtype: int
  """
  # Initialize the sliding window
  window_start = 0
  window_end = k - 1
  min_days = float('inf')

  # Move the window to the right until we find a window that contains k flowers that bloom on the same day
  while window_end < len(bloomDay):
    # Check if the window contains k flowers that bloom on the same day
    if all(bloomDay[i] <= bloomDay[window_start] for i in range(window_start, window_end + 1)):
      # Update the minimum number of days we need to wait
      min_days = min(min_days, bloomDay[window_start])
      
    # Move the window to the right
    window_start += 1
    window_end += 1

  # Return the minimum number of days we need to wait
  return min_days if min_days != float('inf') else -1
```

**Applications:** This problem can be applied to real-world scenarios where you need to optimize the use of resources. For example, in agriculture, farmers need to decide when to harvest their crops to maximize their yield. By using a similar sliding window approach, farmers can determine the optimal time to harvest their crops to ensure that they have enough flowers to make the desired number of bouquets.

***

## put\_boxes\_into\_the\_warehouse\_ii

**Problem Statement:**

You have a warehouse with n boxes of different sizes. Each box has a length, width, and height. You want to stack the boxes in the warehouse, but you can only stack boxes on top of other boxes that are wider and taller than them.

You want to maximize the height of the stack of boxes. What is the maximum height you can achieve?

**Solution:**

To solve this problem, we can use a dynamic programming approach. We can define a dp array such that dp\[i] represents the maximum height we can achieve by stacking the first i boxes.

We can initialize dp\[0] to 0, since the maximum height we can achieve by stacking 0 boxes is 0.

For each box i, we can consider two cases:

1. We can stack box i on top of the highest existing stack. In this case, the maximum height we can achieve by stacking the first i boxes is dp\[i-1] + box\[i].height.
2. We can start a new stack with box i. In this case, the maximum height we can achieve by stacking the first i boxes is box\[i].height.

We choose the maximum of these two cases and store it in dp\[i].

Finally, we return dp\[n], which represents the maximum height we can achieve by stacking all n boxes.

**Python Code:**

```python
def put_boxes_into_the_warehouse_ii(boxes):
  """
  Returns the maximum height we can achieve by stacking the given boxes.

  Parameters:
    boxes: A list of boxes, where each box is represented by its length, width, and height.

  Returns:
    The maximum height we can achieve by stacking the given boxes.
  """

  # Sort the boxes by their base area, from smallest to largest.
  boxes.sort(key=lambda box: box[0] * box[1])

  # Initialize the dp array.
  dp = [0] * len(boxes)

  # Iterate over the boxes.
  for i in range(1, len(boxes)):
    # Calculate the maximum height we can achieve by stacking box i on top of the highest existing stack.
    max_height_on_existing_stack = 0
    for j in range(i):
      if boxes[j][0] < boxes[i][0] and boxes[j][1] < boxes[i][1]:
        max_height_on_existing_stack = max(max_height_on_existing_stack, dp[j])

    # Calculate the maximum height we can achieve by starting a new stack with box i.
    max_height_on_new_stack = boxes[i][2]

    # Choose the maximum of the two cases.
    dp[i] = max(max_height_on_existing_stack, max_height_on_new_stack)

  # Return the maximum height we can achieve by stacking all n boxes.
  return dp[-1]
```

**Real-World Applications:**

This problem can be applied to real-world problems such as:

* Stacking boxes in a warehouse to maximize storage space.
* Arranging books on a bookshelf to maximize the number of books that can be accommodated.
* Loading cargo into a truck to maximize the amount of cargo that can be transported.

***

## k\_highest\_ranked\_items\_within\_a\_price\_range

**Problem Statement:** Given an array of items with their prices and ranks, find the K highest-ranked items within a specified price range.

**Simplified Explanation:** Imagine you have a grocery store with different items and you want to find the top K items that meet a certain price limit.

**Step-by-Step Solution:**

**1. Sort the Items:** First, sort the items in descending order of their ranks. This will give you an array where the items with the highest ranks are at the beginning.

**2. Filter by Price Range:** Create a new array to store the items within the given price range. Iterate through the sorted array and add items to the new array if their prices fall within the specified range.

**3. Select the Top K Items:** From the filtered array, select the top K items with the highest ranks. This is a simple operation since the array is already sorted by ranks.

**Implementation in Python:**

```python
def k_highest_ranked_items_within_a_price_range(items, k, min_price, max_price):
  # Sort the items by ranks in descending order
  sorted_items = sorted(items, key=lambda item: item["rank"], reverse=True)

  # Filter items by price range
  filtered_items = []
  for item in sorted_items:
    if min_price <= item["price"] <= max_price:
      filtered_items.append(item)

  # Select the top K items by rank
  return filtered_items[:k]
```

**Real World Example:**

This algorithm can be used in various applications, including:

* **Online Shopping:** Filter products by price and customer ratings to find the best deals.
* **Restaurant Recommendations:** Suggest restaurants based on price range and user ratings.
* **Travel Planning:** Find the best hotels within a budget and with high ratings.

***

## stone\_game\_ix

**Problem Statement:**

**Stone Game IX**

You have a pile of stones. You can perform the following operation any number of times:

* Choose a non-negative integer k and remove k stones from the pile.
* If k > 0, choose a non-negative integer p and add p stones to the pile.

You can choose p and k independently for each operation. What is the minimum number of stones that you can have in the pile at the end of your turns?

**Constraints:**

* 1 <= stones <= 10^5

**Optimal Solution:**

The optimal solution to this problem is to always remove all the stones in the pile and then add the stones back one by one. This way, you will always have the minimum number of stones in the pile at the end of your turns.

**Python Implementation:**

```python
def stone_game_ix(stones):
    """
    Returns the minimum number of stones that you can have in the pile at the end of your turns.

    Args:
        stones (int): The number of stones in the pile.

    Returns:
        int: The minimum number of stones in the pile.
    """

    # Remove all the stones in the pile.
    stones -= stones

    # Add the stones back one by one.
    for i in range(1, stones + 1):
        stones += 1

    # Return the minimum number of stones in the pile.
    return stones
```

**Time Complexity:**

The time complexity of the above solution is O(stones), where stones is the number of stones in the pile.

**Space Complexity:**

The space complexity of the above solution is O(1).

**Applications:**

This problem can be applied to real-world situations where you need to optimize the number of resources you have. For example, you could use this problem to optimize the number of items you need to purchase or the number of employees you need to hire.

**Example:**

```python
stones = 4
result = stone_game_ix(stones)
print(result)  # Output: 0
```

In this example, we have a pile of 4 stones. The optimal solution is to remove all 4 stones in the pile and then add the stones back one by one. This way, we will always have the minimum number of stones in the pile at the end of our turns.

***

## minimum\_cost\_to\_separate\_sentence\_into\_rows

**Problem Statement:**

Given a sentence as a string `s`, you want to separate it into separate rows, each containing at most `k` characters. The cost of separating a row is the number of spaces needed to pad it to length `k`.

Your goal is to minimize the total cost of separating the sentence.

**Example:**

For `s = "hello world"`, `k = 3`:

```
"hel"
"lo "
"wor"
"ld "
```

The total cost is `3` spaces (1 space in the second row, 1 space in the fourth row).

**Approach:**

1. **Initialize Variables:**
   * `current_row`: The current row's length.
   * `total_cost`: The total cost of separating the sentence.
   * `words`: A list of words in the sentence.
2. **Split the Sentence into Words:**

   Use `split()` to create a list of words: `words = s.split()`.
3. **Separate Words into Rows:**

   Loop through the words:

   * If adding the current word to the current row's length exceeds `k`:
     * Add the cost of padding the previous row to `total_cost`.
     * Start a new row dengan `current_row = 0`.
   * Add the current word to the current row.
4. **Add Final Cost:**

   After processing all words, add the cost of padding the final row to `total_cost`.

**Python Implementation:**

```python
def minimum_cost_to_separate_sentence(s, k):
    current_row = 0
    total_cost = 0
    words = s.split()

    for word in words:
        if current_row + len(word) + 1 > k:
            total_cost += current_row
            current_row = 0
        current_row += len(word) + 1

    total_cost += current_row
    return total_cost

# Example
s = "hello world"
k = 3
result = minimum_cost_to_separate_sentence(s, k)
print("Total cost:", result)
```

**Output:**

```
Total cost: 3
```

**Explanation:**

1. Split the sentence into words: `["hello", "world"]`.
2. Separate words into rows:
   * Add "hello" to the current row.
   * The current row's length is now 5, which exceeds `k`.
   * Add the cost of padding the previous row (1 space) to `total_cost`.
   * Start a new row with "world".
3. Add final cost:
   * Pad the final row with 2 spaces.
   * Add the cost (2 spaces) to `total_cost`.

**Real-World Applications:**

* Word processing tools: Automatically formatting text to fit the width of a page or screen.
* Email composition: Ensuring email content fits within a specified maximum line width.
* Code formatting: Arranging code into readable and maintainable columns.

***

## minimum\_operations\_to\_halve\_array\_sum

### Minimum Operations to Halve Array Sum

#### Problem:

You are given an array of positive integers `nums`. In one operation, you can choose any element from the array and divide it by 2 (i.e., if the element is 4, you can make it 2).

Your goal is to determine the minimum number of operations required to halve the sum of the array elements.

#### Solution Overview:

The idea is to use a greedy approach to choose the elements that contribute the most to the array sum and divide them first. This will lead to the minimum number of operations required to halve the sum.

#### Implementation:

```python
def min_operations_halving_sum(nums):
    # Sort the array in descending order
    nums.sort(reverse=True)

    # Initialize the number of operations and the current sum
    ops = 0
    total_sum = sum(nums)

    # Continue dividing elements until the sum is halved
    while total_sum > (sum(nums) / 2):
        # Divide the first element by 2
        nums[0] /= 2
        
        # Update the number of operations and the current sum
        ops += 1
        total_sum -= nums[0]

    # Return the minimum number of operations
    return ops
```

#### Example:

```python
nums = [5, 3, 2]
result = min_operations_halving_sum(nums)
print(result)  # Output: 2
```

#### Breakdown:

* **Sorting:** We first sort the array in descending order. This ensures that the elements with the highest contributions to the sum come first.
* **Divide and Update:** We repeatedly divide the first element by 2, update the number of operations, and subtract the divided value from the total sum.
* **Halving Condition:** We continue this process until the current sum is less than or equal to half of the original sum.
* **Minimum Operations:** The final value of `ops` represents the minimum number of operations required to halve the sum of the array elements.

#### Real-World Applications:

* **Data Compression:** In data compression algorithms, it's often necessary to reduce the sum of values in a data set to save space. Halving the sum can be an effective way to achieve significant compression.
* **Resource Allocation:** In systems with limited resources, such as CPU time or memory, it may be necessary to reduce the overall load. Halving the sum of resource usage can help distribute the load more evenly.
* **Financial Planning:** When managing a budget, halving the sum of expenses can help reduce overall spending and achieve financial stability.

***

## minimum\_number\_of\_people\_to\_teach

**Problem Statement:**

Given a group of people where some of them already know how to solve a problem, you need to determine the minimum number of people you need to teach to solve the problem.

**Example:**

```
Input: [0, 1, 1, 0]
Output: 1
Explanation: The person at index 0 does not know how to solve the problem, so we need to teach them.
```

**Approach:**

The idea is to use a greedy approach. We start by sorting the people based on their knowledge of the problem. Then, we iterate through the sorted list and teach the first person who does not know how to solve the problem.

**Implementation:**

```python
def minimum_number_of_people_to_teach(people):
  # Sort the people based on their knowledge of the problem.
  people.sort()

  # Iterate through the sorted list and teach the first person who does not know how to solve the problem.
  for person in people:
    if person == 0:
      return people.index(person) + 1

  # If everyone knows how to solve the problem, return 0.
  return 0
```

**Time Complexity:**

The time complexity of the above solution is O(n log n), where n is the number of people. This is because we need to sort the people, which takes O(n log n) time.

**Space Complexity:**

The space complexity of the above solution is O(1).

**Real World Applications:**

This problem can be applied in any situation where you need to determine the minimum number of people you need to teach to solve a problem. For example, you could use this algorithm to determine the minimum number of students you need to teach to solve a math problem, or the minimum number of employees you need to train to perform a new task.

***

## number\_of\_substrings\_with\_only\_1s

**Problem Statement**: Given a binary string `s`, return the number of non-empty substrings that contain only 1s.

**Example 1**:

```
Input: s = "01101001"
Output: 15
Explanation: The substrings with only 1s are:
"1"
"11"
"110"
"1101"
"11010"
"110100"
"1101001"
"101"
"1010"
"10100"
"101001"
"1001"
"10010"
"100100"
"1001001"
```

**Example 2**:

```
Input: s = "10101"
Output: 4
Explanation: The substrings with only 1s are:
"1"
"101"
"1010"
"10101"
```

**Solution**:

1. **Prefix Sum Array**:
   * Create a prefix sum array `ps` where `ps[i]` stores the number of 1s in the substring `s[0:i]`.
   * This can be calculated by initializing `ps[0]` to 0 and iterating over `s`:
     * If `s[i] == '1'`, then `ps[i+1] = ps[i] + 1`.
     * Otherwise, `ps[i+1] = ps[i]`.
2. **Count Substrings**:
   * Initialize `count` to 0.
   * Iterate over the prefix sum array:
     * If `ps[i] == i`, then the substring `s[0:i]` contains only 1s. Increment `count` by `i`.
     * Otherwise, the substring `s[0:i]` does not contain only 1s. So, consider the substring `s[ps[i]:i]`. If `ps[i] == 0`, then the substring contains only 1s. Increment `count` by `i - ps[i]`. Otherwise, the substring does not contain only 1s.

**Simplified Explanation**:

1. **Prefix Sum Array**:
   * We create an array that keeps track of how many 1s there are in each prefix of the string. For example, if `s = "1101"`, the prefix sum array would be `[1, 2, 2, 3]`.
2. **Count Substrings**:
   * We iterate over the prefix sum array and count the number of substrings that contain only 1s.
   * If the number of 1s in the current prefix is equal to the index, then the entire prefix contains only 1s.
   * Otherwise, we count the number of 1s in the current substring by subtracting the number of 1s in the previous prefix from the number of 1s in the current prefix.

**Real-World Applications**:

This algorithm can be used in various real-world applications, such as:

* **DNA Analysis**: Counting the number of consecutive "1"s in a DNA sequence can help identify genetic markers and mutations.
* **Image Processing**: Counting the number of consecutive white pixels in an image can help detect objects and boundaries.
* **Data Compression**: This algorithm can be used to optimize data compression by identifying and compressing sequences of consecutive 1s.

**Complete Python Implementation**:

```python
def count_binary_substrings(s):
    # Create a prefix sum array
    ps = [0] * len(s)
    ps[0] = 1 if s[0] == '1' else 0

    for i in range(1, len(s)):
        if s[i] == '1':
            ps[i] = ps[i-1] + 1
        else:
            ps[i] = ps[i-1]

    # Count substrings with only 1s
    count = 0
    for i in range(0, len(s)):
        if ps[i] == i:
            count += i
        elif ps[i] == 0:
            count += i - ps[i]

    return count
```

***

## minimum\_average\_difference

**Problem Statement**

Given an array of integers, find the minimum average difference between any two contiguous subarrays.

**Example**

For example, given the array `[4, 2, 1, 3]`, the minimum average difference is `0.5`, which is the difference between the subarrays `[4, 2]` and `[1, 3]`.

**Solution**

**Brute Force Approach:**

The brute force approach is to compute the average difference of all possible contiguous subarrays and then find the minimum. This approach has a time complexity of O(n^3), where n is the length of the array.

```python
def minimum_average_difference_brute_force(arr):
    min_diff = float('inf')
    for i in range(1, len(arr)):
        for j in range(i):
            avg_diff = abs(sum(arr[j:i]) / (i - j) - sum(arr[i:]) / (len(arr) - i))
            if avg_diff < min_diff:
                min_diff = avg_diff
    return min_diff
```

**Prefix Sum Approach:**

A more efficient approach uses prefix sums to compute the sum of each subarray in constant time. This reduces the time complexity to O(n^2).

```python
def minimum_average_difference_prefix_sum(arr):
    prefix_sums = [0] * len(arr)
    for i in range(len(arr)):
        prefix_sums[i] = arr[i] if i == 0 else prefix_sums[i - 1] + arr[i]

    min_diff = float('inf')
    for i in range(1, len(arr)):
        avg_diff = abs((prefix_sums[i - 1] / i) - ((prefix_sums[-1] - prefix_sums[i - 1]) / (len(arr) - i)))
        if avg_diff < min_diff:
            min_diff = avg_diff
    return min_diff
```

**Sliding Window Approach:**

The sliding window approach maintains a window of size 2 and computes the average difference of the current window. As the window slides over the array, the average difference is updated. This approach has a time complexity of O(n).

```python
def minimum_average_difference_sliding_window(arr):
    min_diff = float('inf')
    window_sum = sum(arr[:2])
    for i in range(2, len(arr)):
        avg_diff = abs((window_sum / 2) - ((sum(arr) - window_sum) / (len(arr) - 2)))
        if avg_diff < min_diff:
            min_diff = avg_diff
        window_sum += arr[i] - arr[i - 2]
    return min_diff
```

**Applications**

The minimum average difference problem has applications in:

* **Data analysis:** Finding the best split point for a dataset to minimize the difference between the two resulting subsets.
* **Machine learning:** Determining the optimal number of clusters for a dataset by minimizing the average distance between points within each cluster.
* **Finance:** Measuring the volatility of a stock or portfolio by calculating the average difference between the closing prices of consecutive trading days.

***

## stock\_price\_fluctuation

**Problem Statement:** Given an array of integers representing stock prices on different days, find the maximum profit that can be obtained by buying and selling the stock any number of times.

**Implementation:** **Brute Force Approach:** This approach is straightforward but inefficient. We iterate over all possible pairs of days and calculate the profit for each pair. The maximum profit among all these is the answer.

```python
def max_profit_brute_force(prices):
    max_profit = 0
    for i in range(len(prices)):
        for j in range(i + 1, len(prices)):
            profit = prices[j] - prices[i]
            if profit > max_profit:
                max_profit = profit
    return max_profit
```

**Time Complexity:** O(n^2), where n is the length of the prices array. **Space Complexity:** O(1).

**Kadane's Algorithm (Greedy Approach):** This algorithm iterates over the array and maintains the maximum profit encountered so far as well as the minimum price observed till that point. The difference between the current price and the minimum price is the potential profit at that point. The maximum of these potential profits is the answer.

```python
def max_profit_kadane(prices):
    max_profit = 0
    min_price = prices[0]
    for price in prices:
        min_price = min(min_price, price)
        max_profit = max(max_profit, price - min_price)
    return max_profit
```

**Time Complexity:** O(n), where n is the length of the prices array. **Space Complexity:** O(1).

**Simplified Explanation:**

**Brute Force Approach:** Imagine you have a list of stock prices. You want to find the maximum profit you can make by buying and selling the stock multiple times.

* Create a two-dimensional table where each row represents a start day and each column represents an end day.
* Calculate the profit for each possible pair of start and end days.
* Keep track of the maximum profit.

**Kadane's Algorithm:**

* Keep a running track of the minimum price encountered so far.
* For each day, the potential profit is the difference between the current price and the minimum price.
* Keep track of the maximum potential profit.

**Applications in Real World:**

* Stock trading: Determine the best times to buy and sell stocks to maximize profit.
* Financial analysis: Analyze historical stock prices to identify trading opportunities.
* Investment planning: Determine the optimal allocation of funds for investment in stocks or other financial instruments.

***

## number\_of\_laser\_beams\_in\_a\_bank

**Leetcode Problem:**

**Number of Laser Beams in a Bank**

Given an array of zeros and ones, representing the status of laser beams in a bank. Each zero represents a laser beam that is turned off, and each one represents a laser beam that is turned on.

A laser beam is considered to be reflected if it meets the boundary of the array or another laser beam. Two laser beams are considered to be intersected if they meet in the same cell.

Return the number of laser beams that are not reflected or intersected.

**Solution:**

**Breakdown:**

* Iterate over each cell in the array.
* Check if the current cell is a laser beam (i.e., equal to 1).
* If it is a laser beam, check if it is blocked in either the horizontal or vertical direction.
* If it is not blocked, increment the count of unblocked laser beams.

**Simplified Explanation:**

Imagine a grid of laser beams. Each laser beam can only travel in one direction: horizontally or vertically.

If a laser beam encounters a wall or another laser beam, it will stop traveling.

We want to count the number of laser beams that can travel unimpeded.

**Implementation:**

```python
def count_laser_beams(grid):
    m, n = len(grid), len(grid[0])
    unblocked_beams = 0

    for i in range(m):
        for j in range(n):
            beam = grid[i][j]

            # Check if the beam is turned on.
            if beam == 1:
                # Check if the beam is blocked horizontally.
                blocked_horizontally = False
                for k in range(j + 1, n):
                    if grid[i][k] == 1:
                        blocked_horizontally = True
                        break

                # Check if the beam is blocked vertically.
                blocked_vertically = False
                for k in range(i + 1, m):
                    if grid[k][j] == 1:
                        blocked_vertically = True
                        break

                # If the beam is not blocked, increment the count.
                if not blocked_horizontally and not blocked_vertically:
                    unblocked_beams += 1

    return unblocked_beams
```

**Potential Applications:**

* Designing laser security systems
* Simulating the spread of light in a room
* Designing optical communication systems

***

## lowest\_common\_ancestor\_of\_a\_binary\_tree\_iii

**Lowest Common Ancestor of a Binary Tree III**

**Problem Statement:** Given a binary tree and two nodes, find the lowest common ancestor (LCA) of the two nodes. The constraint is that each node has a parent pointer.

**Example:** Consider the following binary tree:

```
           A
          / \
         B   C
        / \ / \
       D   E F   G
```

If we need to find the LCA of nodes D and E, it would be node B.

**Intuition:** We can start from the two nodes and move up the tree until we find their LCA. Since each node has a parent pointer, we can easily access its parent.

**Algorithm:**

1. Start from the two nodes.
2. If one of the nodes is the parent of the other, then that node is the LCA.
3. If not, move up one level for both nodes.
4. Repeat steps 2 and 3 until we find the LCA.

**Implementation:**

```python
def lowestCommonAncestor(root, p, q):
    if not root or root == p or root == q:
        return root

    left = lowestCommonAncestor(root.left, p, q)
    right = lowestCommonAncestor(root.right, p, q)

    if left and right:
        return root
    return left or right
```

**Breakdown:**

The function `lowestCommonAncestor` takes three arguments:

* `root`: The root of the binary tree.
* `p`: The first node.
* `q`: The second node.

The function first checks if the `root` is `None` or if it is equal to either `p` or `q`. If so, it returns `root`. Otherwise, it calls the function recursively on the left and right subtrees of the `root`. If both `left` and `right` are not `None`, then the current `root` is the LCA. Otherwise, the function returns `left` or `right` depending on which one is not `None`.

**Example Usage:**

```python
# Create the binary tree
root = TreeNode('A')
root.left = TreeNode('B')
root.right = TreeNode('C')
root.left.left = TreeNode('D')
root.left.right = TreeNode('E')
root.right.left = TreeNode('F')
root.right.right = TreeNode('G')

# Find the LCA of nodes D and E
lca = lowestCommonAncestor(root, D, E)

# Print the LCA
print(lca.val)  # B
```

**Real-World Applications:**

The LCA algorithm has many applications in real-world scenarios, such as:

* Finding the common ancestor of two nodes in a file system hierarchy.
* Finding the common ancestor of two users in a social network.
* Finding the common ancestor of two commits in a version control system.

***

## maximum\_matrix\_sum

**Problem Statement**

Given a matrix of integers, find the maximum sum of all the elements in the matrix.

**Example**

Input:

```
[[1, 2, 3],
 [4, 5, 6],
 [7, 8, 9]]
```

Output:

```
45
```

**Solution**

The best approach to solve this problem is to use a nested loop to iterate over the matrix and add each element to a running sum.

```python
def maximum_matrix_sum(matrix):
  """
  Computes the maximum sum of all the elements in a matrix.

  Parameters:
    matrix: A list of lists of integers.

  Returns:
    The maximum sum of all the elements in the matrix.
  """

  # Initialize the running sum to 0.
  sum = 0

  # Iterate over the matrix.
  for row in matrix:
    for element in row:
      # Add each element to the running sum.
      sum += element

  # Return the running sum.
  return sum
```

**Time Complexity**

The time complexity of this solution is O(mn), where m is the number of rows in the matrix and n is the number of columns. This is because the algorithm iterates over each element in the matrix once.

**Space Complexity**

The space complexity of this solution is O(1). This is because the algorithm does not use any additional space beyond the input matrix.

**Applications**

This algorithm can be used in a variety of applications, including:

* Finding the maximum sum of a subarray.
* Finding the maximum sum of an array of weighted elements.
* Finding the maximum profit from a series of trades.

***

## furthest\_building\_you\_can\_reach

**Problem Statement:**

You are given a total of n buildings numbered from 0 to n-1, with each building having a certain amount of energy. Initially, all the buildings have zero energy.

You have an unlimited supply of energy units. For each unit of energy, you can perform one of the following two actions:

1. Increase the energy of a building by 1.
2. Increase the energy of two adjacent buildings by 1.

Your task is to determine the maximum total energy that can be stored in all the buildings.

**Solution:**

The maximum total energy can be stored by applying the following greedy algorithm:

1. Sort the buildings in ascending order of their initial energy.
2. Start from the building with the lowest initial energy.
3. If the energy of the current building is less than the energy of the next building, then increase the energy of the current building by 1.
4. Else, increase the energy of the current building and its adjacent building by 1.
5. Repeat steps 3 and 4 until all the buildings have been processed.

**Simplified Explanation:**

* **Step 1: Sorting** helps us prioritize buildings with lower energy.
* **Steps 3 and 4:** We allocate energy to buildings efficiently. If the current building has lower energy than its neighbor, we prioritize it. Otherwise, we distribute energy to both the current building and its neighbor.
* **Repeating until all buildings processed:** Ensures we maximize energy for all buildings.

**Real-World Applications:**

This algorithm can be used in various real-world scenarios:

* **Load balancing:** Distributing workload across multiple servers efficiently.
* **Resource allocation:** Optimally allocating resources among different tasks or users.
* **Power grid optimization:** Optimizing energy distribution in power grids.

**Example:**

Consider three buildings with initial energies \[1, 5, 2].

* **Step 1: Sorting:** \[1, 2, 5]
* **Step 2:** Starting with the lowest energy building (1).
* **Step 3:** Current building (1) has lower energy than the next building (2). Increase energy of the current building by 1 to \[2, 2, 5].
* **Step 4:** Current building (2) has equal energy to the next building (5). Increase energy of both buildings by 1 to \[3, 3, 5].
* **Final energy distribution:** The maximum total energy is 11 (2 + 3 + 5 + 1 unit allocated to building 1).

**Complete Code Implementation (Python):**

```python
def max_total_energy(buildings):
  """Returns the maximum total energy that can be stored in the buildings."""

  # Sort the buildings in ascending order of their initial energy.
  buildings.sort()

  # Initialize the total energy.
  total_energy = 0

  # Start from the building with the lowest initial energy.
  for building in buildings:
    # If the energy of the current building is less than the energy of the next building, then increase the energy of the current building by 1.
    if building < buildings[min(len(buildings) - 1, buildings.index(building) + 1)]:
      total_energy += 1
      building += 1

    # Else, increase the energy of the current building and its adjacent building by 1.
    else:
      total_energy += 2
      building += 1
      buildings[min(len(buildings) - 1, buildings.index(building) + 1)] += 1

  return total_energy
```

***

## count\_the\_hidden\_sequences

**Count the Hidden Sequences** Given a string, your task is to count the number of hidden sequences in the given string.

**Definition of Hidden Sequence:** A hidden sequence is a sequence of characters that is not visible when you read the string from left to right. A character is hidden if it is surrounded by two characters that are the same.

**Example:** Given string: "ababa" Hidden sequences: "b", "b" Output: 2

**Approach:** To count the hidden sequences, you can use the following steps:

1. **Iterate over the string:** Iterate over the characters of the string from left to right.
2. **Check if the current character is hidden:** Check if the current character is surrounded by two characters that are the same. If it is, increment the count of hidden sequences.
3. **Repeat steps 1 and 2:** Repeat steps 1 and 2 for all characters in the string.

**Here's the Python code for this approach:**

```python
def count_hidden_sequences(string):
  """
  Counts the number of hidden sequences in a given string.

  Parameters:
    string: The string to search for hidden sequences.

  Returns:
    The number of hidden sequences in the string.
  """

  # Initialize the count of hidden sequences.
  count = 0

  # Iterate over the characters in the string.
  for i in range(1, len(string) - 1):
    # Check if the current character is hidden.
    if string[i - 1] == string[i + 1]:
      # Increment the count of hidden sequences.
      count += 1

  # Return the count of hidden sequences.
  return count
```

**Time Complexity:** O(n), where n is the length of the string.

**Here's an example usage of the code:**

```python
string = "ababa"
result = count_hidden_sequences(string)
print(result)  # Output: 2
```

**Potential Applications in the Real World:** The concept of hidden sequences can be applied in various real-world scenarios, such as:

* **Text analysis:** Identifying hidden sequences in text can help in extracting meaningful insights from the text.
* **Natural language processing:** Hidden sequences can be used to detect named entities or other grammatical structures in natural language text.
* **Data mining:** Hidden sequences can be used to uncover patterns and relationships in large datasets.

***

## two\_best\_non\_overlapping\_events

**Problem Statement:**

Given a set of events with start and end times, find the two events that can be attended without overlapping.

**Brute Force Approach:**

The simplest approach is to try all possible pairs of events and check if they overlap. This takes O(n^2) time, where n is the number of events.

**Optimal Approach:**

The optimal approach is to sort the events by their end times. This takes O(n log n) time. Then, we can iterate through the sorted events and greedily select the next event that does not overlap with the previous event. This takes O(n) time.

**Python Implementation:**

```python
def two_best_non_overlapping_events(events):
  """
  Finds the two events that can be attended without overlapping.

  Args:
    events: A list of events. Each event is represented as a tuple (start, end).

  Returns:
    A list of two events.
  """

  # Sort the events by their end times.
  sorted_events = sorted(events, key=lambda event: event[1])

  # Initialize the two best events.
  best_event1 = None
  best_event2 = None

  # Iterate through the sorted events.
  for event in sorted_events:
    # Check if the event overlaps with the previous event.
    if best_event1 is not None and event[0] < best_event1[1]:
      continue

    # Update the two best events.
    best_event2 = best_event1
    best_event1 = event

  # Return the two best events.
  return [best_event1, best_event2]
```

**Real-World Applications:**

This problem has applications in scheduling and resource allocation. For example, it can be used to schedule meetings or to allocate resources to different tasks.

**Simplified Explanation:**

We can think of each event as a block of time. Our goal is to find two blocks of time that do not overlap.

We can sort the blocks of time by their end times. This will make it easier to find the next block of time that does not overlap with the previous block of time.

We can then iterate through the sorted blocks of time and greedily select the next block of time that does not overlap with the previous block of time.

This approach is efficient because it only considers the blocks of time that are relevant to the current block of time.

***

## count\_collisions\_on\_a\_road

**Problem Statement:**

Imagine a road with multiple cars driving in the same direction. Suppose a car is considered crashed if it is directly behind another car and traveling at a slower speed.

Given an array of integers `cars` representing the speed of each car in km/h, return the total number of crashed cars on the road.

**Example:**

```
Input: cars = [1,2,3,4,5]
Output: 0
Explanation: No cars are crashing into each other because each car is going faster than the car in front of it.

Input: cars = [3,0,1,2]
Output: 2
Explanation: The car with speed 0 is crashed into the car with speed 3, and the car with speed 1 is crashed into the car with speed 3.
```

**Python Implementation:**

```python
def count_collisions_on_a_road(cars):
  """
  Counts the number of crashed cars on a road.

  Parameters:
    cars: A list of integers representing the speed of each car in km/h.

  Returns:
    The total number of crashed cars on the road.
  """

  # Initialize the number of crashed cars to 0.
  crashed_cars = 0
  
  # Initialize the speed of the car in front of the current car to 0.
  prev_speed = 0

  # Iterate over the cars in the array.
  for car_speed in cars:
    # If the current car is traveling at a slower speed than the car in front of it, then it is crashed.
    if car_speed < prev_speed:

      # Increment the number of crashed cars.
      crashed_cars += 1
    
    # Otherwise, if the current car is traveling at the same speed as the car in front of it, 
    # then it is not crashed.
    elif car_speed == prev_speed:
      crashed_cars += 1

    # Update the speed of the car in front of the current car.
    prev_speed = car_speed

  # Return the total number of crashed cars on the road.
  return crashed_cars
```

**Explanation:**

* We initialize the `crashed_cars` variable to 0.
* We initialize the `prev_speed` variable to 0.
* We iterate over the `cars` array.
* For each car, we check if its speed is less than the speed of the car in front of it.
  * If it is, then we increment the `crashed_cars` variable.
* We update the `prev_speed` variable to the speed of the current car.
* We return the `crashed_cars` variable.

**Real-World Applications:**

This problem can be applied in real-world situations to help optimize traffic flow and reduce the number of accidents. By understanding the patterns of traffic collisions, we can design better road systems and implement measures to prevent crashes.

***

## find\_original\_array\_from\_doubled\_array

**Problem:** Given an array `nums` that is sorted in ascending order and has been doubled, we want to find the original array from which `nums` was obtained.

**Solution:**

**Step 1: Initialize Variables**

```python
original = []  # Empty list to store the original array
```

**Step 2: Iterate Over nums**

We iterate over the elements in `nums`. For each element `num`, we check if it is even. If it is:

```python
if num % 2 == 0:
    # If num is even, we divide it by 2 to get the corresponding element in the original array.
    original.append(num // 2)
```

If `num` is not even, it means that it is already the original element. We append it directly to `original`.

**Step 3: Return the Original Array**

After iterating over `nums`, we return the `original` list.

**Code:**

```python
def find_original_array_from_doubled_array(nums):
    original = []
    for num in nums:
        if num % 2 == 0:
            original.append(num // 2)
        else:
            original.append(num)
    return original
```

**Example:**

```python
nums = [1, 2, 4, 8, 16]
result = find_original_array_from_doubled_array(nums)
print(result)  # Output: [1, 2, 4, 8]
```

**Explanation:**

The input array `nums` is a doubled version of the original array `[1, 2, 4, 8]`. The elements in `nums` are all even and appear twice. Our function iterates over `nums` and divides each even element by 2 to get the corresponding element in the original array. The original array is returned as `[1, 2, 4, 8]`.

**Applications in the Real World:**

* **Data Analysis:** Identifying the original data set from a processed or transformed data set.
* **Fraud Detection:** Detecting fraudulent transactions by comparing a set of transactions to a known "normal" baseline.
* **Image Processing:** Recovering the original image from a distorted or compressed version.

***

## range\_sum\_of\_sorted\_subarray\_sums

**Problem Statement:**

Given an array of positive integers, return the maximum sum of all contiguous subarrays of the array.

**For Example:**

```
Input: nums = [5, 4, 3, 2, 1]
Output: 15
Explanation: The contiguous subarray [5, 4, 3, 2, 1] has the maximum sum of 15.
```

**Implementation in Python:**

```python
def max_subarray_sum(nums):
  """
  Returns the maximum sum of all contiguous subarrays of the given array.

  Parameters:
    nums: A list of positive integers.

  Returns:
    The maximum sum of all contiguous subarrays of the given array.
  """

  # Initialize the current and maximum sums to 0.
  current_sum = 0
  max_sum = 0

  # Iterate over the elements in the array.
  for num in nums:
    # Add the current element to the current sum.
    current_sum += num

    # Update the maximum sum if the current sum is greater.
    max_sum = max(max_sum, current_sum)

    # If the current sum is negative, reset it to 0.
    if current_sum < 0:
      current_sum = 0

  # Return the maximum sum.
  return max_sum
```

**Explanation:**

The `max_subarray_sum()` function takes an array of positive integers as input and returns the maximum sum of all contiguous subarrays of the array.

The function initializes the current and maximum sums to 0. Then, it iterates over the elements in the array, adding each element to the current sum. If the current sum is greater than the maximum sum, the function updates the maximum sum. If the current sum is negative, the function resets it to 0.

Finally, the function returns the maximum sum.

**Real-World Applications:**

The `max_subarray_sum()` function can be used in a variety of real-world applications, such as:

* Finding the maximum profit from a stock price series.
* Finding the maximum score in a game.
* Finding the maximum distance that a car can travel on a given amount of fuel.

***

## find\_the\_kth\_largest\_integer\_in\_the\_array

#### Problem Statement

Given an integer array `nums` and an integer `k`, return the `kth` largest element in the array.

#### Solution

The best & performant solution to find the kth largest element in the array is to use the quick select algorithm. Quick select is a divide-and-conquer algorithm that is similar to quicksort, but instead of sorting the entire array, it only finds the kth largest element.

Here is how quick select works:

1. Pick a pivot element, which is the element in the middle of the array.
2. Partition the array into two subarrays: one containing elements that are less than the pivot, and one containing elements that are greater than or equal to the pivot.
3. Recursively apply quick select to the subarray that contains elements that are less than the pivot.
4. If the number of elements in the subarray that contains elements that are greater than or equal to the pivot is equal to k, then the pivot is the kth largest element.
5. Otherwise, recursively apply quick select to the subarray that contains elements that are greater than or equal to the pivot.

Here is the Python implementation of quick select:

```python
def find_kth_largest(nums, k):
  """
  Finds the kth largest element in an array.

  Args:
    nums: The array to search.
    k: The index of the largest element to find.

  Returns:
    The kth largest element in the array.
  """

  # If the array is empty, return None.
  if not nums:
    return None

  # If k is out of range, return None.
  if k > len(nums):
    return None

  # Pick the pivot element.
  pivot = nums[len(nums) // 2]

  # Partition the array into two subarrays.
  left = [num for num in nums if num < pivot]
  right = [num for num in nums if num >= pivot]

  # If the number of elements in the left subarray is equal to k, then the pivot is the kth largest element.
  if len(left) == k:
    return pivot

  # If the number of elements in the left subarray is less than k, then the kth largest element is in the right subarray.
  elif len(left) < k:
    return find_kth_largest(right, k - len(left) - 1)

  # Otherwise, the kth largest element is in the left subarray.
  else:
    return find_kth_largest(left, k)
```

#### Example

Here is an example of how to use quick select to find the kth largest element in an array:

```python
nums = [3, 2, 1, 5, 6, 4]
k = 2
result = find_kth_largest(nums, k)
print(result)  # Output: 5
```

#### Applications

Quick select can be used to find the kth largest element in a variety of applications, including:

* Finding the median of an array.
* Finding the top k elements in a dataset.
* Selecting the kth best candidate in a job interview.

***

## lexicographically\_smallest\_string\_after\_applying\_operations

**Problem Statement**

Given a string 's' and an integer 'a', you can perform the following operation any number of times:

* Choose any 'i' (1 <= i <= s.length) and change s\[i] to any letter from a to z.

Return the lexicographically smallest string that you can obtain after applying the operation any number of times.

**Breakdown**

1. **Understanding the Problem:**
   * We need to find the smallest possible string by changing some characters in 's' to letters from 'a' to 'z'.
   * The string we obtain after performing the operations should be lexicographically smaller than the original string.
2. **Creating a Solution Plan:**
   * We can start by iterating through the string 's' from left to right.
   * For each character 's\[i]', we check if it can be changed to a smaller letter ('a' to 'z').
   * If it can, we update 's\[i]' to the smallest letter that is greater than or equal to 'a'.
   * This ensures that the final string will be lexicographically smaller than the original string.
3. **Real-World Example:**

   Suppose we have a string 's' = "abc" and 'a' = 2.

   * We can change 's\[1]' to 'a', resulting in "aac".
   * We cannot change 's\[2]' to a letter less than 'a', so we leave it unchanged.
   * The final string is "aac", which is lexicographically smaller than the original string.

**Simplified Code Implementation:**

```python
def lexicographically_smallest_string_after_applying_operations(s: str, a: int) -> str:
    """
    Returns the lexicographically smallest string after applying the given operations.

    Args:
    s: The original string.
    a: The minimum letter that can be used to replace characters in s.

    Returns:
    The lexicographically smallest string.
    """
    result = []
    for char in s:
        new_char = chr(ord(char) - a + ord('a'))
        if new_char >= 'a' and new_char <= 'z':
            result.append(new_char)
        else:
            result.append(char)
    return ''.join(result)
```

**Potential Applications**

* **Data compression:** Strings can be compressed by changing some characters to smaller ones.
* **Error correction:** Strings can be corrected by changing corrupted characters to the nearest valid characters.
* **String transformations:** Strings can be transformed into other strings by applying various operations.

***

## best\_team\_with\_no\_conflicts

**Problem Statement:**

You are given a list of employees and their teammates. For each employee, you need to find the best possible team for them, considering the following criteria:

* The team should have no conflicts (teammates should not be on the same team).
* The team should have the highest average skill level.

**Solution:**

To solve this problem, we can use a combination of graph algorithms and sorting.

**Step 1: Create a Graph**

We create a graph where the nodes represent employees, and the edges represent conflicts between employees. If two employees have a conflict, we add an edge between their nodes.

**Step 2: Find all Conflict-Free Teams**

We use a depth-first search (DFS) to find all conflict-free teams. Starting from each node, we explore the graph and add employees to our team until we reach a conflict. We then backtrack and try a different path.

**Step 3: Calculate Average Skill Level**

For each conflict-free team, we calculate the average skill level of its members.

**Step 4: Select the Best Team**

We select the conflict-free team with the highest average skill level.

**Example:**

Let's say we have the following employees and their teammates:

```
Employee | Teammates
---------|----------
A        | B, C
B        | A, D
C        | A
D        | B
```

We create a graph with the following edges:

```
A -- B
A -- C
B -- D
```

Using DFS, we find the following conflict-free teams:

```
Team 1: A, C
Team 2: B
Team 3: D
```

We calculate the average skill level of each team:

```
Team 1: (A's skill + C's skill) / 2
Team 2: B's skill
Team 3: D's skill
```

We select Team 1 as the best team because it has the highest average skill level.

**Real-World Applications:**

This problem can be used in real-world scenarios to:

* **Scheduling:** Assign employees to teams for maximum productivity.
* **Project Management:** Form teams with minimal conflicts and optimal skill sets.
* **Resource Allocation:** Distribute resources to teams based on their capabilities.

***

## lowest\_common\_ancestor\_of\_a\_binary\_tree\_ii

**Problem Statement:**

Given a binary tree and three nodes, find the lowest common ancestor (LCA) of the three nodes. The LCA is the node with the largest depth that is a descendant of all the given nodes.

**Solution:**

The solution is based on the following idea:

1. Find the LCA of the first two nodes, let's call it `lca1`.
2. Find the LCA of `lca1` and the third node, let's call it `lca2`.
3. `lca2` is the LCA of the three nodes.

The implementation of this solution is as follows:

```python
def lowest_common_ancestor_of_a_binary_tree_ii(root, nodes):
  """
  Finds the lowest common ancestor of a binary tree given a list of nodes.

  :param root: The root node of the binary tree.
  :param nodes: The list of nodes to find the LCA of.
  :return: The LCA of the given nodes.
  """

  # If the root is null, then there is no LCA.
  if root is None:
    return None

  # If the root is one of the given nodes, then it is the LCA.
  if root in nodes:
    return root

  # Recursively find the LCA of the left and right subtrees.
  left_lca = lowest_common_ancestor_of_a_binary_tree_ii(root.left, nodes)
  right_lca = lowest_common_ancestor_of_a_binary_tree_ii(root.right, nodes)

  # If the LCA is found in both subtrees, then it is the root.
  if left_lca and right_lca:
    return root

  # If the LCA is found in one of the subtrees, then it is the LCA of the subtree.
  if left_lca:
    return left_lca
  if right_lca:
    return right_lca

  # Otherwise, there is no LCA.
  return None
```

**Real-World Applications:**

The LCA can be used in a variety of real-world applications, including:

* **Finding the common ancestor of two or more people in a family tree.**
* **Finding the common ancestor of two or more files in a file system.**
* **Finding the common ancestor of two or more versions of a software program.**

**Potential Applications:**

* **Data mining:** The LCA can be used to find the common ancestor of two or more data points.
* **Machine learning:** The LCA can be used to find the common ancestor of two or more features.
* **Databases:** The LCA can be used to find the common ancestor of two or more tables.
* **Networks:** The LCA can be used to find the common ancestor of two or more nodes in a network.

***

## stone\_game\_vii

**Problem Statement:**

**Stone Game VII**

There are n piles of stones, where the ith pile has stones\[i] stones.

Two players play a game where they alternate turns, with player 1 moving first.

On each turn, a player can do one of the following:

* Remove a stone from the top of a pile.
* Remove two stones from the top of one pile.
* Remove a stone from the top of each of two different piles.

The player who removes the last stone wins the game.

Return **true** if the first player wins the game, otherwise return **false**.

**Example 1:**

```
Input: stones = [5,3,4,5]
Output: true
Explanation:  First player can remove a stone from the first pile, then remove a stone from the second pile, and then remove two stones from the fourth pile.
```

**Example 2:**

```
Input: stones = [3,7,2,3]
Output: true
Explanation:  First player can remove two stones from the first pile, then remove two stones from the second pile, and then remove one stone from the fourth pile.
```

**Example 3:**

```
Input: stones = [5,3,4,7]
Output: false
Explanation:  First player cannot win the game.
```

**Breakdown of the Best and Performant Solution:**

The best and performant solution for this problem is based on **Dynamic Programming**. The basic idea is to calculate the outcome of the game for all possible states of the game, and then use these outcomes to determine the outcome of the current state.

More specifically, we can define the dp array as follows:

* dp\[i]\[j] = true if the first player wins the game starting with the subarray stones\[i:j+1].
* dp\[i]\[j] = false otherwise.

We can calculate the dp array using the following recurrence relation:

* dp\[i]\[j] = true if any of the following conditions is true:
  * dp\[i+1]\[j] = false (i.e., the second player loses starting with the subarray stones\[i+1:j+1])
  * dp\[i+2]\[j] = false (i.e., the second player loses starting with the subarray stones\[i+2:j+1])
  * dp\[i]\[j-1] = false (i.e., the second player loses starting with the subarray stones\[i:j])
  * dp\[i+1]\[j-1] = false (i.e., the second player loses starting with the subarray stones\[i+1:j])
* dp\[i]\[j] = false otherwise.

We can calculate the dp array in bottom-up order, starting with the base cases:

* dp\[i]\[i] = true for all i.
* dp\[i]\[i+1] = (stones\[i] == stones\[i+1]).

Once we have calculated the dp array, we can simply return dp\[0]\[n-1] to determine the outcome of the game.

**Implementation in Python:**

```python
def stoneGameVII(stones):
    n = len(stones)
    dp = [[None] * n for _ in range(n)]

    for i in range(n):
        dp[i][i] = True

    for i in range(n-1,-1,-1):
        for j in range(i+1,n):
            if stones[i] == stones[j]:
                dp[i][j] = True
            else:
                dp[i][j] = (not dp[i+1][j]) or (not dp[i][j-1]) or (not dp[i+1][j-1])

    return dp[0][n-1]
```

**Applications in the Real World:**

This algorithm can be applied to any game where players alternate turns and must make decisions based on the current state of the game. For example, it can be used to solve games such as Nim and Connect Four.

***

## find\_missing\_observations

**Problem Statement:**

You have an array of integers `nums` that represents the observations of a sequence. Unfortunately, some observations are missing. Your task is to find the missing observations in the sequence.

**Example:**

For `nums = [1, 2, 3, 5, 7]`, the missing observations are `4` and `6`.

**Best & Performant Solution (Python):**

```python
def find_missing_observations(nums):
  result = []  # To store the missing observations

  # Iterate over the observations
  for i in range(1, len(nums)):
    # Calculate the difference between current and previous observation
    diff = nums[i] - nums[i - 1]

    # If the difference is greater than 1, there are missing observations
    if diff > 1:
      # Iterate and append the missing observations
      for missing in range(nums[i-1] + 1, nums[i]):
        result.append(missing)

  return result
```

**Breakdown:**

* Initialize an empty list `result` to store the missing observations.
* Iterate over the observations in `nums` starting from the second one.
* Calculate the difference between the current and previous observations.
* If the difference is greater than 1, it means there are missing observations.
* Iterate from the previous observation to the current observation, excluding the current observation, and append the missing observations to `result`.

**Time Complexity:** O(n), where n is the number of observations.

**Real-World Applications:**

* Identifying missing data in sensor measurements
* Detecting gaps in financial transactions
* Monitoring the progress of a project by comparing actual observations to expected observations

***

## number\_of\_good\_ways\_to\_split\_a\_string

**Number of Good Ways to Split a String**

**Problem Statement:** Given a string, determine the number of ways to split it into two non-empty strings that are palindrome strings.

**Example:**

* Input: "abccba"
* Output: 9
  * Split: "ab" and "ccba"
  * Split: "a" and "bccba"
  * Split: "ab" and "ccb"
  * Split: "abc" and "cba"
  * Split: "ab" and "ccba"
  * Split: "a" and "bccb"
  * Split: "abc" and "cb"
  * Split: "a" and "bccba"
  * Split: "abcc" and "ba"

**Solution:**

1. **Define a Dynamic Programming Array:**
   * Create a 2D array `dp` with dimensions (n+1, n+1), where `n` is the length of the input string.
2. **Base Case:**
   * For `dp[i][j]`, where `i <= j`, set `dp[i][j] = 1` if the substring from index `i` to index `j` is a palindrome. Otherwise, set `dp[i][j] = 0`.
3. **Populate the DP Array:**
   * Use a nested loop to iterate over all possible substrings.
   * For each substring from index `i` to index `j`, check if it is a palindrome by using a helper function `is_palindrome`.
   * If it is a palindrome, set `dp[i][j] = 1`. Otherwise, set it to 0.
4. **Count the Good Splits:**
   * Initialize a variable `good_splits` to 0.
   * For each index `i` from 1 to `n`, iterate over all substrings that start at index `i`.
   * Check if the current substring is a palindrome and the substring before it is also a palindrome.
   * If both substrings are palindromes, increment `good_splits`.
5. **Return the Result:**
   * After populating the DP array, return the value of `good_splits`.

**Helper Function: `is_palindrome`**

* **Input:** A string `s`.
* **Output:** True if `s` is a palindrome, False otherwise.
* **Implementation:**
  * Convert `s` to lowercase.
  * Reverse `s` and store the result in `reversed_s`.
  * Check if `s` is equal to `reversed_s`.
  * Return the result.

**Implementation:**

```python
def count_good_ways_to_split_a_string(s):
    # Create the DP array
    n = len(s)
    dp = [[0] * (n+1) for _ in range(n+1)]

    # Base case
    for i in range(n):
        for j in range(i, n):
            if is_palindrome(s[i:j+1]):
                dp[i][j] = 1

    # Populate the DP array
    for i in range(n-1, -1, -1):
        for j in range(i, n):
            if is_palindrome(s[i:j+1]):
                dp[i][j] = 1
            else:
                for k in range(i, j):
                    dp[i][j] |= (dp[i][k] & dp[k+1][j])

    # Count the good splits
    good_splits = 0
    for i in range(1, n):
        if dp[0][i-1] and dp[i][n-1]:
            good_splits += 1

    return good_splits

def is_palindrome(s):
    s = s.lower()
    reversed_s = s[::-1]
    return s == reversed_s
```

**Real-World Application:**

* Splitting text into palindrome substrings can be useful in natural language processing (NLP) applications, such as text summarization and plagiarism detection.
* It can also be used in genome analysis, where palindrome sequences can indicate the presence of regulatory elements or genetic disorders.
* In cryptography, palindromes can be used to create secure hash functions and random number generators.

***

## design\_browser\_history

**Problem Statement:**

Design a browser history class that can perform the following operations:

* **Browse:** Visit a new website.
* **Back:** Go back to the previous website visited.
* **Forward:** Go forward to the next website visited.
* **Get History:** Return the current browsing history as a list of URLs.

**Simplified Explanation:**

Imagine you're designing a browser like Chrome or Safari. You need to create a way to keep track of the websites the user visits and allow them to move back and forward through the history.

**Solution:**

We can use a doubly linked list to represent the browsing history. Each node in the list represents a website visit, and it has pointers to both the previous and next websites.

```python
class Node:
    def __init__(self, url):
        self.url = url
        self.prev = None
        self.next = None
```

The `BrowserHistory` class maintains a current pointer to the current website visit and a `head` and `tail` pointer to the first and last website visits in the history.

```python
class BrowserHistory:
    def __init__(self):
        self.current = None
        self.head = None
        self.tail = None
```

**Operations:**

* **Browse(url):**
  * Create a new node for the website visit.
  * If the current pointer is not None, set the `next` pointer of the current node to the new node.
  * If the current pointer is None, this is the first visit, so set the `head` to the new node.
  * Set the `prev` pointer of the new node to the current node.
  * Set the current pointer to the new node.
  * Set the `tail` to the new node.
* **Back():**
  * If the current pointer is not the `head`, move the current pointer to the previous node.
* **Forward():**
  * If the current pointer is not the `tail`, move the current pointer to the next node.
* **GetHistory():**
  * Create a list to store the URLs.
  * Start at the `head` and traverse the list until the `tail` is reached.
  * Add the URL of each node to the list.

**Time Complexity:**

* **Browse:** O(1)
* **Back:** O(1)
* **Forward:** O(1)
* **GetHistory:** O(n)

**Space Complexity:**

* O(n), where n is the number of website visits in the history.

**Applications:**

* **Web Browsers:** To keep track of the user's browsing history.
* **Undo/Redo Systems:** To allow users to undo and redo actions in applications.
* **Navigation Controls:** To provide back and forward buttons in applications.

***

## last\_moment\_before\_all\_ants\_fall\_out\_of\_a\_plank

To solve the problem "last moment before all ants fall out of a plank", we need to understand the problem statement and design an algorithm to solve it.

**Problem Statement:**

You have n ants on a plank of length L. The ants are initially positioned at points a\_1, a\_2, ..., a\_n on the plank (i.e. the position of the ith ant is a\_i). Each ant walks with a constant speed of 1 unit per second. The ants move either to the left or the right, initially, all ants move to the right. At any moment, any ant can turn around and start moving in the opposite direction.

Return the last moment when at least one ant is still on the plank.

**Input:**

```
n = 4
a = [4, 3, 2, 1]
L = 5
```

**Output:**

```
4
```

**Python Implementation:**

```python
def last_moment(n, a, L):
    # Sort the ants by their initial position
    a.sort()
  
    # Calculate the last moment when at least one ant is still on the plank
    last_moment = 0
    for i in range(n):
        # Calculate the time taken by the ith ant to reach the end of the plank
        time_to_end = L - a[i]
  
        # Calculate the time taken by the ith ant to reach the start of the plank
        time_to_start = a[i]
  
        # Update the last moment if the ith ant reaches the end of the plank at a later time
        last_moment = max(last_moment, time_to_end)
  
        # Update the last moment if the ith ant reaches the start of the plank at a later time
        last_moment = max(last_moment, time_to_start)
  
    return last_moment
```

**Example Usage:**

```python
n = 4
a = [4, 3, 2, 1]
L = 5
result = last_moment(n, a, L)
print(result)  # Output: 4
```

**Explanation:**

1. **Sort the ants by their initial position:** Sorting the ants helps us to calculate the time taken by each ant to reach the end and start of the plank more efficiently.
2. **Calculate the last moment when at least one ant is still on the plank:** We iterate through each ant and calculate the time taken by that ant to reach the end and start of the plank. We update the `last_moment` variable if the ant reaches the end or start of the plank at a later time.
3. **Return the last moment:** After iterating through all the ants, we return the `last_moment`.

**Time Complexity:** The time complexity of the algorithm is O(n log n), where n is the number of ants. Sorting the ants takes O(n log n) time, and calculating the last moment for each ant takes O(n) time.

**Real-World Applications:**

The problem of ants falling off a plank can be applied in various real-world situations, such as:

1. **Crowd simulation:** Simulating the movement of a crowd of people in a confined space, such as a stadium or a concert hall. Ants represent individuals moving around, and the plank represents the physical boundaries of the space. The last moment before all ants fall off the plank can represent the time when the crowd becomes too dense and starts to disperse.
2. **Resource allocation:** Allocating resources among multiple entities, such as servers or CPUs. Ants represent the entities requesting resources, and the plank represents the total available resources. The last moment before all ants fall off the plank can represent the time when the resources become exhausted and the entities can no longer be served.
3. **Traffic management:** Optimizing traffic flow in a road network. Ants represent vehicles moving through the network, and the plank represents the capacity of the network. The last moment before all ants fall off the plank can represent the time when the network becomes congested and traffic starts to slow down.

***

## average\_waiting\_time

**Problem Description**

The "Average Waiting Time" problem asks you to find the average amount of time customers spend waiting in a queue.

**Input**

The input consists of:

* `customers`: A list of integers representing the arrival times of customers.
* `n`: The number of customers.

**Output**

The output is a floating-point number representing the average waiting time.

**Examples**

```
Input:
customers = [1, 2, 3, 4, 5]
n = 5

Output:
2.5
```

```
Input:
customers = [1, 3, 5, 7, 9]
n = 5

Output:
4.0
```

**Algorithm**

We can use a simple algorithm to solve this problem:

1. Sort the customer arrival times in ascending order.
2. Calculate the waiting time for each customer. The waiting time for a customer is the difference between the time they arrived and the time the previous customer left.
3. Calculate the average waiting time by summing up the waiting times for all customers and dividing by the number of customers.

**Python Implementation**

```python
def average_waiting_time(customers, n):
  """
  Calculates the average waiting time for customers in a queue.

  Args:
    customers: A list of integers representing the arrival times of customers.
    n: The number of customers.

  Returns:
    A floating-point number representing the average waiting time.
  """

  # Sort the customer arrival times in ascending order.
  customers.sort()

  # Calculate the waiting time for each customer.
  waiting_times = []
  for i in range(1, n):
    waiting_times.append(customers[i] - customers[i - 1])

  # Calculate the average waiting time by summing up the waiting times for all customers and dividing by the number of customers.
  average_waiting_time = sum(waiting_times) / n

  return average_waiting_time
```

**Real-World Applications**

This problem has many real-world applications, such as:

* **Customer service**: Calculating the average waiting time for customers in a call center.
* **Traffic management**: Calculating the average waiting time for vehicles at a traffic light.
* **Scheduling**: Calculating the average waiting time for jobs in a scheduling system.

**Conclusion**

The "Average Waiting Time" problem is a common problem in competitive coding and has many real-world applications. The algorithm presented in this solution is simple and efficient, and can be used to solve this problem in O(n) time.

***

## find\_root\_of\_n\_ary\_tree

#### Leetcode Problem

> Given a n-ary tree, find its root node.

#### Breakdown of the Problem

1. **What is a n-ary tree?**

A n-ary tree is a tree data structure in which each node can have an arbitrary number of children. In contrast, a binary tree can have at most two children per node.

2. **What is the root node?**

The root node is the topmost node in the tree. It is the node from which all other nodes descend.

3. **How to find the root node?**

There are two common approaches to find the root node of a n-ary tree:

* **Recursive approach:** Start from any node in the tree and recursively visit all of its children. The first node that does not have any children is the root node.
* **Iterative approach:** Use a queue to store all the nodes in the tree. Then, repeatedly dequeue nodes from the queue and visit all of their children. The last node that remains in the queue is the root node.

#### A Simplified Explanation

Imagine you have a family tree. Each person in the family tree is a node in the n-ary tree. The parents of each person are the children of the person's node. The topmost person in the family tree, the ancestor of all other people, is the root node of the n-ary tree.

#### Real-World Applications

N-ary trees are used in a variety of real-world applications, including:

* **File systems:** The file system on your computer can be represented as a n-ary tree. Each directory is a node in the tree, and each file or subdirectory is a child of the directory node.
* **XML documents:** XML documents are represented as n-ary trees. Each element in the XML document is a node in the tree, and each attribute or child element is a child of the element node.
* **Social networks:** Social networks can be represented as n-ary trees. Each person in the social network is a node in the tree, and each friend of the person is a child of the person's node.

#### Python Code Implementation

```python
# Node class for n-ary tree
class Node:
  def __init__(self, data):
    self.data = data
    self.children = []

# Function to find the root node of a n-ary tree
def find_root(tree):
  # Use a queue to store all the nodes in the tree
  queue = [tree]

  # Repeatedly dequeue nodes from the queue and visit all of their children
  while queue:
    # Dequeue a node from the queue
    node = queue.pop(0)

    # Check if the node has any children
    if not node.children:
      # If the node has no children, it is the root node
      return node

    # Otherwise, add all of the node's children to the queue
    else:
      for child in node.children:
        queue.append(child)

# Example usage
tree = Node(1)
tree.children.append(Node(2))
tree.children.append(Node(3))
tree.children[0].children.append(Node(4))
tree.children[0].children.append(Node(5))

root_node = find_root(tree)
print(root_node.data)  # Output: 1
```

***

## detect\_cycles\_in\_2d\_grid

**Problem Statement**: Given a 2D grid, determine if there are any cycles in the grid. A cycle is a sequence of adjacent cells that can be traversed in a closed loop, without visiting any cell more than once.

**Solution**: To detect cycles in a 2D grid, we can use a depth-first search (DFS) algorithm. DFS is a recursive algorithm that explores a graph by traversing all possible paths from a given starting point, until all paths have been explored or a cycle is found.

**Algorithm**:

1. Start with any cell in the grid as the current cell.
2. Check if there is a cycle from the current cell. To do this, recursively call the DFS function on all adjacent cells that have not been visited yet.
3. During the DFS, keep track of the cells that have been visited.
4. If any adjacent cell has already been visited, then there is a cycle.
5. Repeat steps 2 and 3 for all unvisited cells in the grid.

**Python Code**:

```python
def detect_cycles_in_2d_grid(grid):
  """
  Detects cycles in a 2D grid.

  Args:
    grid: A 2D grid represented as a list of lists of integers.

  Returns:
    True if there is a cycle in the grid, False otherwise.
  """

  # Create a dictionary to keep track of visited cells.
  visited = {}

  # Iterate over all cells in the grid.
  for i in range(len(grid)):
    for j in range(len(grid[0])):

      # If the current cell has not been visited, check for cycles from it.
      if (i, j) not in visited:
        if dfs(grid, i, j, visited, None):
          return True

  # If no cycles are found, return False.
  return False


def dfs(grid, i, j, visited, parent):
  """
  Performs a depth-first search on the grid from the given cell.

  Args:
    grid: A 2D grid represented as a list of lists of integers.
    i: The row index of the current cell.
    j: The column index of the current cell.
    visited: A dictionary keeping track of visited cells.
    parent: The parent cell of the current cell.

  Returns:
    True if a cycle is found, False otherwise.
  """

  # Mark the current cell as visited.
  visited[(i, j)] = True

  # Iterate over all adjacent cells.
  for dx, dy in [(1, 0), (-1, 0), (0, 1), (0, -1)]:
    # Calculate the adjacent cell's coordinates.
    x = i + dx
    y = j + dy

    # Check if the adjacent cell is within the grid boundaries.
    if x < 0 or x >= len(grid) or y < 0 or y >= len(grid[0]):
      continue

    # Check if the adjacent cell has already been visited.
    if (x, y) in visited:
      # If the adjacent cell is the parent of the current cell, then there is no cycle.
      if (x, y) == parent:
        continue
      else:
        return True

    # If the adjacent cell has not been visited, perform DFS on it.
    if dfs(grid, x, y, visited, (i, j)):
      return True

  # If no cycles are found, return False.
  return False
```

**Real-World Applications**: Detecting cycles in a 2D grid has applications in various real-world scenarios, such as:

* **Maze solving**: To find a path through a maze without getting stuck in a loop.
* **Circuit analysis**: To identify loops in electrical circuits.
* **Graph theory**: To study the properties of graphs, which are used to represent networks and other interconnected systems.

***

## maximum\_number\_of\_non\_overlapping\_subarrays\_with\_sum\_equals\_target

**Problem:**

You are given an array of integers and a target value. Find the maximum number of non-overlapping subarrays such that the sum of elements in each subarray equals the target.

**Intuition:**

We can use a sliding window approach to solve this problem. Here's the basic idea:

* Start with a window that spans from index 0 to index 0.
* Calculate the sum of the elements in the current window.
* If the sum equals the target, increment the count of non-overlapping subarrays.
* Move the window to the right by increasing the end index by 1.
* Repeat steps 2-4 until the end index reaches the end of the array.

**Implementation:**

```python
def max_non_overlapping_subarrays(nums, target):
  """
  Finds the maximum number of non-overlapping subarrays whose sum equals the target.

  Parameters:
    nums: The input array of integers.
    target: The target sum.

  Returns:
    The maximum number of non-overlapping subarrays.
  """

  # Initialize variables
  max_count = 0
  current_sum = 0
  start_index = 0

  # Iterate over the array
  for end_index in range(len(nums)):
    # Add the current element to the current sum
    current_sum += nums[end_index]

    # Check if the current sum equals the target
    while current_sum == target:
      # Increment the count of non-overlapping subarrays
      max_count += 1

      # Move the start index of the window
      current_sum -= nums[start_index]
      start_index += 1

    # Check if the current sum is greater than the target
    if current_sum > target:
      # Move the start index of the window
      current_sum -= nums[start_index]
      start_index += 1

  # Return the maximum count of non-overlapping subarrays
  return max_count
```

**Example:**

```python
nums = [1, 2, 1, 2, 1]
target = 3
result = max_non_overlapping_subarrays(nums, target)
print(result)  # Output: 2
```

**Explanation:**

The function `max_non_overlapping_subarrays` takes an array of integers `nums` and a target value `target` as input. It initializes the maximum count of non-overlapping subarrays `max_count` to 0, the current sum `current_sum` to 0, and the start index of the window `start_index` to 0.

The function then iterates over the array using a for loop, starting at index 0. For each index `end_index`, the function adds the current element `nums[end_index]` to the current sum.

If the current sum equals the target, the function increments the maximum count of non-overlapping subarrays `max_count` by 1. The function then moves the start index of the window to the next index by updating `start_index` to `start_index + 1` and subtracting the element at the start index from the current sum.

If the current sum is greater than the target, the function moves the start index of the window to the next index by updating `start_index` to `start_index + 1` and subtracting the element at the start index from the current sum.

After the for loop completes, the function returns the maximum count of non-overlapping subarrays `max_count`.

In the example given, the input array `nums` is \[1, 2, 1, 2, 1] and the target value `target` is 3. The function `max_non_overlapping_subarrays` finds two non-overlapping subarrays whose sum equals the target: \[1, 2] and \[1, 2]. Therefore, the function returns 2 as the result.

**Real-World Applications:**

The problem of finding the maximum number of non-overlapping subarrays with a given sum can be applied to a variety of real-world problems, such as:

* **Data compression:** By finding the maximum number of non-overlapping subarrays, we can compress data by replacing each subarray with a single value that represents the sum of the subarray.
* **Signal processing:** In signal processing, we can use this problem to find the maximum number of non-overlapping segments of a signal that have a given amplitude.
* **Image segmentation:** In image segmentation, we can use this problem to find the maximum number of non-overlapping regions of an image that have a given color or texture.

***

## decode\_the\_slanted\_ciphertext

**Problem Statement:**

Given a string of lowercase letters, you need to decode it using a slanted cipher. In a slanted cipher, the letters are shifted down by a certain number of positions based on their position in the alphabet. The amount the letter is shifted is equal to its position in the alphabet.

For example:

* 'a' is shifted down 1 position, becoming 'b'
* 'z' is shifted down 26 positions, wrapping around to 'a'

**Optimal Solution:**

**Step 1: Create a Dictionary for Shifted Letters**

Create a dictionary to store the mapping of original letters to their shifted counterparts. Iterate through the lowercase letters and shift each one down the alphabet by its corresponding position.

```python
shifted_letters = {}
for i in range(ord('a'), ord('z') + 1):
    shifted_letters[chr(i)] = chr((i - ord('a') + 1) % 26 + ord('a'))
```

**Step 2: Decode the Ciphertext**

Iterate through the characters in the ciphertext and translate them using the shifted letters dictionary.

```python
def decode_slanted_ciphertext(ciphertext):
    decoded_text = ""
    for char in ciphertext:
        if char.isalpha():
            char = shifted_letters.get(char, char)
        decoded_text += char
    return decoded_text
```

**Time Complexity:** O(n), where n is the length of the ciphertext.

**Example:**

```python
ciphertext = "sfrvjh"
decoded_text = decode_slanted_ciphertext(ciphertext)
print(decoded_text)  # Output: hello
```

**Real-World Applications:**

Slanted ciphers can be used for simple text encryption, such as in puzzles or games. For example, a secret message in a scavenger hunt could be encrypted using a slanted cipher.

***

## remove\_all\_ones\_with\_row\_and\_column\_flips

**Problem Statement:**

Given a binary matrix `M` of size `n x m`, remove all the rows and columns with at least one 1 in them. Return the modified matrix without the removed rows and columns.

**Implementation:**

```python
def remove_all_ones_with_row_and_column_flips(matrix):
    """
    Removes all rows and columns with at least one 1 in them.

    Parameters:
    matrix (list[list[int]]): A binary matrix.

    Returns:
    list[list[int]]: The modified matrix without the removed rows and columns.
    """

    # Create sets to track rows and columns with at least one 1.
    rows_to_remove = set()
    cols_to_remove = set()

    # Iterate over the matrix.
    for row in range(len(matrix)):
        for col in range(len(matrix[0])):
            if matrix[row][col] == 1:
                rows_to_remove.add(row)
                cols_to_remove.add(col)

    # Create a new matrix without the removed rows and columns.
    modified_matrix = [[matrix[row][col] for col in range(len(matrix[0])) if col not in cols_to_remove] for row in range(len(matrix)) if row not in rows_to_remove]

    return modified_matrix
```

**Simplified Explanation:**

1. We iterate over the entire matrix and check if any element is equal to 1.
2. If we find an element equal to 1, we add the corresponding row and column to sets called `rows_to_remove` and `cols_to_remove`.
3. We then create a new matrix called `modified_matrix` that will contain only the rows and columns that do not have any 1s.
4. Finally, we return the `modified_matrix`.

**Example:**

Input:

```
matrix = [[0, 0, 0],
         [1, 1, 0],
         [0, 0, 1]]
```

Output:

```
[[0, 0]]
```

**Applications in the Real World:**

This algorithm can be used in various applications where we need to remove incomplete or corrupted data. For instance, it can be used to:

* Remove rows or columns of a dataset that contain missing values.
* Remove items from a list that do not meet certain criteria.
* Remove unused items from a directory or database.

***

## simple\_bank\_system

### LeetCode Problem: Design a Simple Banking System

#### Problem Statement

Design a simple banking system where:

* You have one bank with multiple users.
* Each user has an account with an account number, a balance, and a list of transactions.
* The bank needs to allow users to:
  * Open an account
  * Close an account
  * Deposit money into an account
  * Withdraw money from an account
  * Get the account balance
  * Get the list of transactions for an account

#### Simplified Breakdown and Explanation

**1. Bank Model**

The bank is represented as a class with a list of user accounts.

**2. User Model**

Each user has an account number, balance, and list of transactions. The account number is unique for each user.

**3. User Account Operations**

The bank class provides methods to perform user account operations:

* **Open Account:** Creates a new account for a user with a specified account number.
* **Close Account:** Closes an existing account by removing it from the bank's list of accounts.
* **Deposit Money:** Adds a specified amount to the account balance and creates a new transaction record.
* **Withdraw Money:** Subtracts a specified amount from the account balance and creates a new transaction record.
* **Get Account Balance:** Returns the current balance of the account.
* **Get Transactions:** Returns a list of all transactions for the account.

#### Python Implementation

```python
class Bank:
    def __init__(self):
        self.accounts = {}

    def open_account(self, account_number):
        if account_number in self.accounts:
            raise ValueError("Account already exists")
        self.accounts[account_number] = {"balance": 0, "transactions": []}

    def close_account(self, account_number):
        if account_number not in self.accounts:
            raise ValueError("Account does not exist")
        del self.accounts[account_number]

    def deposit_money(self, account_number, amount):
        if account_number not in self.accounts:
            raise ValueError("Account does not exist")
        self.accounts[account_number]["balance"] += amount
        self.accounts[account_number]["transactions"].append({"type": "deposit", "amount": amount})

    def withdraw_money(self, account_number, amount):
        if account_number not in self.accounts:
            raise ValueError("Account does not exist")
        if amount > self.accounts[account_number]["balance"]:
            raise ValueError("Insufficient funds")
        self.accounts[account_number]["balance"] -= amount
        self.accounts[account_number]["transactions"].append({"type": "withdraw", "amount": amount})

    def get_account_balance(self, account_number):
        if account_number not in self.accounts:
            raise ValueError("Account does not exist")
        return self.accounts[account_number]["balance"]

    def get_transactions(self, account_number):
        if account_number not in self.accounts:
            raise ValueError("Account does not exist")
        return self.accounts[account_number]["transactions"]
```

**Example Usage:**

```python
# Create a bank
bank = Bank()

# Open an account for user A with account number 1234
bank.open_account(1234)

# Deposit 100 into account 1234
bank.deposit_money(1234, 100)

# Withdraw 50 from account 1234
bank.withdraw_money(1234, 50)

# Get the balance of account 1234
balance = bank.get_account_balance(1234)  # 50

# Get the transactions for account 1234
transactions = bank.get_transactions(1234)
```

#### Performance Analysis

The time complexity of the banking system operations is **O(1)**, as each operation directly accesses the account information for the specified account number. The list of transactions for an account is stored in an array, so getting the transactions also has a time complexity of **O(1)**.

***

## maximize\_the\_confusion\_of\_an\_exam

You are a teacher who needs to give an exam to the students, but you want to maximize the confusion of the exam. You know the questions that will be asked and the answers to those questions. You can rearrange the questions as you want. For example, suppose the questions are:

* What is the capital of France?
* What is the currency of the United Kingdom?
* What is the tallest mountain in the world? And the answers are:
* Paris
* Pound sterling
* Mount Everest You can rearrange the questions and answers as follows:

1. What is the capital of France?

* Paris

2. What is the tallest mountain in the world?

* Pound sterling

3. What is the currency of the United Kingdom?

* Mount Everest By rearranging the questions and answers, the confusion of the exam will be maximized. Given an array of questions and an array of answers, output the rearrangement of the questions and answers that will maximize the confusion of the exam.

**Explanation:** The confusion of the exam is maximized when the questions and answers are arranged in such a way that the students are most likely to get the answers wrong. To achieve this, we can use the following strategy:

1. For each question, find the answer that is most different from the correct answer.
2. Arrange the questions and answers such that the most different answer is given for each question.

This strategy will maximize the confusion of the exam because the students are most likely to choose the most different answer, which is the wrong answer.

**Implementation:**

````python
def maximize_confusion(questions, answers):
  """Maximizes the confusion of an exam.

  Args:
    questions: A list of questions.
    answers: A list of answers.

  Returns:
    A rearrangement of the questions and answers that will maximize the confusion of
    the exam.
  """

  # Find the most different answer for each question.
  most_different_answers = []
  for question in questions:
    most_different_answer = None
    min_distance = float('inf')
    for answer in answers:
      distance = edit_distance(question, answer)
      if distance > min_distance:
        most_different_answer = answer
        min_distance = distance
    most_different_answers.append(most_different_answer)

  # Arrange the questions and answers such that the most different answer is given
  # for each question.
  confused_questions = []
  confused_answers = []
  for i in range(len(questions)):
    confused_questions.append(questions[i])
    confused_answers.append(most_different_answers[i])

  return confused_questions, confused_answers

def edit_distance(string1, string2):
  """Computes the edit distance between two strings.

  Args:
    string1: The first string.
    string2: The second string.

  Returns:
    The edit distance between the two strings.
  """

  # Initialize the edit distance matrix.
  d = [[i + j for j in range(len(string2) + 1)] for i in range(len(string1) + 1)]

  # Populate the edit distance matrix.
  for i in range(1, len(string1) + 1):
    for j in range(1, len(string2) + 1):
      if string1[i - 1] == string2[j - 1]:
        d[i][j] = d[i - 1][j - 1]
      else:
        d[i][j] = min(d[i - 1][j], d[i][j - 1], d[i - 1][j - 1]) + 1

  # Return the edit distance.
  return d[len(string1)][len(string


---
# minimum_cost_to_reach_city_with_discounts

**Problem Statement:**
You are given a list of `n` cities, and a list of `m` pairs of cities that have discounts. The cost of traveling between any two cities that have a discount is reduced by a certain percentage (specified with each discount). You need to find the minimum cost to reach the last city from the first city.

**Breakdown:**
The problem can be broken down into two main steps:

1. **Create a graph:**
   - Create a graph with `n` vertices, representing the cities.
   - For each pair of cities with a discount, add an edge between them with a weight equal to the reduced cost.

2. **Run Dijkstra's algorithm:**
   - Start Dijkstra's algorithm from the first city.
   - As you process each city, update the minimum cost to reach that city.
   - Finally, the minimum cost to reach the last city will be in the output.

**Applications:**
This problem has applications in various real-world scenarios, such as:

- **Transportation planning:** Finding optimal routes for buses, trains, or airplanes.
- **Logistics:** Optimizing delivery routes and costs.
- **Scheduling:** Finding the most efficient way to schedule tasks or appointments.

**Code Implementation:**
```python
import heapq

def minimum_cost_to_reach_city_with_discounts(cities, discounts):
    # Create a graph with n vertices
    graph = [[] for _ in range(cities)]

    # Add edges to the graph with discounted costs
    for city1, city2, discount in discounts:
        graph[city1].append((city2, discount))
        graph[city2].append((city1, discount))

    # Run Dijkstra's algorithm
    min_cost = [float('inf')] * cities
    min_cost[0] = 0
    pq = [(0, 0)]  # (cost, city)

    while pq:
        cost, city = heapq.heappop(pq)

        if cost > min_cost[city]:
            continue

        for neighbor, discount in graph[city]:
            new_cost = cost + (1 - discount) * 100  # Calculate new cost with discount
            if new_cost < min_cost[neighbor]:
                min_cost[neighbor] = new_cost
                heapq.heappush(pq, (new_cost, neighbor))

    # Return the minimum cost to reach the last city
    return min_cost[cities - 1]

# Example usage:
cities = 5
discounts = [(0, 1, 0.1), (1, 2, 0.2), (2, 3, 0.3), (3, 4, 0.4)]
print(minimum_cost_to_reach_city_with_discounts(cities, discounts))  # Output: 360 (100 * (1 - 0.1) * (1 - 0.2) * (1 - 0.3) * (1 - 0.4))
````

**Explanation:**

* The `create_graph()` function creates a graph with `n` vertices and adds edges with discounted costs.
* The `Dijkstra()` function runs Dijkstra's algorithm to find the minimum cost to reach each city.
* The `minimum_cost_to_reach_city_with_discounts()` function combines these two functions to find the minimum cost to reach the last city.

***

## merge\_nodes\_in\_between\_zeros

**Problem Statement:**

Given the head of a linked list, remove all the nodes with a value of `0` and their adjacent non-zero nodes.

**Brute Force Approach:**

This approach iterates through the list and checks each node's value. If it's 0, we remove it and the next node (if it exists).

```python
def merge_nodes_in_between_zeros(head):
  dummy = ListNode(0)
  dummy.next = head
  prev = dummy
  curr = head

  while curr:
    if curr.val == 0:
      prev.next = curr.next
      curr = curr.next
    else:
      prev = curr
      curr = curr.next

  return dummy.next
```

**Time Complexity:** O(N), where N is the number of nodes in the list.

**Optimized Approach:**

This approach maintains a pointer to the previous node and checks if the current node is 0. If it is, we skip the current node and the next node (if it exists) and update the previous pointer to point to the node after the skipped nodes.

```python
def merge_nodes_in_between_zeros(head):
  dummy = ListNode(0)
  dummy.next = head
  prev = dummy
  curr = head

  while curr:
    if curr.val == 0:
      prev.next = curr.next.next if curr.next else None
    else:
      prev = curr
    curr = curr.next if curr else None

  return dummy.next
```

**Time Complexity:** O(N), where N is the number of nodes in the list.

**Example:**

```python
# Input: 0 -> 1 -> 0 -> 2 -> 0 -> 3 -> 0
# Output: 1 -> 2 -> 3
```

**Real-World Applications:**

Removing nodes with specific values and their adjacent nodes can be useful in data processing. For example, removing zeros and their adjacent non-zeros can be used in digital signal processing to remove noise or interference from a signal.

***

## check\_if\_word\_can\_be\_placed\_in\_crossword

**Problem:** Given a crossword puzzle with missing letters and a list of available words, determine if any of the available words can be placed in one of the empty rows or columns to complete the crossword.

**Solution:**

1. **Check if the grid has valid dimensions:** Ensure that the crossword grid is rectangular with positive dimensions.
2. **Create a set of available words:** Convert the list of available words into a set for faster lookup.
3. **Iterate over rows and columns:** Loop through each row and column of the crossword grid.
4. **Check if the word fits:** Determine if any of the words in the available set can fit in the current row or column. This involves matching the length of the word with the number of empty cells and ensuring that the letters in the word match any existing letters in the grid.
5. **Check for conflicts:** If a word fits, check if it conflicts with any of the existing letters in the grid.
6. **Update the grid (optional):** If a word is found to fit and there are no conflicts, you can update the crossword grid with the new word.

**Python Implementation:**

```python
def check_if_word_can_be_placed_in_crossword(crossword, available_words):
  # Check if the grid has valid dimensions
  if len(crossword) == 0 or len(crossword[0]) == 0:
    return False

  # Convert available words to a set
  available_words_set = set(available_words)

  # Iterate over rows and columns
  for i in range(len(crossword)):
    for j in range(len(crossword[0])):
      # Check if there is an empty cell
      if crossword[i][j] == '#':
        # Check if any available word fits in the current row or column
        for word in available_words_set:
          if len(word) == len(crossword[0]) - j:
            # Check if the word fits
            fits = True
            for k in range(len(word)):
              if crossword[i][j+k] != '#' and crossword[i][j+k] != word[k]:
                fits = False
                break
            if fits:
              return True

        # Check if any available word fits in the current column
        for word in available_words_set:
          if len(word) == len(crossword) - i:
            # Check if the word fits
            fits = True
            for k in range(len(word)):
              if crossword[i+k][j] != '#' and crossword[i+k][j] != word[k]:
                fits = False
                break
            if fits:
              return True

  # No word can be placed in the crossword
  return False
```

**Real-World Applications:** This algorithm can be used in various applications, such as:

* **Crossword puzzle solvers:** Automatically solve crossword puzzles by finding words that fit into the empty cells.
* **Word search games:** Find words hidden in a grid of letters.
* **Medical imaging:** Analyze and interpret medical images by detecting patterns and anomalies.

***

## find\_good\_days\_to\_rob\_the\_bank

**Problem:**

You are planning to rob houses along a street. Each house has a certain amount of money stashed, the only constraint stopping you from robbing each of them is that adjacent houses have security systems connected, and if you rob one house, you cannot rob the adjacent house. Given a list of non-negative integers representing the amount of money of each house, determine the maximum amount of money you can rob without alerting the security systems.

**Solution:**

**Brute Force:**

* Try all possible combinations of houses to rob and calculate the total amount of money for each combination.
* Select the combination with the maximum total amount.

**Optimized Solution (Dynamic Programming):**

Dynamic programming stores the maximum amount of money that can be robbed up to a certain house. Let `dp[i]` represent the maximum amount that can be robbed up to house `i`.

* **Base Cases:**
  * `dp[0] = arr[0]` (Robbing only house 0)
  * `dp[1] = max(arr[0], arr[1])` (Robbing house 0 or house 1)
* **Recursive Formula:**
  * `dp[i] = max(dp[i-1], dp[i-2] + arr[i])`
    * `dp[i-1]` is the maximum amount that can be robbed up to house `i-1` without robbing house `i`.
    * `dp[i-2] + arr[i]` is the maximum amount that can be robbed up to house `i-2` with the addition of the money at house `i`.
* **Final Result:**
  * `max(dp[len(arr) - 1], dp[len(arr) - 2])`

**Python Implementation:**

```python
def rob(nums):
    """
    :type nums: List[int]
    :rtype: int
    """
    if not nums:
        return 0
    if len(nums) == 1:
        return nums[0]

    dp = [0] * len(nums)
    dp[0] = nums[0]
    dp[1] = max(nums[0], nums[1])
    for i in range(2, len(nums)):
        dp[i] = max(dp[i-1], dp[i-2] + nums[i])
    return max(dp[len(nums)-1], dp[len(nums)-2])
```

**Example:**

```python
nums = [1, 2, 3, 1]
result = rob(nums)  # Output: 4
```

**Real-World Applications:**

* **Job Scheduling:** Maximizing the number of jobs completed while minimizing conflicts or dependencies.
* **Investment Planning:** Determining the best sequence of investments to maximize returns while considering market fluctuations.
* **Inventory Management:** Optimizing the order of items to sell or restock to minimize costs or maximize profits.
* **Ride-Sharing:** Scheduling drivers efficiently to maximize both driver revenue and customer satisfaction.

***

## maximum\_compatibility\_score\_sum

**Problem Statement:**

Suppose you have a list of students and a list of mentors. Each student has a skill level, and each mentor has a mentoring capacity. You want to pair each student with a mentor such that the total sum of the compatibility scores between each student-mentor pair is maximized.

The compatibility score between a student and a mentor is calculated as follows:

* If the student's skill level is less than or equal to the mentor's mentoring capacity, the compatibility score is equal to the student's skill level.
* Otherwise, the compatibility score is equal to 0.

**Solution:**

The following Python code implements a simple greedy algorithm to solve this problem:

```python
def maximum_compatibility_score_sum(students, mentors):
  """
  Calculates the maximum sum of compatibility scores between a list of
  students and a list of mentors.

  Args:
    students: A list of students, where each student is represented as an
      integer representing their skill level.
    mentors: A list of mentors, where each mentor is represented as an integer
      representing their mentoring capacity.

  Returns:
    The maximum sum of compatibility scores.
  """

  # Sort the students and mentors in ascending order of skill level and mentoring
  # capacity, respectively.
  students.sort()
  mentors.sort()

  # Initialize the score sum and the current student and mentor indices.
  score_sum = 0
  student_idx = 0
  mentor_idx = 0

  # Loop through the sorted students and mentors.
  while student_idx < len(students) and mentor_idx < len(mentors):
    # Calculate the compatibility score between the current student and mentor.
    compatibility_score = min(students[student_idx], mentors[mentor_idx])

    # Add the compatibility score to the score sum.
    score_sum += compatibility_score

    # Increment the current student and mentor indices.
    student_idx += 1
    mentor_idx += 1

  # Return the score sum.
  return score_sum
```

**Example:**

```python
students = [1, 2, 3, 4, 5]
mentors = [3, 4, 5, 6, 7]

result = maximum_compatibility_score_sum(students, mentors)
print(result)  # Output: 15
```

**Explanation:**

The algorithm works as follows:

1. Sort the students and mentors in ascending order of skill level and mentoring capacity, respectively.
2. Initialize the score sum and the current student and mentor indices to 0.
3. Loop through the sorted students and mentors.
4. Calculate the compatibility score between the current student and mentor.
5. Add the compatibility score to the score sum.
6. Increment the current student and mentor indices.
7. Repeat steps 3-6 until all students and mentors have been paired.
8. Return the score sum.

In the example above, the students are \[1, 2, 3, 4, 5] and the mentors are \[3, 4, 5, 6, 7]. The algorithm pairs the students with the mentors as follows:

* Student 1 with mentor 3 (compatibility score: 1)
* Student 2 with mentor 4 (compatibility score: 2)
* Student 3 with mentor 5 (compatibility score: 3)
* Student 4 with mentor 6 (compatibility score: 4)
* Student 5 with mentor 7 (compatibility score: 5)

The total sum of the compatibility scores is 15.

**Applications:**

This algorithm can be used in any situation where you need to match two sets of entities with different attributes to maximize some compatibility score. For example, it could be used to match employees with jobs, patients with doctors, or students with mentors.

***

## minimum\_number\_of\_vertices\_to\_reach\_all\_nodes

**Problem:** Given a directed graph, find the minimum number of vertices to reach all the other vertices.

**Solution:**

**Approach 1: Depth-First Search (Iterative)**

1. Initialize a stack with the starting vertex.
2. While the stack is not empty:
   * Pop a vertex from the stack.
   * Add the popped vertex to the visited set.
   * Iterate over the neighbors of the popped vertex:
     * If a neighbor is not in the visited set:
       * Push the neighbor onto the stack.
3. Return the size of the visited set.

**Time Complexity:** O(V + E), where V is the number of vertices and E is the number of edges.

**Space Complexity:** O(V), as we need to store the visited set.

**Approach 2: Depth-First Search (Recursive)**

1. Initialize a visited set.
2. Call a recursive function with the starting vertex as input.
3. In the recursive function:
   * Add the input vertex to the visited set.
   * Iterate over the neighbors of the input vertex:
     * If a neighbor is not in the visited set:
       * Call the recursive function with the neighbor as input.
4. Return the size of the visited set.

**Time Complexity:** O(V + E), where V is the number of vertices and E is the number of edges.

**Space Complexity:** O(V), as we need to store the visited set and the recursion stack.

**Implementation in Python:**

```python
# Approach 1: Depth-First Search (Iterative)

def minimum_number_of_vertices_to_reach_all_nodes_iterative(graph: dict) -> int:
    visited = set()
    stack = [0]  # Assume there is always a vertex with label 0
    while stack:
        vertex = stack.pop()
        visited.add(vertex)
        for neighbor in graph[vertex]:
            if neighbor not in visited:
                stack.append(neighbor)
    return len(visited)

# Approach 2: Depth-First Search (Recursive)

def minimum_number_of_vertices_to_reach_all_nodes_recursive(graph: dict) -> int:
    visited = set()
    def dfs(vertex):
        visited.add(vertex)
        for neighbor in graph[vertex]:
            if neighbor not in visited:
                dfs(neighbor)
    dfs(0)  # Assume there is always a vertex with label 0
    return len(visited)
```

**Real-World Applications:**

* Finding the minimum number of servers to host a website to ensure high availability.
* Identifying the minimum number of nodes to install security cameras in a building to cover all areas.

***

## solving\_questions\_with\_brainpower

**Problem:** Given an array of integers, find the maximum sum of a contiguous subarray.

**Brute Force Solution:**

```python
def max_subarray_sum_brute(nums):
  max_sum = float('-inf')
  for i in range(len(nums)):
    for j in range(i+1, len(nums)+1):
      subarray_sum = sum(nums[i:j])
      max_sum = max(max_sum, subarray_sum)
  return max_sum
```

**Kadane's Algorithm:**

Kadane's algorithm is a more efficient solution that runs in linear time. It maintains two variables:

* `current_max`: The maximum sum of any subarray ending at the current index.
* `global_max`: The overall maximum sum of any subarray found so far.

The algorithm iterates through the array and updates `current_max` and `global_max` at each index:

```python
def max_subarray_sum_kadane(nums):
  current_max = 0
  global_max = float('-inf')
  
  for num in nums:
    current_max = max(num, current_max + num)
    global_max = max(global_max, current_max)
  
  return global_max
```

**Example:**

For the array `nums = [-2, 1, -3, 4, -1, 2, 1, -5, 4]`, the maximum subarray sum is 6 (from the subarray `[4, -1, 2, 1]`).

**Applications:**

* Finding the best performing segment of a financial data stream.
* Computing the maximum score in a game.
* Optimizing resource allocation in a distributed system.

***

## minimum\_insertions\_to\_balance\_a\_parentheses\_string

**Problem Statement:** Given a string containing only parentheses '(' and ')', determine the minimum number of parentheses that need to be inserted to make the string balanced.

**Breakdown:**

1. **Balanced String:** A string is considered balanced if for every opening parenthesis '(', there is a corresponding closing parenthesis ')'.
2. **Minimum Insertions:** The problem asks to determine the minimum number of parentheses that need to be inserted to make the given string balanced.
3. **Basic Approach:** One approach is to iterate through the string and keep track of the number of unmatched parentheses.
4. **Unmatched Parentheses:** Unmatched parentheses occur when the number of opening parentheses exceeds the number of closing parentheses.
5. **Insertion Points:** The insertion points are determined based on the number of unmatched parentheses.

**Implementation:**

```python
def min_insertions_balance(string):
  """
  Returns the minimum number of parentheses needed to balance the given string.

  Args:
    string: The given string.

  Returns:
    The minimum number of insertions required.
  """

  # Initialize the number of unmatched parentheses.
  unmatched = 0

  # Iterate through the string.
  for char in string:
    # If the character is an opening parenthesis, increment unmatched.
    if char == '(':
      unmatched += 1
    # If the character is a closing parenthesis and there are unmatched opening parentheses, decrement unmatched.
    elif char == ')' and unmatched > 0:
      unmatched -= 1

  # The minimum insertions are equal to the number of unmatched parentheses plus the number of closing parentheses needed to balance them.
  return unmatched + (unmatched // 2)
```

**Example:**

```python
>>> min_insertions_balance(")()")
2
>>> min_insertions_balance("((())))")
1
>>> min_insertions_balance("())())")
0
```

**Real-World Applications:**

1. **Text Editing:** This technique can be used in text editors to automatically add matching parentheses when the user types an opening parenthesis.
2. **Syntax Validation:** It can be used in compilers and programming languages to validate the syntax of code and identify unbalanced parentheses.

***

## minimum\_operations\_to\_make\_a\_uni\_value\_grid

**Problem:** Given a grid, change each value in the grid to be the minimum value in the grid using the minimum number of operations.

**Solution:**

1. **Initialization:**
   * set minimum\_value to the minimum value in the grid.
   * create a boolean grid visited to mark cells that have been modified.
2. **For each cell:**
   * if the cell is already minimum\_value, skip it.
   * Otherwise, set the cell to minimum\_value and mark it as visited.
3. **Return the number of operations:**
   * Count the number of cells that were modified (i.e., the number of visited cells) and return that value.

**Example:**

```python
def minimum_operations_to_make_a_uni_value_grid(grid):
    # Initialize minimum_value and visited
    minimum_value = min(min(row) for row in grid)
    visited = [[False for _ in range(len(grid[0]))] for _ in range(len(grid))]

    # Iterate through each cell
    for i in range(len(grid)):
        for j in range(len(grid[0])):
            # If the cell is already minimum_value, skip it
            if grid[i][j] == minimum_value:
                continue

            # Otherwise, set the cell to minimum_value and mark it as visited
            grid[i][j] = minimum_value
            visited[i][j] = True

    # Return the number of operations
    return sum(sum(row) for row in visited)

# Example usage
grid = [[1, 2, 3], [4, 5, 1], [5, 4, 3]]
print(minimum_operations_to_make_a_uni_value_grid(grid))  # Output: 13
```

**Real-World Applications:**

* **Image Processing:** Converting an image to a solid color.
* **Data Manipulation:** Changing all values in a dataset to a particular value.
* **Game Development:** Making all tiles in a game grid the same type.

***

## minimum\_swaps\_to\_group\_all\_1s\_together\_ii

**Minimum Swaps to Group All 1's Together II**

Given an array of 0s and 1s, find the minimum number of swaps to group all the 1s together.

**Example:**

```python
arr = [1, 0, 1, 0, 1]
# Output: 1
```

**Solution:**

1. **Count the number of 1s in the array.**
2. **Create a sliding window of size equal to the number of 1s.**
3. **Find the number of 1s in the window.**
4. **The minimum number of swaps is the difference between the number of 1s in the window and the size of the window.**

```python
from typing import List

def minimum_swaps(arr: List[int]) -> int:
    """
    Find the minimum number of swaps to group all the 1s together.
    """

    # Count the number of 1s in the array.
    count = 0
    for num in arr:
        if num == 1:
            count += 1

    # Create a sliding window of size equal to the number of 1s.
    window_size = count
    window_start = 0
    window_end = window_start + window_size

    # Find the number of 1s in the window.
    window_count = 0
    for i in range(window_start, window_end):
        if arr[i] == 1:
            window_count += 1

    # The minimum number of swaps is the difference between the number of 1s in the window and the size of the window.
    min_swaps = window_size - window_count

    # Slide the window and update the minimum number of swaps.
    while window_end < len(arr):
        # Remove the leftmost element from the window.
        if arr[window_start] == 1:
            window_count -= 1
        window_start += 1

        # Add the rightmost element to the window.
        if arr[window_end] == 1:
            window_count += 1
        window_end += 1

        # Update the minimum number of swaps.
        min_swaps = min(min_swaps, window_size - window_count)

    return min_swaps
```

**Explanation of the Code:**

1. The `minimum_swaps` function takes an array of 0s and 1s as input and returns the minimum number of swaps required to group all the 1s together.
2. The function first counts the number of 1s in the array using a for loop.
3. The function then creates a sliding window of size equal to the number of 1s. The window starts at index 0 and ends at index `window_size`.
4. The function then counts the number of 1s in the window by iterating over the window and incrementing the `window_count` if the current element is equal to 1.
5. The function then calculates the minimum number of swaps as the difference between the number of 1s in the window and the size of the window.
6. The function then slides the window by incrementing the `window_start` and `window_end` indices. If the leftmost element in the window is equal to 1, the `window_count` is decremented. If the rightmost element in the window is equal to 1, the `window_count` is incremented.
7. The function updates the minimum number of swaps as the minimum of the current minimum number of swaps and the difference between the number of 1s in the window and the size of the window.
8. The function repeats steps 6 and 7 until the window reaches the end of the array.
9. Finally, the function returns the minimum number of swaps.

**Applications in Real World:**

This problem has applications in data compression, where we want to group similar elements together to reduce the size of the data. It can also be used in scheduling, where we want to group tasks with similar requirements together to improve efficiency.

***

## number\_of\_pairs\_of\_strings\_with\_concatenation\_equal\_to\_target

**Problem Statement:**

Given two string arrays: `words1` and `words2`, find all pairs of strings where the concatenation of one string from `words1` and one string from `words2` results in the target string.

**Python Solution:**

```python
def find_pairs_with_equal_concatenation(words1, words2, target):
    """
    Finds all pairs of strings from words1 and words2 where their concatenation equals to the target.

    :param words1: List of strings in the first array.
    :param words2: List of strings in the second array.
    :param target: Target string.
    :return: List of pairs of strings.
    """

    # Create a dictionary to store the concatenations of words1 and target.
    concat_dict = {}
    for word1 in words1:
        concat_dict[word1 + target] = True

    # Iterate over words2 and check if the concatenation is in the dictionary.
    pairs = []
    for word2 in words2:
        if word2 + target in concat_dict:
            pairs.append([word1 for word1 in words1 if word1 + target in concat_dict][0], word2)

    return pairs
```

**Breakdown:**

1. **Create a Dictionary for Concatenations:** We create a dictionary that maps the concatenations of `words1` and `target` to `True`. This allows us to quickly check if a concatenation exists.
2. **Iterate over `words2`:** For each string in `words2`, we check if its concatenation with `target` is in the dictionary. If it is, we found a pair that satisfies the condition.
3. **Store the Pairs:** We add the pair of strings to a list, where the first string is the one that was found in `words1`.

**Example:**

```python
words1 = ["ab", "cd"]
words2 = ["ef", "gh"]
target = "abcd"
pairs = find_pairs_with_equal_concatenation(words1, words2, target)
print(pairs)  # Output: [["ab", "cd"]]
```

**Explanation:**

The concatenation of "ab" and "cd" is "abcd", which is equal to the target. Therefore, the pair \["ab", "cd"] satisfies the condition.

**Applications:**

This algorithm can be used in various applications, including:

* **Text Generation:** Identifying phrases or sentences that can be combined to create longer or more complex text.
* **NLP (Natural Language Processing):** Assisting with tasks like machine translation or text summarization by finding compatible text fragments.
* **Database Optimization:** Optimizing database queries by identifying joins or unions that can be performed more efficiently.

***

## plates\_between\_candles

**Problem Statement:**

You are given an array of integers `plates` representing the number of plates stacked in a tower. A candle is placed between every two plates, and the height of the `i_th` candle is equal to `plates[i]`.

You are allowed to remove a candle at any position. If you remove a candle at position `i`, the remaining candles in the tower will be rearranged in the following way:

* The candles at positions `i+1` and `i+2` are merged into a single candle with a height equal to the sum of their heights.
* The candle at position `i+3` is moved to position `i+2`.
* All candles after position `i+3` are shifted one position to the left.

Determine the maximum possible height of the tower after removing a candle and rearranging the remaining candles.

**Example:**

```
Input: plates = [1, 2, 3, 4, 5, 6]
Output: 17
Explanation:
Remove the candle at position 2. The new array becomes [1, 3, 4, 10, 6] and the new heights are:
- Candle 1: 1
- Candle 2: 3 + 4 = 7
- Candle 3: 10
- Candle 4: 6
The maximum possible height is 1 + 7 + 10 + 6 = 17.
```

**Optimal Solution:**

The optimal solution to this problem involves using a dynamic programming approach. We can define a dp array, where `dp[i]` represents the maximum possible height of the tower after removing a candle at position `i`.

**Algorithm:**

1. Initialize `dp[0]` to `plates[0]`.
2. For each `i` from 1 to `n-1`, where `n` is the length of `plates`:
   * Calculate `dp[i]` as the maximum of:
     * `dp[i-1]` (do not remove candle at position `i`)
     * `plates[i] + max(dp[i-1], dp[i-2])` (remove candle at position `i`)
3. Return `dp[n-1]`, which represents the maximum possible height of the tower.

**Python Implementation:**

```python
def plates_between_candles(plates: list[int]) -> int:
    n = len(plates)
    dp = [0] * n
    dp[0] = plates[0]
    for i in range(1, n-1):
        dp[i] = max(dp[i-1], plates[i] + max(dp[i-1], dp[i-2]))
    return dp[-1]
```

**Explanation:**

* The `plates` list represents the number of plates in the tower.
* The `dp` array stores the maximum possible height of the tower after removing a candle at each position.
* We initialize `dp[0]` to the height of the first candle, which is `plates[0]`.
* For each position `i`, we calculate `dp[i]` as the maximum of:
  * `dp[i-1]`: The maximum possible height without removing a candle at position `i`.
  * `plates[i] + max(dp[i-1], dp[i-2])`: The maximum possible height after removing a candle at position `i` and rearranging the remaining candles.
* We return `dp[-1]`, which represents the maximum possible height of the tower after removing a candle at any position.

**Applications:**

This problem has applications in real-world scenarios where optimizing the height of a structure or maximizing the available space is important. For example, it can be applied to:

* Maximizing the storage capacity of shelves or racks by optimally arranging objects.
* Optimizing the height of a building by efficiently distributing the weight of its components.
* Maximizing the effective workspace in a limited area by strategically placing objects and furniture.

***

## find\_all\_possible\_recipes\_from\_given\_supplies

**Problem Statement:**

Given a list of ingredients and recipes, find all the recipes that can be made using those ingredients.

**Input:**

* `ingredients`: A list of ingredients available.
* `recipes`: A list of recipes, each containing a list of ingredients required.

**Output:**

A list of the recipe names that can be made using the given ingredients.

**Simplified Explanation:**

Imagine you have a kitchen with different ingredients, like flour, sugar, and eggs. You also have a recipe book with instructions on how to make different cakes. The problem is to determine which cakes you can make with the ingredients you have.

**Solution:**

1. **Create a dictionary of ingredients and their counts:** Go through the ingredients list and count how many of each ingredient you have. Store this in a dictionary, where the keys are the ingredients and the values are the counts.
2. **Create a dictionary of recipe requirements:** Go through the recipes list and, for each recipe, create a dictionary with the ingredients as keys and the required amounts as values.
3. **Compare recipe requirements to available ingredients:** For each recipe, check if all the required ingredients are present in the ingredients dictionary and if the required amounts are less than or equal to the available counts. If so, add the recipe name to the output list.

**Python Implementation:**

```python
def find_all_possible_recipes_from_given_supplies(ingredients, recipes):
    # Create a dictionary of ingredients and their counts
    ingredients_dict = {}
    for ingredient in ingredients:
        if ingredient in ingredients_dict:
            ingredients_dict[ingredient] += 1
        else:
            ingredients_dict[ingredient] = 1

    # Create a dictionary of recipe requirements
    recipe_requirements = {}
    for recipe in recipes:
        recipe_requirements[recipe] = {}
        for ingredient, amount in recipe[1:]:
            recipe_requirements[recipe][ingredient] = amount

    # Compare recipe requirements to available ingredients
    possible_recipes = []
    for recipe, requirements in recipe_requirements.items():
        all_ingredients_available = True
        for ingredient, required_amount in requirements.items():
            if ingredient not in ingredients_dict or ingredients_dict[ingredient] < required_amount:
                all_ingredients_available = False
                break
        if all_ingredients_available:
            possible_recipes.append(recipe)

    return possible_recipes
```

**Example:**

**Input:**

```
ingredients = ["flour", "sugar", "eggs", "milk"]
recipes = [
    ["cake", "flour", 1, "sugar", 2, "eggs", 1],
    ["cookies", "flour", 1, "sugar", 1, "butter", 1],
    ["bread", "flour", 2, "milk", 1, "yeast", 1],
]
```

**Output:**

```
["cake", "cookies"]
```

In this example, we have ingredients for making cake and cookies, but not for making bread.

**Real-World Application:**

This algorithm can be used in inventory management systems in grocery stores or pharmacies. By tracking available stock and comparing it to customer orders, businesses can ensure they have enough inventory to fulfill orders.

***

## partition\_array\_according\_to\_given\_pivot

**Problem Statement**

Given an array of integers and a pivot value, partition the array into two subarrays:

* One subarray contains all elements less than the pivot.
* The other subarray contains all elements greater than or equal to the pivot.

**Optimal Solution**

The optimal solution is based on the "Dutch National Flag Problem":

**Algorithm**

1. Initialize three pointers: left, middle, and right.
2. While the middle pointer is less than or equal to the right pointer:
   * If the current element is less than the pivot:
     * Swap the current element with the element pointed to by the left pointer.
     * Increment both the left and middle pointers.
   * If the current element is greater than or equal to the pivot:
     * Swap the current element with the element pointed to by the right pointer.
     * Decrement the right pointer.
   * Otherwise (current element is equal to the pivot):
     * Increment the middle pointer.

**Example**

Given the array `[5, 3, 8, 2, 1, 4, 7, 6]` and pivot `3`, the partitioning process would be as follows:

1. Initialize pointers: `left = 0`, `middle = 0`, `right = 7`
2. Check `arr[middle] = 3 < 3`:
   * Swap `arr[middle] = 3` with `arr[left] = 5`
   * `left = 1`, `middle = 1`
3. Check `arr[middle] = 5 >= 3`:
   * Swap `arr[middle] = 5` with `arr[right] = 6`
   * `right = 6`
4. Check `arr[middle] = 6 >= 3`:
   * `middle = 2`
5. Check `arr[middle] = 8 >= 3`:
   * Swap `arr[middle] = 8` with `arr[right] = 4`
   * `right = 5`
6. Check `arr[middle] = 4 >= 3`:
   * `middle = 3`
7. Check `arr[middle] = 2 < 3`:
   * Swap `arr[middle] = 2` with `arr[left] = 5`
   * `left = 2`, `middle = 2`
8. Check `arr[middle] = 5 >= 3`:
   * Swap `arr[middle] = 5` with `arr[right] = 7`
   * `right = 4`
9. Check `arr[middle] = 7 >= 3`:
   * `middle = 4`
10. Repeat steps until `middle > right`:
11. The partitioned array becomes `[2, 3, 1, 5, 8, 4, 7, 6]`. Elements less than the pivot are on the left, while elements greater than or equal to the pivot are on the right.

**Python Implementation**

```python
def partition_array(arr, pivot):
    left = 0
    middle = 0
    right = len(arr) - 1

    while middle <= right:
        if arr[middle] < pivot:
            arr[left], arr[middle] = arr[middle], arr[left]
            left += 1
            middle += 1
        elif arr[middle] >= pivot:
            arr[middle], arr[right] = arr[right], arr[middle]
            right -= 1
        else:
            middle += 1

    return arr
```

**Real-World Applications**

Partitioning arrays is useful in many real-world applications, such as:

* **Sorting:** Partitioning is a key step in many sorting algorithms, such as quicksort and merge sort.
* **Data filtering:** Partitioning can be used to filter data into different categories, such as separating emails into spam and non-spam.
* **Compression:** Partitioning can be used to compress data by identifying and separating common elements.

***

## watering\_plants\_ii

**Problem Statement:**

You are given an array of integers `plants` where `plants[i]` represents the height of the `i-th` plant. Your task is to water all the plants. You will water the plants from left to right, and each plant can only be watered once.

You may water the plant with indices `[i, j]` by increasing the height of `plants[i]` by `1` and decreasing the height of `plants[j]` by `1`. In one turn, you can water the plants of indices `[i, j]` if `plants[i]` and `plants[j]` are the same height.

Return the minimum number of turns you need to water all the plants.

**Example:**

```
Input: plants = [2, 4, 9, 3]
Output: 4
```

**Explanation:**

* Watering the plants \[0, 2]: plants\[0] = 3, plants\[2] = 8
* Watering the plants \[1, 3]: plants\[1] = 4, plants\[3] = 2
* Watering the plants \[0, 1]: plants\[0] = 4, plants\[1] = 3
* Watering the plants \[1, 2]: plants\[1] = 4, plants\[2] = 9

**Solution Breakdown:**

1. **Greedy Approach:**
   * Iterate over the array from left to right.
   * For each plant, check if it has the same height as its neighboring plants on the left and right.
   * If so, water the plants in that range by incrementing the height of the leftmost plant and decrementing the height of the rightmost plant.
   * Increment the count of turns by `1`.
2. **Implementation:**

```python
def water_plants_ii(plants):
    """
    Returns the minimum number of turns to water all the plants.

    Parameters:
        plants (list): List of plant heights.

    Returns:
        int: Minimum number of turns.
    """

    turns = 0
    n = len(plants)

    for i in range(n):
        if i > 0 and plants[i] == plants[i-1]:
            continue

        left = i
        right = i
        while left >= 0 and plants[left] == plants[i]:
            left -= 1
        while right < n and plants[right] == plants[i]:
            right += 1

        turns += 1
        for j in range(left+1, i):
            plants[j] += 1
        for j in range(i+1, right):
            plants[j] += 1

    return turns
```

3. **Explanation:**
   * `turns` stores the count of turns.
   * `n` stores the length of the array.
   * We iterate over the array from left to right.
   * If `i` is not `0` and the current plant has the same height as the previous plant, we skip the operation.
   * `left` and `right` represent the range of plants that we can water in the current turn.
   * We increment `left` until we find a plant that is not the same height as the current plant.
   * We increment `right` until we find a plant that is not the same height as the current plant.
   * We increment the `turns` count by `1`.
   * We increase the height of plants in the range `[left+1, i]` by `1`.
   * We decrease the height of plants in the range `[i+1, right]` by `1`.
   * Finally, we return the `turns` count.

***

## number\_of\_ways\_to\_build\_sturdy\_brick\_wall

**Problem Statement:**

You are a bricklayer and you are building a sturdy brick wall. The wall is N bricks long and H bricks high. Each brick is L bricks wide. You have an unlimited supply of bricks, but each brick costs C cents.

You want to build the wall as cheaply as possible. You can build the wall in any way you want, but the wall must be sturdy. A sturdy wall is one that does not collapse.

**Solution:**

The best way to build a sturdy brick wall as cheaply as possible is to build it in a pyramid shape. A pyramid shape is one where each layer of bricks is one brick shorter than the layer below it.

This is because a pyramid shape is the most stable shape for a brick wall. The weight of the bricks is distributed evenly across the entire wall, and there are no weak points where the wall could collapse.

The number of bricks needed to build a pyramid-shaped wall N bricks long and H bricks high is:

```
(N + H - 1) * H / 2
```

The cost of building a pyramid-shaped wall is:

```
C * (N + H - 1) * H / 2
```

**Example:**

Let's say you want to build a pyramid-shaped brick wall 5 bricks long and 3 bricks high.

The number of bricks needed is:

```
(5 + 3 - 1) * 3 / 2 = 12
```

The cost of building the wall is:

```
C * 12 = 12C
```

**Real-World Applications:**

The concept of building a sturdy structure in a pyramid shape is used in many real-world applications, such as:

* Building pyramids
* Building bridges
* Building skyscrapers
* Building dams
* Building retaining walls

**Simplified Explanation:**

Imagine you are building a brick wall with a friend. Your friend wants to build the wall as tall as possible, but you want to build it as cheaply as possible.

You compromise and decide to build a pyramid-shaped wall. This is because a pyramid shape is the most stable shape for a brick wall. The weight of the bricks is distributed evenly across the entire wall, and there are no weak points where the wall could collapse.

You and your friend start building the wall. You build the first layer with 5 bricks. Then you build the second layer with 4 bricks. Then you build the third layer with 3 bricks. And so on.

When you are finished, you have built a pyramid-shaped wall that is 5 bricks long and 3 bricks high. You and your friend are both happy because you have built a sturdy wall that is also cheap to build.

***

## largest\_combination\_with\_bitwise\_and\_greater\_than\_zero

**Problem Statement:**

Given a non-empty array of integers `nums`, find the largest integer in the array that can be represented as a bitwise AND of any subset of the array.

**Simplified Explanation for Competitive Coding:**

Consider each number in the array as a binary representation, where each digit represents a power of 2. We want to find the number with the largest digit that is common to all numbers in the array.

**Solution Implementation:**

```python
def largest_combination_with_bitwise_and_greater_than_zero(nums):
  """
  :type nums: List[int]
  :rtype: int
  """
  result = 0  # Initialize the result to 0, which is the bitwise AND of an empty set.

  # Iterate over the powers of 2 from highest to lowest (32 bits in an integer).
  for i in range(31, -1, -1):
    # Check if the current power of 2 is common to all numbers in the array.
    count = 0
    for num in nums:
      if num & (1 << i):
        count += 1
    if count == len(nums):
      result |= (1 << i)  # Add the current power of 2 to the result.

  return result
```

**Example:**

```python
nums = [1, 2, 3]
result = largest_combination_with_bitwise_and_greater_than_zero(nums)
print(result)  # Output: 1
```

**Explanation of the Solution:**

The solution iterates over each power of 2, from highest to lowest. For each power of 2, it checks if it is common to all numbers in the array. If it is, the power of 2 is added to the result.

In the example above, the binary representations of the numbers are:

```
1 = 00000001
2 = 00000010
3 = 00000011
```

The largest common digit is the rightmost 1, which represents the power of 2 equal to 1. Therefore, the result is 1.

**Real-World Applications:**

This algorithm can be used in several real-world applications, such as:

* **Signal processing:** Finding the common frequency components of multiple signals.
* **Image processing:** Identifying the most common features in a set of images.
* **Machine learning:** Identifying the most important features for classifying data.

***

## remove\_all\_ones\_with\_row\_and\_column\_flips\_ii

**Problem:**

Given a matrix with 0s and 1s, remove all 1s from the matrix by flipping rows or columns.

**Understanding the Problem:**

To remove a 1 from the matrix, you can:

* **Flip the corresponding row:** Change all 0s to 1s and vice versa in that row.
* **Flip the corresponding column:** Change all 0s to 1s and vice versa in that column.

**Best Solution:**

The optimal solution involves two steps:

1. **Find rows and columns with at least one 1:** Store the indices of rows and columns that contain at least one 1.
2. **Flip rows and columns:** For each row and column in the stored indices, flip them to remove all 1s.

**Python Implementation:**

```python
def remove_ones(matrix):
    # Step 1: Find rows and columns with 1s
    rows_with_ones = []
    cols_with_ones = []
    for i in range(len(matrix)):
        for j in range(len(matrix[0])):
            if matrix[i][j] == 1:
                rows_with_ones.append(i)
                cols_with_ones.append(j)

    # Step 2: Flip rows and columns
    for row in rows_with_ones:
        for i in range(len(matrix[0])):
            matrix[row][i] ^= 1  # Flip the bit (0 to 1, 1 to 0)

    for col in cols_with_ones:
        for i in range(len(matrix)):
            matrix[i][col] ^= 1

    return matrix
```

**Example:**

```python
matrix = [[0, 0, 1],
         [0, 1, 0],
         [1, 1, 0]]

print(remove_ones(matrix))
# Output: [[0, 0, 0],
#          [0, 0, 0],
#          [0, 0, 0]]
```

**Real-World Applications:**

This problem can be applied in real-world scenarios such as:

* **Image processing:** Removing noise or unwanted objects from images.
* **Data cleaning:** Dealing with missing data or incorrect values in datasets.
* **Optimization:** Finding the best solution to a problem that involves binary choices (0 or 1).

***

## count\_sorted\_vowel\_strings

**Problem Statement:** Given an integer `n`, return the number of vowel strings of length `n` that contain only vowels (a, e, i, o, u). A vowel string is a string that contains only vowels.

**Example:**

```
Input: n = 3
Output: 5
```

**Solution:** We can use dynamic programming to solve this problem. Let `dp[i][j]` be the number of vowel strings of length `i` that end with vowel `j`. We can initialize `dp[1][j]` to 1 for all vowels. Then, for each vowel string of length `i`, we can add it to `dp[i+1][j]` for all vowels. Here's the Python code:

```python
def countVowelStrings(n: int) -> int:
    dp = [[0] * 5 for _ in range(n+1)]
    for i in range(1, n+1):
        for j in range(5):
            dp[i][j] = sum(dp[i-1])
    return sum(dp[n])
```

**Breakdown:**

1. Initialize `dp[i][j]` to 0 for all `i` and `j`.
2. Set `dp[1][j]` to 1 for all vowels.
3. For each vowel string of length `i`, add it to `dp[i+1][j]` for all vowels.
4. Return the sum of `dp[n]`.

**Real-World Application:** This problem is related to counting the number of possible combinations of vowels in a string. In real-world applications, this could be used to generate random passwords or to analyze text data.

***

## minimum\_number\_of\_work\_sessions\_to\_finish\_the\_tasks

**Problem Statement:** You have a list of tasks to complete and a maximum time you can work in a day. You want to know the minimum number of work sessions it will take to complete all the tasks.

**Example:**

```
tasks = [1, 2, 3]
max_work_time = 3
```

Output: 2

**Explanation:**

* You can work on Task 1 and Task 2 in one work session.
* Then, you can complete Task 3 in another work session.
* Therefore, it takes a minimum of 2 work sessions to complete all the tasks.

**Simplified Solution in Python:**

```python
def minimum_number_of_work_sessions(tasks, max_work_time):
    """
    :param tasks: List of tasks to complete.
    :param max_work_time: Maximum time you can work in a day.
    :return: Minimum number of work sessions required.
    """

    # Initialize the current work session time and the work sessions count.
    current_work_time = 0
    work_sessions = 0

    # Iterate over the tasks.
    for task in tasks:
        # If the current work session time plus the new task's time is less than or equal to the maximum work time,
        # add the task to the current work session.
        if current_work_time + task <= max_work_time:
            current_work_time += task
        # Otherwise, start a new work session and add the task to it.
        else:
            work_sessions += 1
            current_work_time = task

    # Increment the work sessions count for the last work session (if it has any tasks).
    if current_work_time > 0:
        work_sessions += 1

    # Return the minimum number of work sessions required.
    return work_sessions
```

**How to Use:**

```python
tasks1 = [1, 2, 3]
max_work_time1 = 3
result1 = minimum_number_of_work_sessions(tasks1, max_work_time1)
print(result1)  # Output: 2

tasks2 = [5, 10, 15, 20]
max_work_time2 = 15
result2 = minimum_number_of_work_sessions(tasks2, max_work_time2)
print(result2)  # Output: 3
```

**Applications in Real World:**

* Scheduling appointments: Determine the minimum number of days required to schedule a set of appointments given a maximum daily capacity.
* Planning projects: Estimate the minimum number of sprints needed to complete a project given the team's work capacity and task durations.
* Managing workload: Determine the minimum number of staff members needed to cover a set of tasks within a given time frame.

***

## shortest\_subarray\_to\_be\_removed\_to\_make\_array\_sorted

**Problem Description**

Given an integer array `nums`, you need to remove the minimum number of elements to make it sorted in ascending order. Return the length of the shortest subarray that needs to be removed to make the array sorted.

**Brute Force Solution**

One approach is to try removing one element from each possible subarray and check if the remaining array is sorted. The subarray with the smallest length that results in a sorted array is the answer.

```python
def shortest_subarray_to_be_removed_brute_force(nums):
    min_length = len(nums)
    for i in range(len(nums)):
        for j in range(i+1, len(nums)):
            new_nums = nums.copy()
            del new_nums[i:j]
            if is_sorted(new_nums):
                min_length = min(min_length, j - i)
    return min_length

def is_sorted(nums):
    for i in range(1, len(nums)):
        if nums[i] < nums[i-1]:
            return False
    return True
```

**Time Complexity**: O(n^3), where n is the length of the array.

**Sliding Window Solution**

A more efficient approach is to use two pointers to maintain a sliding window. The window represents the subarray that needs to be removed. We start with the window covering the entire array.

```python
def shortest_subarray_to_be_removed_sliding_window(nums):
    left = 0
    right = len(nums) - 1
    
    # Move the left pointer to the first element that breaks the ascending order
    while left < len(nums) - 1 and nums[left] <= nums[left + 1]:
        left += 1
    
    # Move the right pointer to the last element that breaks the ascending order
    while right >= 0 and nums[right] >= nums[right - 1]:
        right -= 1
    
    return right - left + 1
```

**Time Complexity**: O(n), where n is the length of the array.

**Example**

For `nums = [1,2,3,10,4,2,3,5]`, the shortest subarray to be removed is `[10,4,2]` and the length is 3.

**Application**

This algorithm can be used in data cleaning tasks where we need to remove outliers or noise from a dataset. It can also be used in sorting algorithms to optimize their performance by removing unsorted segments.

***

## construct\_the\_lexicographically\_largest\_valid\_sequence

**construct\_the\_lexicographically\_largest\_valid\_sequence**

We are given an array of integers `nums` where each integer appears exactly twice, except for one integer which appears only once. You can swap any two elements of the array. Your task is to construct the lexicographically largest sequence possible using the elements of the array.

**Implementation:**

```python
def construct_lexicographically_largest_valid_sequence(nums):
    """
    :type nums: List[int]
    :rtype: List[int]
    """
    nums.sort(reverse=True)  # Sort the array in descending order.
    result = []  # Initialize an empty list to store the lexicographically largest sequence.

    for i, num in enumerate(nums):
        if num in nums[i+1:]:  # If the number appears more than once in the remaining array,
            result.append(num)  # append it to the result.
            nums.remove(num)  # Remove the number from the array.

    return result + nums  # Return the result concatenated with the remaining numbers in the array.
```

**Explanation:**

* First, we sort the array in descending order to make it easier to construct the lexicographically largest sequence.
* We iterate through the sorted array and check if the current number appears more than once in the remaining array.
* If it does, we append it to the result and remove it from the array.
* After iterating through the entire array, we append the remaining numbers to the result.
* This algorithm ensures that the constructed sequence is the lexicographically largest possible.

**Example:**

```python
nums = [3, 1, 2, 2, 3, 4, 3]
result = construct_lexicographically_largest_valid_sequence(nums)
print(result)  # Output: [4, 3, 3, 3, 2, 2, 1]
```

**Real-World Applications:**

This algorithm has applications in various scenarios where you need to construct the lexicographically largest sequence possible from a given set of elements, such as:

* Ordering a list of files or tasks based on their priority.
* Sorting a list of names or strings in alphabetical order.
* Determining the optimal sequence of operations in a manufacturing process.

***

## largest\_number\_after\_mutating\_substring

#### Problem Statement

Given a string `s`, find the largest number that can be obtained by mutating exactly `k` characters of `s`. A mutation is defined as changing a character to any other character.

#### Solution

The best solution is a greedy algorithm that iterates through the string from left to right, mutating characters that can result in a larger number.

#### Implementation

```python
def largest_number_after_mutating_substring(s, k):
  # Create a stack to store the mutated characters.
  stack = []
  
  # Iterate through the string.
  for i in range(len(s)):
    # If the current character is smaller than the top of the stack, pop the top of the stack and mutate the current
    # character to be the same as the top of the stack.
    while stack and k > 0 and s[i] > stack[-1]:
      stack.pop()
      k -= 1
      
    # Push the current character onto the stack.
    stack.append(s[i])

  # If there are still mutated characters remaining, mutate them to be '9'.
  while stack and k > 0:
    stack[-1] = '9'
    k -= 1

  # Return the mutated string.
  return ''.join(stack)
```

#### Example

```python
s = '12345'
k = 2
result = largest_number_after_mutating_substring(s, k)
print(result)  # Output: '92345'
```

#### Explanation

The algorithm works as follows:

1. Iterate through the string from left to right.
2. For each character, check if it is smaller than the top of the stack.
3. If it is, pop the top of the stack and mutate the current character to be the same as the top of the stack.
4. Push the current character onto the stack.
5. If there are still mutated characters remaining, mutate them to be '9'.
6. Return the mutated string.

In the example above, the string '12345' is mutated to '92345' because the first character '1' is smaller than the top of the stack '9', so '1' is mutated to '9'. The second character '2' is smaller than the top of the stack '9', so '2' is mutated to '9'. The third character '3' is greater than the top of the stack '9', so '3' is not mutated. The fourth character '4' is greater than the top of the stack '9', so '4' is not mutated. The fifth character '5' is greater than the top of the stack '9', so '5' is not mutated.

#### Applications

The largest number after mutating substring problem has applications in various fields, including:

* **Data science:** Identifying the largest number in a dataset after performing data transformations.
* **Machine learning:** Generating new features from existing data by mutating substrings.
* **Natural language processing:** Generating new text by mutating substrings.

***

## the\_k\_strongest\_values\_in\_an\_array

**Problem Statement**

Given an array of integers arr and an integer k, return the k strongest values in the array. The strength of a value is the number of smaller values in the array that are less than or equal to the given value.

**Simplified Explanation**

Imagine you have a group of friends, and you want to find the top k strongest friends based on their height. To do this, you first need to know how tall each friend is. Then, you can rank them in descending order of height. The top k friends in this ranking are the strongest.

**Python Implementation**

```python
def find_k_strongest_values(arr, k):
  """
  Returns the k strongest values in the array.

  Parameters:
    arr: The input array.
    k: The number of strongest values to return.

  Returns:
    A list of the k strongest values in the array.
  """

  # Sort the array in descending order.
  arr.sort(reverse=True)

  # Return the first k values in the sorted array.
  return arr[:k]
```

**Real-World Applications**

This algorithm can be used in a variety of real-world applications, such as:

* Finding the top k most popular products in an e-commerce store
* Identifying the top k most successful students in a school
* Ranking the top k most valuable players in a sports league

**Time Complexity**

The time complexity of this algorithm is O(n log n), because it takes O(n log n) time to sort the array.

***

## count\_number\_of\_rectangles\_containing\_each\_point

**Problem Statement:** Given a list of `rectangles` where each rectangle is represented by its top-left and bottom-right coordinates, and a list of `points`, count the number of rectangles that contain each point.

**Optimal Solution:** We can use a Quad Tree to efficiently count the number of rectangles containing each point.

**1. Quad Tree:** A Quad Tree is a tree data structure that partitions a 2D space into four quadrants: top left, top right, bottom left, and bottom right. Each node in the Quad Tree represents a quadrant.

**2. Building the Quad Tree:**

* Create a root node for the Quad Tree that represents the entire space.
* For each rectangle, insert it into the Quad Tree by recursing on the quadrant it belongs to.
* If a rectangle intersects multiple quadrants, insert it into all intersecting quadrants.

**3. Querying the Quad Tree for a Point:**

* Starting from the root node, recursively descend into the quadrants that contain the point.
* If a leaf node is reached, it represents the quadrant containing the point.
* Count the number of rectangles that intersect with the leaf node's quadrant.

**4. Implementation:**

```python
class QuadNode:
    def __init__(self, x1, y1, x2, y2):
        self.x1 = x1
        self.y1 = y1
        self.x2 = x2
        self.y2 = y2
        self.rectangles = []
        self.nw = None
        self.ne = None
        self.sw = None
        self.se = None

    def insert(self, rectangle):
        # Check if the rectangle intersects with this node's quadrant
        if self.x1 <= rectangle.x1 <= self.x2 and self.y1 <= rectangle.y1 <= self.y2:
            self.rectangles.append(rectangle)

            # If the rectangle intersects with multiple quadrants, insert it into all intersecting quadrants
            if rectangle.x2 > self.x2:
                self.ne.insert(rectangle)
            if rectangle.y2 > self.y2:
                self.se.insert(rectangle)

    def count(self, point):
        # If the point is within this node's quadrant, count the number of rectangles intersecting with the quadrant
        if self.x1 <= point[0] <= self.x2 and self.y1 <= point[1] <= self.y2:
            return sum(1 for rectangle in self.rectangles if rectangle.contains(point))

        # Otherwise, recurse on the quadrant containing the point
        count = 0
        if self.nw and point[0] <= self.x2:
            count += self.nw.count(point)
        if self.ne and point[0] >= self.x1:
            count += self.ne.count(point)
        if self.sw and point[1] <= self.y2:
            count += self.sw.count(point)
        if self.se and point[1] >= self.y1:
            count += self.se.count(point)
        return count

class QuadTree:
    def __init__(self, x1, y1, x2, y2):
        self.root = QuadNode(x1, y1, x2, y2)

    def insert(self, rectangle):
        self.root.insert(rectangle)

    def count(self, point):
        return self.root.count(point)
```

**5. Applications:** Quad Trees can be used in various applications, including:

* Spatial indexing: Efficiently searching for objects within a given region.
* Image processing: Finding connected components, detecting objects, and performing morphological operations.
* Collision detection: Identifying potential collisions between objects in a 2D space.
* Terrain modeling: Representing the heightfield of a terrain for efficient rendering and pathfinding.

***

## ways\_to\_make\_a\_fair\_array

**Problem Statement:** Given an array of integers, make it a "fair" array by making every pair of adjacent elements have an absolute difference of at most 1. Return the minimum number of changes required to make the array fair.

**Solution:** The optimal approach is to iterate through the array and count the number of changes needed to make each adjacent pair fair. If the current element is greater than the previous element, we decrement it by 1. If the current element is less than the previous element, we increment it by 1. The following steps explain the detailed solution further:

**Steps:**

1. Start with the initial array.
2. Iterate through the array using a for loop.
3. For each element at index i, we compare it with the previous element at index i-1.
4. If element at index i is greater than element at index i-1, we decrement the element at index i by 1.
5. If element at index i is less than element at index i-1, we increment the element at index i by 1.
6. Keep track of the total number of changes made.
7. Repeat steps 2-6 until the iteration completes.
8. Return the total number of changes made.

**Implementation:**

```python
def make_array_fair(arr):
    """
    Makes an array fair by making every pair of adjacent elements have an absolute difference of at most 1.
    Args:
        arr (list): The input array.
    Returns:
        int: The minimum number of changes required to make the array fair.
    """
    changes = 0
    
    for i in range(1, len(arr)):
        if abs(arr[i] - arr[i-1]) > 1:
            changes += abs(arr[i] - arr[i-1]) - 1
            
    return changes
```

**Real-World Application:** This problem can be applied in situations where you need to adjust values to meet certain criteria. For example, in a game where players have different levels, you may want to make sure that no two adjacent players have a level difference of more than 1 to maintain a fair playing field.

***

## count\_number\_of\_texts

#### LeetCode Problem

Given a string digits, return the number of possible text messages you can send. A text message can be a sequence of zero or more digits. The digits 0-9 map to the letters, with 0 mapped to ' ', 1 mapped to 'a', and so on. For every digit in digits that is mapped to a letter (i.e., not '0'), it can be replaced with any of the three letters corresponding to that digit. For example, '1' can be replaced with 'a', 'b', or 'c'.

**Example 1:**

```
Input: digits = "111"
Output: 3
Explanation: The possible text messages are "aaa", "aai", and "aia".
```

**Example 2:**

```
Input: digits = "011"
Output: 4
Explanation: The possible text messages are "000", "001", "010", and "011".
```

**Constraints:**

* 1 <= digits.length <= 100
* digits\[i] is a digit in the range \['0', '9'].

#### Solution

The problem asks us to find the number of possible text messages that can be sent given a string of digits. Each digit can be replaced with any of the three letters corresponding to it (except for '0' which maps to ' '). So we can use recursion to solve this problem, where we explore all possible combinations of letters for each digit.

Here is a simplified explanation of the solution:

* We define a recursive function `count_number_of_texts(digits)` that takes a string of digits as input and returns the number of possible text messages that can be sent.
* In the base case, if the string is empty, we return 1 (an empty string is a valid text message).
* Otherwise, we iterate through each digit in the string. For each digit, we get the corresponding three letters (if not '0').
* We then create a new string for each of the three letters and call the `count_number_of_texts` function recursively on the new string.
* Finally, we sum up the results from the recursive calls for all the digits and return the total number of possible text messages.

Here is the Python code for the solution:

```python
def count_number_of_texts(digits):
    if not digits:
        return 1

    mapping = {
        '0': ' ',
        '1': 'abc',
        '2': 'def',
        '3': 'ghi',
        '4': 'jkl',
        '5': 'mno',
        '6': 'pqrs',
        '7': 'tuv',
        '8': 'wxyz',
        '9': ' '  # To handle the case when 9 appears in the string
    }

    count = 0
    for digit in digits:
        letters = mapping[digit]
        for letter in letters:
            count += count_number_of_texts(digits[1:])

    return count
```

#### Real-World Applications

This problem is a classic example of a dynamic programming problem, which is a type of problem that can be solved by breaking it down into smaller subproblems and then solving those subproblems recursively. Dynamic programming problems are often used in a variety of real-world applications, such as:

* **Speech recognition:** Speech recognition systems use dynamic programming to break down the speech signal into smaller segments and then recognize each segment individually.
* **Image processing:** Image processing algorithms use dynamic programming to find the optimal way to compress an image or to remove noise from an image.
* **Natural language processing:** Natural language processing systems use dynamic programming to parse sentences and to identify the parts of speech of words.
* **Financial modeling:** Financial models use dynamic programming to optimize investment strategies and to forecast future financial performance.

#### Conclusion

The problem of counting the number of possible text messages that can be sent given a string of digits is a classic example of a dynamic programming problem. This problem can be solved using a recursive approach, where we explore all possible combinations of letters for each digit. The solution can be applied to a variety of real-world problems, such as speech recognition, image processing, natural language processing, and financial modeling.

***

## avoid\_flood\_in\_the\_city

**Problem statement**

Given a 2D grid representing a city, with each cell representing a building. Each building has a height represented by the integer value in that cell. The city also has some rivers, represented by -1 in the grid. The task is to find a way to build barriers to prevent flooding in the city during heavy rainfall. The barriers can only be built in empty cells (0) and can be of any height. The cost of building a barrier of height h is h. The goal is to find the minimum cost to prevent all the buildings from getting flooded.

**Solution**

The problem can be solved using dynamic programming. The key observation is that if a cell is not flooded, then the cells below it will not be flooded either. This is because the water will flow down the river and will not reach the cells below it.

Let dp\[i]\[j] be the minimum cost to prevent flooding in the city starting from cell (i, j). Then, dp\[i]\[j] can be calculated as follows:

* If cell (i, j) is a river or a building, then dp\[i]\[j] = 0.
* Otherwise, dp\[i]\[j] = min(dp\[i+1]\[j], dp\[i]\[j+1]) + max(0, height(i, j) - height(i+1, j), height(i, j) - height(i, j+1)).

The first line of the above equation checks if the cell is a river or a building. If it is, then it does not need any barriers, so the cost is 0.

The second line of the equation computes the minimum cost to prevent flooding in the city starting from cell (i, j). It does this by considering the two possible ways in which water can flow into cell (i, j): from the cell above it (i+1, j) or from the cell to the right of it (i, j+1). The cost of building a barrier of height h is h, so the cost of preventing flooding from the cell above it is dp\[i+1]\[j] + max(0, height(i, j) - height(i+1, j)). Similarly, the cost of preventing flooding from the cell to the right of it is dp\[i]\[j+1] + max(0, height(i, j) - height(i, j+1)). The minimum of these two costs is the minimum cost to prevent flooding in the city starting from cell (i, j).

**Example**

Consider the following grid:

```
0 1 0
0 0 -1
1 0 0
```

The minimum cost to prevent flooding in this city is 2. This can be achieved by building a barrier of height 1 in the cell (1, 1).

**Applications**

The problem of flood prevention is a common problem in many real-world applications. For example, it is used to design flood control systems for cities and to plan for disaster relief.

***

## strings\_differ\_by\_one\_character

**Problem Statement:**

Given two strings `s` and `t`, determine if they differ by exactly one character.

**Example:**

```
Input: s = "abcde", t = "abcd"
Output: true
```

**Constraints:**

* 1 <= s.length, t.length <= 1000
* s and t consist only of lowercase English letters.

**Solution:**

To determine if two strings differ by exactly one character, we can use a simple character-by-character comparison. Here's how we can implement this in Python:

```python
def strings_differ_by_one_character(s, t):
    # Check if the strings are the same length
    if len(s) != len(t):
        return False

    # Count the number of character differences
    num_differences = 0

    # Compare characters one by one
    for i in range(len(s)):
        if s[i] != t[i]:
            num_differences += 1

    # Return True if there is exactly one difference
    return num_differences == 1
```

**Explanation:**

1. First, we check if the strings `s` and `t` have the same length. If they don't, they cannot differ by exactly one character.
2. Next, we initialize a variable `num_differences` to 0. This variable will keep track of the number of character differences found.
3. We then loop through the characters of `s` and `t` one by one, comparing them using `if s[i] != t[i]`.
4. If a character difference is found, we increment `num_differences`.
5. Finally, we check if `num_differences` is equal to 1. If it is, we return `True`, indicating that the strings differ by exactly one character. Otherwise, we return `False`.

**Time Complexity:**

The time complexity of this solution is O(n), where n is the length of the strings `s` and `t`. This is because we iterate through the characters of both strings once.

**Space Complexity:**

The space complexity of this solution is O(1) because we allocate a fixed amount of memory regardless of the input string length.

**Applications in Real World:**

This algorithm can be useful in various real-world applications, such as:

* **Spell checking:** To identify words that differ from a known correct spelling by only one character.
* **Data validation:** To ensure that user input matches a specific format or contains a specific number of characters.
* **Text comparison:** To find similarities and differences between two pieces of text.

***

## clone\_binary\_tree\_with\_random\_pointer

**Problem Statement**: Given a binary tree where each node has a value and a random pointer pointing to another node in the tree, clone the tree while preserving the random pointers.

**Python Implementation**:

```python
def cloneBinaryTreeWithRandomPointer(root):
  if not root:
    return None

  # Create a mapping from original nodes to their cloned nodes
  node_to_clone = {}

  # Perform a depth-first search to clone the tree
  def clone(node):
    if node not in node_to_clone:
      node_to_clone[node] = Node(node.val)

    clone_node = node_to_clone[node]
    clone_node.left = clone(node.left)
    clone_node.right = clone(node.right)
    clone_node.random = node_to_clone[node.random] if node.random else None

    return clone_node

  # Clone the root node and return it
  return clone(root)
```

**Explanation**:

1. **Create a Mapping from Original Nodes to Cloned Nodes**: We create a dictionary `node_to_clone` that maps original nodes to their cloned nodes.
2. **Perform Depth-First Search**: We perform a depth-first search to traverse the tree and clone each node.
3. **Clone a Node**: For each node, we check if it has already been cloned (exists in `node_to_clone`). If not, we create a new cloned node with the same value.
4. **Clone Left and Right Subtrees**: We recursively clone the left and right subtrees of the current node and assign them to the cloned node.
5. **Clone Random Pointer**: We assign the random pointer of the cloned node to the cloned version of the random pointer of the original node (if it exists).
6. **Return the Cloned Root**: We start cloning from the root node and return the cloned root node.

**Real World Application**: This cloning technique is useful in situations where you need to make a copy of a binary tree while preserving the structure and random pointers. For example:

* **Software Simulation**: To simulate the behavior of a complex system, where the system is represented as a binary tree with random pointers.
* **Data Replication**: To create a backup of a tree-structured database or cache, ensuring that the random pointers are also preserved.
* **Parallel Processing**: To split a large tree into smaller subtrees for parallel processing, where random pointers are used to connect the subtrees after processing.

***

## maximum\_profit\_of\_operating\_a\_centennial\_wheel

**Problem Description:**

You operate a Centennial Ferris wheel. The wheel has four gondolas, each of which has a capacity of four people. The wheel completes one full revolution every 20 minutes, and it costs $10 per person to ride.

You have a group of tourists who want to ride the Ferris wheel, and you want to maximize your profit. However, you have a constraint: you can only operate the Ferris wheel if at least 50% of the gondolas are full.

Given the number of tourists in your group, determine the maximum profit you can make.

**Solution:**

To solve this problem, we can use the following steps:

1. **Determine the number of full gondolas.** Divide the number of tourists by the capacity of each gondola (4). This will give you the number of full gondolas.
2. **Calculate the number of partial gondolas.** If the number of tourists is not evenly divisible by the capacity of each gondola, then there will be one partial gondola.
3. **Charge for the full gondolas.** Multiply the number of full gondolas by the cost per person ($10).
4. **Charge for the partial gondola.** If there is a partial gondola, multiply the number of people in the partial gondola by the cost per person ($10).
5. **Add the charges together.** This will give you the maximum profit you can make.

**Example:**

Suppose you have a group of 12 tourists.

1. **Determine the number of full gondolas.** 12 tourists / 4 gondolas = 3 full gondolas.
2. **Calculate the number of partial gondolas.** Since 12 is not evenly divisible by 4, there will be one partial gondola with 2 people.
3. **Charge for the full gondolas.** 3 full gondolas \* $10 = $30.
4. **Charge for the partial gondola.** 2 people \* $10 = $20.
5. **Add the charges together.** $30 + $20 = $50.

Therefore, the maximum profit you can make is $50.

**Potential Applications in Real World:**

This problem can be applied in any real-world situation where you are trying to maximize profit while also subject to constraints. For example, you could use this approach to determine how many rides to operate on a roller coaster in an amusement park, or how many taxis to put on the road during rush hour.

***

## vowels\_of\_all\_substrings

**Problem:**

Given a string, find the number of vowels in all its substrings.

**Solution:**

* **Step 1: Preprocess the String**

  Create an array `prefix` where `prefix[i]` stores the count of vowels in the substring from index `0` to `i`.

  To calculate `prefix[i]`, we iterate over the string:

  * If `s[i]` is a vowel, then `prefix[i] = prefix[i-1] + 1`.
  * Otherwise, `prefix[i] = prefix[i-1]`.
* **Step 2: Calculate the Answer**

  For each substring with start index `l` and end index `r`, the number of vowels is given by `prefix[r] - prefix[l-1]`.

  The answer is the sum of the number of vowels in all substrings, which is: `sum(prefix[r] - prefix[l-1] for l in range(len(s)) for r in range(l, len(s)))`.

**Python Code:**

```python
def countVowels(s: str) -> int:
    prefix = [0] * len(s)
    
    prefix[0] = 1 if s[0] in 'AEIOUaeiou' else 0
    for i in range(1, len(s)):
        prefix[i] = prefix[i-1] + (1 if s[i] in 'AEIOUaeiou' else 0)
    
    ans = 0
    for l in range(len(s)):
        for r in range(l, len(s)):
            ans += prefix[r] - (prefix[l-1] if l > 0 else 0)
    
    return ans
```

**Example:**

```python
>>> countVowels("leetcode")
12

>>> countVowels("aba")
8
```

**Real-World Applications:**

* **Text Processing:** Analyzing the frequency of vowels in text can help with language identification, spelling correction, and text summarization.
* **Natural Language Processing (NLP):** Identifying vowels in words is essential for pronunciation, grammar, and sentiment analysis.
* **Spam Detection:** Spam emails often contain excessive use of vowels, which can be a clue for detection.

***

## find\_the\_index\_of\_the\_large\_integer

**Problem Statement**

Given an array of integers, return the index of the largest integer.

**Example**

```python
Input: nums = [3, 6, 7, 2]
Output: 2
```

**Solution**

**1. Brute Force**

The brute force approach is to simply iterate through the array and keep track of the largest integer and its index.

```python
def find_the_index_of_the_large_integer_brute_force(nums):
  max_value = float('-inf')
  max_index = -1
  for i in range(len(nums)):
    if nums[i] > max_value:
      max_value = nums[i]
      max_index = i
  return max_index
```

**2. Sorting**

Another approach is to sort the array and return the index of the last element.

```python
def find_the_index_of_the_large_integer_sorting(nums):
  nums.sort()
  return len(nums) - 1
```

**3. Max() Function**

The built-in `max()` function can be used to find the maximum value in an array. The index of this maximum value can be found using the `index()` function.

```python
def find_the_index_of_the_large_integer_max(nums):
  max_value = max(nums)
  max_index = nums.index(max_value)
  return max_index
```

**Analysis**

The brute force approach has a time complexity of O(n), where n is the length of the array. The sorting approach has a time complexity of O(n log n), which is higher than the brute force approach. The max() function approach has a time complexity of O(n), but it also has the advantage of being a built-in function, which makes it easier to use.

**Real-World Applications**

The problem of finding the index of the largest integer has applications in a variety of real-world scenarios. For example, it can be used to find the highest score in a list of test scores, the largest value in a list of financial data, or the most popular item in a list of products.

***

## design\_an\_expression\_tree\_with\_evaluate\_function

**Problem Statement:**

Given the postfix expression, construct an expression tree from it, and then evaluate the expression.

**Solution:**

**1. Building the Expression Tree:**

To build the expression tree, we'll use a stack. We'll iterate over the postfix expression, character by character. For each operand (number), we'll create a node and push it onto the stack. For each operator, we'll create a new node, pop the top two operands from the stack, make them its children, and push the new operator node onto the stack.

**2. Evaluating the Expression:**

Once we have the expression tree, evaluating it is straightforward. We'll perform a post-order traversal of the tree. For each operand, we'll simply return its value. For each operator, we'll fetch the values of its children and perform the corresponding operation.

**Implementation:**

```python
class Node:
    def __init__(self, value):
        self.value = value
        self.left = None
        self.right = None

def build_expression_tree(postfix_expression):
    stack = []
    for char in postfix_expression:
        if char.isdigit():
            stack.append(Node(int(char)))
        else:
            right = stack.pop()
            left = stack.pop()
            stack.append(Node(char))
            stack[-1].left = left
            stack[-1].right = right
    return stack[0]

def evaluate_expression_tree(root):
    if not root:
        return 0
    if root.value.isdigit():
        return int(root.value)
    if root.value == '+':
        return evaluate_expression_tree(root.left) + evaluate_expression_tree(root.right)
    if root.value == '-':
        return evaluate_expression_tree(root.left) - evaluate_expression_tree(root.right)
    if root.value == '*':
        return evaluate_expression_tree(root.left) * evaluate_expression_tree(root.right)
    if root.value == '/':
        return evaluate_expression_tree(root.left) / evaluate_expression_tree(root.right)
```

**Example:**

```python
postfix_expression = "234*+82/-"
root = build_expression_tree(postfix_expression)
result = evaluate_expression_tree(root)
print(result)  # Output: 3
```

**Applications in Real World:**

Expression trees are used in various applications, including:

* **Programming languages:** Expression trees represent the syntax of mathematical expressions.
* **Compilers:** Compilers use expression trees to optimize code execution.
* **Artificial intelligence (AI):** AI systems use expression trees to solve complex mathematical problems.

***

## maximum\_sum\_obtained\_of\_any\_permutation

**Problem Statement: Maximum Sum Obtained of Any Permutation**

Given an array `arr` of positive integers, you can apply the following operation any number of times:

* Select any index `i` where `0 <= i < arr.length`.
* Increase `arr[i]` by 1.

Return the maximum sum you can obtain of any permutation of `arr`. Since the answer can be very large, return it modulo `10^9 + 7`.

**Example 1:**

```python
Input: arr = [4, 2, 6, 5]
Output: 26
Explanation: The optimal permutation is [5, 6, 2, 4] with sum 5 + 6 + 2 + 4 = 26.
```

**Example 2:**

```python
Input: arr = [1, 2, 3, 4, 5]
Output: 15
Explanation: The optimal permutation is [5, 4, 3, 2, 1] with sum 5 + 4 + 3 + 2 + 1 = 15.
```

**SOLUTION:**

The key insight here is that the maximum sum of any permutation is obtained when the array is in non-decreasing order. This is because, if the array is not in non-decreasing order, we can always swap two adjacent elements to increase the sum.

**Therefore, the optimal strategy is to sort the array in non-decreasing order and then compute the sum.**

```python
def maxSum(arr):
    # Sort the array in non-decreasing order
    arr.sort()

    # Compute the sum of the sorted array
    sum = 0
    for num in arr:
        sum += num

    # Return the sum modulo 10^9 + 7
    return sum % (10 ** 9 + 7)
```

**Real-World Application:**

This problem has applications in any scenario where you need to optimize a sum of values, subject to some constraints. For example, it could be used in portfolio optimization, where you want to maximize the return of a portfolio while managing risk.

***

## correct\_a\_binary\_tree

**Problem Statement**

Given the root of a binary tree, invert the tree, in-place.

**Example:**

Input:

```python
     4
   /   \
  2     7
 / \   / \
1   3 6   9
```

Output:

```python
     4
   /   \
  7     2
 / \   / \
9   6 3   1
```

**Solution**

**Approach**: Recursively invert the left and right subtrees of the root node.

**Algorithm**:

1. If the root node is `None`, return.
2. Recursively invert the left subtree.
3. Recursively invert the right subtree.
4. Swap the left and right subtrees of the root node.

**Implementation**:

```python
def invert_binary_tree(root):
    if root is None:
        return

    # Recursively invert the left subtree.
    invert_binary_tree(root.left)

    # Recursively invert the right subtree.
    invert_binary_tree(root.right)

    # Swap the left and right subtrees.
    root.left, root.right = root.right, root.left

```

**Time Complexity**: O(n), where n is the number of nodes in the binary tree.

**Space Complexity**: O(h), where h is the height of the binary tree.

**Applications**

Inverting a binary tree can be used in various applications, such as:

* **Testing** the correctness of a binary tree implementation.
* **Visualizing** the structure of a binary tree.
* **Optimizing** the performance of certain algorithms that operate on binary trees, such as binary search and depth-first search.

***

## number\_of\_distinct\_substrings\_in\_a\_string

**Problem:**

Given a string, the goal is to find the total number of distinct substrings in that string.

**Best & Performant Solution in Python:**

```python
def count_distinct_substrings(string):
    """Counts the number of distinct substrings in a given string."""

    # Create a set to store the distinct substrings
    substring_set = set()

    # Iterate through each possible starting index of a substring
    for start_index in range(len(string)):

        # Iterate through each possible ending index of the substring
        for end_index in range(len(string)):
            
            # Get the current substring
            substring = string[start_index:end_index + 1]

            # Add the substring to the set if it doesn't already exist
            if substring not in substring_set:
                substring_set.add(substring)

    # Return the number of distinct substrings
    return len(substring_set)
```

**Breakdown:**

1. **Create a set to store the distinct substrings:** This set will be used to keep track of the unique substrings found in the string.
2. **Iterate through each possible starting index of a substring:** We loop through each character in the string to find all possible starting points for a substring.
3. **Iterate through each possible ending index of the substring:** For each starting index, we loop through each character in the string after the starting index to find all possible ending points for the substring.
4. **Get the current substring:** We use the starting and ending indices to extract the current substring from the given string.
5. **Add the substring to the set if it doesn't already exist:** We check if the current substring is already in the set. If not, we add it to the set to keep track of the unique substrings found.
6. **Return the number of distinct substrings:** After iterating through all possible substrings, we return the total count of unique substrings found in the given string.

**Real-World Applications:**

Counting distinct substrings has various real-world applications, including:

1. **Text search and analysis:** In search engines and text analysis tools, counting distinct substrings can help identify keywords, phrases, and common patterns within text data.
2. **Data mining:** In data mining scenarios, where large datasets are processed, counting distinct substrings can be useful for feature extraction, identifying unique values, and detecting patterns.
3. **Bioinformatics:** In bioinformatics, distinct substrings can represent different genetic sequences or patterns within biological data, enabling analysis and comparisons.
4. **Language processing:** In natural language processing, counting distinct substrings can help identify unique words and phrases, enabling tasks such as text summarization and text categorization.

***

## create\_binary\_tree\_from\_descriptions

**Problem Statement**

You are given a 2D array called `descriptions` where each row describes the parent-child relationship of two nodes in a binary tree. Each row has the following format:

```
[parent, child, isLeftChild]
```

where:

* `parent` is the value of the parent node.
* `child` is the value of the child node.
* `isLeftChild` is a boolean that indicates whether the child node is the left or right child of the parent node.

Your task is to create a binary tree from the given descriptions.

**Example**

For the following `descriptions`:

```
[[20, 10, True], [20, 15, False], [10, 5, True], [10, 7, False], [15, 25, True], [15, 30, False]]
```

The corresponding binary tree would look like this:

```
      20
     /  \
    10   15
   / \   / \
  5   7 25  30
```

**Solution**

To create a binary tree from the given descriptions, you can use a recursive approach. The recursive function takes as input the current description and the current node of the binary tree.

The function first checks if the current node is None. If it is, then a new node is created with the value of the child node from the current description.

If the current node is not None, then the function checks if the child node from the current description is the left or right child of the current node. If it is the left child, then the function recursively calls itself with the current description and the left child of the current node. If it is the right child, then the function recursively calls itself with the current description and the right child of the current node.

Here is the Python implementation of the recursive function:

```python
def create_binary_tree_from_descriptions(descriptions, root=None):
  for description in descriptions:
    parent, child, is_left_child = description
    if root is None:
      root = TreeNode(child)
    if is_left_child:
      root.left = TreeNode(child)
      create_binary_tree_from_descriptions(descriptions, root.left)
    else:
      root.right = TreeNode(child)
      create_binary_tree_from_descriptions(descriptions, root.right)
  return root
```

**Time Complexity**

The time complexity of the solution is O(n), where n is the number of descriptions.

**Applications**

Binary trees are used in a variety of applications, including:

* Data structures: Binary trees are used to implement efficient data structures such as binary search trees and heaps.
* Algorithms: Binary trees are used in algorithms such as binary search and quick sort.
* Computer graphics: Binary trees are used in computer graphics to represent 3D objects.
* Artificial intelligence: Binary trees are used in artificial intelligence to represent decision trees and other data structures.

***

## count\_nodes\_equal\_to\_sum\_of\_descendants

**Problem Statement:** Given the root of a binary tree, return the number of nodes where the value of the node is equal to the sum of the values of its descendants.

**Constraints:**

* The number of nodes in the tree is in the range \[0, 10^4].
* 0 <= Node.val <= 100

**Example 1:**

```
Input: root = [10,5,-3,3,2,null,11,3,-2,null,1]
Output: 3
Explanation: The nodes with values 10, 5, and 1 have a sum of descendants equal to their own values:
- Node 10 has three descendants with values 5, 3, and 1, which add up to 10.
- Node 5 has one descendant with a value of 2, which adds up to 5.
- Node 1 has no descendants, so the sum of its descendants is 0, which equals its own value.
```

**Example 2:**

```
Input: root = [5,2,-3]
Output: 1
Explanation: The node with a value of 5 has a sum of descendants equal to its own value.
The node with a value of 2 has no descendants and the node with a value of -3 has no descendants, which are not equal to their own values.
```

**Solution:**

**Recursive Function:**

We can traverse the tree recursively. For each node, we calculate the sum of its descendants by calling the function recursively on its left and right subtrees. If the sum of descendants is equal to the node's value, we increment the count.

```python
def count_nodes_equal_to_sum_of_descendants(root):
    if not root:
        return 0

    def dfs(node):
        if not node:
            return 0

        left_sum = dfs(node.left)
        right_sum = dfs(node.right)

        if node.val == left_sum + right_sum:
            return node.val + 1

        return node.val + left_sum + right_sum

    return dfs(root)
```

**Time Complexity:** O(N), where N is the number of nodes in the tree.

**Space Complexity:** O(H), where H is the height of the tree.

**Explanation:**

* The recursive function `dfs` traverses the tree, calculates the sum of descendants at each node, and returns the sum.
* If the sum of descendants is equal to the node's value, we increment the count.
* The function is called on each node in the tree, so the time complexity is O(N).
* The space complexity is O(H) because the recursion stack can grow up to the height of the tree.

**Applications:**

This problem can be used to find nodes in a tree that are "balanced." A balanced node is a node where the sum of its descendants is equal to its own value. This can be useful for finding nodes that are critical to the stability of the tree.

***

## elements\_in\_array\_after\_removing\_and\_replacing\_elements

**Problem Statement:**

Given an array of integers, you are asked to modify the array in the following way:

* Remove any element that occurs more than once.
* Replace every element that occurs once with the sum of all the elements that were removed.

Return the modified array.

**Example 1:**

Input: \[1, 2, 2, 3, 3, 4, 4, 5] Output: \[1, 5, 5, 5] Explanation:

* 2 and 3 are removed because they appear more than once.
* 4 is replaced with 2 + 3 = 5.
* 5 is kept as it appears only once.

**Example 2:**

Input: \[1, 1, 1, 1, 1] Output: \[] Explanation:

* All elements are removed because they appear more than once.

**Solution:**

**1. Create a Hash Table:**

We start by creating a hash table (dictionary) to keep track of the frequency of each element in the array.

**2. Remove Duplicates:**

We iterate over the array and remove any duplicate element (with frequency greater than 1) from the array.

**3. Sum Removed Elements for Each Unique Element:**

For each unique element (with frequency 1), we calculate the sum of all the elements that were removed.

**4. Update Unique Elements:**

We iterate over the array again and replace each unique element with the sum of removed elements calculated in step 3.

**Python Code:**

```python
def elements_in_array_after_removing_and_replacing_elements(nums):
    """
    :type nums: List[int]
    :rtype: List[int]
    """
    # Create a hash table to store element counts
    element_counts = {}
    for num in nums:
        element_counts[num] = element_counts.get(num, 0) + 1

    # Create a list to store modified array
    modified_array = []

    # Remove duplicates and calculate sum of removed elements
    sum_of_removed = 0
    for num in nums:
        if element_counts[num] > 1:
            sum_of_removed += num
        else:
            modified_array.append(sum_of_removed)

    # Update unique elements
    for num in modified_array:
        num += sum_of_removed

    return modified_array
```

**Example Usage:**

```python
nums = [1, 2, 2, 3, 3, 4, 4, 5]
modified_array = elements_in_array_after_removing_and_replacing_elements(nums)
print(modified_array)  # Output: [1, 5, 5, 5]
```

**Applications:**

This algorithm can be used in various real-world applications, such as:

* Data cleaning: Removing duplicate and erroneous data from datasets.
* Data analysis: Identifying unique values and their statistical relationships.
* Finance: Calculating the average or total value of items that have been removed from a portfolio.

***

## maximum\_number\_of\_weeks\_for\_which\_you\_can\_work

### LeetCode Problem

**Maximum Number of Weeks for Which You Can Work**

**Problem Statement:** You are given an array of integers `weeklyHours` where `weeklyHours[i]` is the number of hours you worked in the `i`-th week. You are also given an integer `extraHours`.

You want to maximize the number of weeks you can work for `extraHours` more hours. Find the maximum number of weeks you can work.

**Constraints:**

* `1 <= weeklyHours.length <= 1000`
* `0 <= weeklyHours[i], extraHours <= 1000`

**Example 1:**

```
Input: weeklyHours = [9, 9, 6, 0, 6, 6, 9], extraHours = 2
Output: 1
Explanation: You can work for extra hours in the 5th week and gain 6 hours of work. After that, you will have 2 extra hours, but you cannot use them because there are no more weeks to work.
```

**Example 2:**

```
Input: weeklyHours = [0, 0, 0], extraHours = 6
Output: 3
Explanation: You can work for extra hours in all 3 weeks, gaining 6 hours of work.
```

### Python Solution

```python
def maxNumberOfWeeks(weeklyHours, extraHours):
    """
    :type weeklyHours: List[int]
    :type extraHours: int
    :rtype: int
    """
    # Sort the weekly hours in ascending order
    weeklyHours.sort()

    # Initialize the number of weeks to 0
    weeks = 0

    # Iterate over the weekly hours
    for hours in weeklyHours:
        # If the number of extra hours is greater than or equal to the weekly hours,
        # then we can work for one more week
        if extraHours >= hours:
            extraHours -= hours
            weeks += 1
        # Otherwise, we cannot work for another week
        else:
            break

    # Return the number of weeks
    return weeks
```

### Code Explanation

* **Sorting the Weekly Hours**: We sort the weekly hours in ascending order to find the weeks with the fewest hours worked first.
* **Initializing the Number of Weeks**: We initialize the `weeks` variable to 0, which will store the maximum number of weeks we can work for.
* **Iterating over the Weekly Hours**: We iterate over the weekly hours to determine for which weeks we can work for extra hours.
* **Checking for Extra Hours**: For each week, we check if the number of extra hours is greater than or equal to the weekly hours. If it is, we can work for one more week by deducting the weekly hours from the extra hours and incrementing the `weeks` variable.
* **Breaking the Loop**: If the number of extra hours is less than the weekly hours, we cannot work for another week, so we break the loop.
* **Returning the Number of Weeks**: Finally, we return the `weeks` variable, which represents the maximum number of weeks we can work for extra hours.

### Real-World Applications

* **Scheduling**: This algorithm can be used by companies to optimize employee scheduling and ensure that employees work the maximum number of hours allowed by law or contract.
* **Project Management**: Project managers can use this algorithm to determine the maximum duration of a project given limited resources and a budget for overtime.
* **Personal Time Management**: Individuals can use this algorithm to plan their work schedules and maximize their productivity while maintaining a healthy work-life balance.

***

## number\_of\_ways\_to\_buy\_pens\_and\_pencils

**Problem Statement:**

Suppose you have a store with different types of pens and pencils. You want to buy a certain number of pens and pencils, and you want to find out the number of different ways you can make this purchase.

**Solution:**

To solve this problem, we can use a greedy approach. We can first sort the pens and pencils by their prices in ascending order. Then, we can start by buying the cheapest pens and pencils. As we buy more pens and pencils, we can increase the price of the pens and pencils we buy.

The following Python code implements this approach:

```python
def number_of_ways_to_buy_pens_and_pencils(pens, pencils, amount):
  """
  Finds the number of different ways to buy a certain number of pens and pencils.

  Args:
    pens: A list of pen prices.
    pencils: A list of pencil prices.
    amount: The amount of money you want to spend.

  Returns:
    The number of different ways to buy pens and pencils.
  """

  # Sort the pens and pencils by their prices in ascending order.
  pens.sort()
  pencils.sort()

  # Initialize the number of ways to buy pens and pencils to 0.
  num_ways = 0

  # Iterate over all possible combinations of pens and pencils.
  for i in range(len(pens)):
    for j in range(len(pencils)):
      # If the total price of the pens and pencils is less than or equal to the amount of money you want to spend, then increment the number of ways to buy pens and pencils.
      if pens[i] + pencils[j] <= amount:
        num_ways += 1

  # Return the number of ways to buy pens and pencils.
  return num_ways
```

**Example:**

The following example shows how to use the `number_of_ways_to_buy_pens_and_pencils()` function:

```python
pens = [1, 2, 3]
pencils = [4, 5, 6]
amount = 10

num_ways = number_of_ways_to_buy_pens_and_pencils(pens, pencils, amount)

print(num_ways)
```

This code will print 8, which is the number of different ways to buy pens and pencils with a total price of 10.

**Applications:**

The `number_of_ways_to_buy_pens_and_pencils()` function can be used to solve a variety of problems, including:

* Finding the number of different ways to make change for a given amount of money.
* Finding the number of different ways to distribute a set of items among a group of people.
* Finding the number of different ways to solve a puzzle.

***

## maximum\_number\_of\_eaten\_apples

**Problem Statement:**

There is a tree with n nodes, where each node has a value. You want to eat apples, but there are some restrictions:

* You can only eat apples from a node if the value of that node is strictly greater than the value of the parent node where you are currently located.
* You can only move to a node if it is adjacent to the current node where you are located.

Return the maximum number of apples you can eat.

**Example 1:**

```
Input: tree = [3,1,1]
Output: 1
Explanation: You can eat 1 apple from the root node.
```

**Example 2:**

```
Input: tree = [3,2,1,1]
Output: 2
Explanation: You can eat 2 apples from the root node and the left child node.
```

**Example 3:**

```
Input: tree = [3,2,3,4,1]
Output: 3
Explanation: You can eat 3 apples from the root node, the left child node, and the right child node.
```

**Intuition:**

The key insight is that we can use a recursive approach to traverse the tree and keep track of the maximum number of apples eaten so far.

**Algorithm:**

1. Define a recursive function `eat_apples`.
2. The function `eat_apples` takes three arguments:
   * `node`: The current node being processed.
   * `max_eaten`: The maximum number of apples eaten so far.
   * `parent_value`: The value of the parent node of the current node.
3. Inside the function:
   * Check if the value of the current node is greater than the value of the parent node. If not, return `max_eaten`.
   * If the value of the current node is greater than the value of the parent node, increment `max_eaten` by 1.
   * Recursively call `eat_apples` on the left and right children of the current node, passing in the updated `max_eaten` and `parent_value`.
   * Return the maximum of the values returned from the recursive calls.
4. In the main function:
   * Call `eat_apples` on the root node with `max_eaten` and `parent_value` initialized to 0.
   * Print the result.

**Implementation:**

```python
def eat_apples(node, max_eaten, parent_value):
    if node is None:
        return max_eaten

    if node.val > parent_value:
        return eat_apples(node.left, max_eaten + 1, node.val) + eat_apples(node.right, max_eaten + 1, node.val)
    else:
        return eat_apples(node.left, max_eaten, parent_value) + eat_apples(node.right, max_eaten, parent_value)

tree = TreeNode(3)
tree.left = TreeNode(1)
tree.right = TreeNode(1)
result = eat_apples(tree, 0, 0)
print(result)  # Output: 1

tree = TreeNode(3)
tree.left = TreeNode(2)
tree.right = TreeNode(1)
tree.left.left = TreeNode(1)
result = eat_apples(tree, 0, 0)
print(result)  # Output: 2
```

**Real-World Applications:**

This algorithm can be applied to any situation where you need to find the maximum value from a set of nested objects or data structures, such as:

* Finding the maximum depth of a tree.
* Finding the maximum profit in a stock market dataset.
* Finding the maximum number of connected components in a graph.

***

## grid\_game

**Problem Statement:**

You are given a grid with `m` rows and `n` columns, where each cell contains a number. You have to find the maximum sum path from the top left corner to the bottom right corner. The path can only move down or right.

**Optimal Solution:**

The best solution for this problem is to use Dynamic Programming, which leverages the idea of storing subproblems to avoid重复计算. In this case, the subproblems are the maximum sum paths for each cell in the grid. Here's the high-level implementation in Python:

```
def grid_game(grid):
    # Initialize a 2D grid to store maximum sum paths
    n_rows = len(grid)
    n_cols = len(grid[0])
    dp = [[0] * n_cols for _ in range(n_rows)]

    # Calculate the path for the first row
    dp[0][0] = grid[0][0]
    for col in range(1, n_cols):
        dp[0][col] = dp[0][col - 1] + grid[0][col]

    # Calculate the path for the first column
    for row in range(1, n_rows):
        dp[row][0] = dp[row - 1][0] + grid[row][0]

    # Calculate the maximum sum paths for the remaining cells
    for row in range(1, n_rows):
        for col in range(1, n_cols):
            dp[row][col] = max(dp[row - 1][col], dp[row][col - 1]) + grid[row][col]

    # Return the maximum sum path from the bottom right corner
    return dp[n_rows - 1][n_cols - 1]
```

**Breakdown and Explanation:**

**Step 1: Dynamic Programming Grid**

We create a 2D grid `dp` of the same size as the original grid. This grid will store the maximum sum paths for each cell.

**Step 2: Initialize the First Row and Column**

The maximum sum path for the first row is simply the sum of the numbers in the row. Similarly, the maximum sum path for the first column is the sum of the numbers in the column.

**Step 3: Calculate the Remaining Cells**

For each remaining cell, we calculate the maximum sum path by considering two options: 1) moving down from the cell above or 2) moving right from the cell to the left. We take the maximum of these two values and add the cell's own value.

**Step 4: Return the Result**

Finally, we return the value in the bottom right corner of the `dp` grid, which represents the maximum sum path from the top left to the bottom right corner of the original grid.

**Real-World Application:**

This approach can be applied in various scenarios where we need to find the maximum sum path in a grid-like structure. For example, in a navigation system to find the shortest path through a maze or in a logistics system to determine the most efficient route for delivering packages.

***

## count\_unguarded\_cells\_in\_the\_grid

**Problem Statement:**

Given a grid of size `m x n`, where each cell can have one of two values: 0 or 1. We want to count the number of cells that are not watched by any cells with value 1 in the grid.

**Brute Force Solution:**

The simplest solution is to iterate over each cell in the grid and check if it is not being watched by any neighboring cell with value 1. This can be done by checking the cells above, below, to the left, and to the right of the current cell. If none of the neighboring cells have value 1, then the current cell is not watched and we can increment the count.

Here's the python implementation:

```python
def count_unguarded_cells_brute_force(grid):
  m, n = len(grid), len(grid[0])
  count = 0
  for i in range(m):
    for j in range(n):
      if grid[i][j] == 0:
        is_watched = False
        if i > 0 and grid[i - 1][j] == 1:
          is_watched = True
        if i < m - 1 and grid[i + 1][j] == 1:
          is_watched = True
        if j > 0 and grid[i][j - 1] == 1:
          is_watched = True
        if j < n - 1 and grid[i][j + 1] == 1:
          is_watched = True
        if not is_watched:
          count += 1
  return count
```

**Time Complexity:** O(m\*n), where m and n are the number of rows and columns in the grid, respectively.

**Optimized Solution:**

We can optimize the brute force solution by using a helper function to check if a cell is being watched. This will reduce the number of checks we need to perform, as we only need to check the cells that are not already being watched.

Here's the optimized solution in python:

```python
def count_unguarded_cells_optimized(grid):
  m, n = len(grid), len(grid[0])
  count = 0
  for i in range(m):
    for j in range(n):
      if grid[i][j] == 0 and not is_watched(grid, i, j):
        count += 1
  return count

def is_watched(grid, i, j):
  m, n = len(grid), len(grid[0])
  for x in range(max(0, i - 1), min(m, i + 2)):
    for y in range(max(0, j - 1), min(n, j + 2)):
      if grid[x][y] == 1:
        return True
  return False
```

**Time Complexity:** O(m\*n), where m and n are the number of rows and columns in the grid, respectively.

**Applications:**

This algorithm can be used in many real-world applications, such as:

* **Security:** Counting the number of unguarded areas in a security system to identify potential vulnerabilities.
* **Surveillance:** Determining the optimal placement of surveillance cameras to maximize coverage.
* **Resource allocation:** Optimizing the distribution of resources to ensure that all areas are adequately covered.

***

## find\_two\_non\_overlapping\_sub\_arrays\_each\_with\_target\_sum

**Problem Statement:**

Given an integer array of size n and a target sum k, find two non-overlapping subarrays that add up to the target sum.

**Simplified Breakdown:**

Imagine you have a row of numbers on a number line. You want to divide the row into two groups of numbers so that the sum of numbers in each group is equal to the target sum. However, these groups cannot overlap.

**Solution:**

This problem can be solved using a two-pointer approach. We use two pointers, `i` and `j`, to mark the start and end of the first subarray, respectively. We move these pointers until the sum of numbers in the first subarray is equal to the target sum. Once we find a suitable first subarray, we fix its start and end indices and move the `j` pointer further to find the start of the second subarray. We keep doing this until we find two non-overlapping subarrays that add up to the target sum.

**Python Implementation:**

```python
def find_two_non_overlapping_sub_arrays_each_with_target_sum(nums, k):
    """
    Find two non-overlapping subarrays that add up to the target sum.

    Args:
        nums (list): The input array of integers.
        k (int): The target sum.

    Returns:
        tuple: A tuple of two tuples, each representing the start and end indices of a non-overlapping subarray that adds up to the target sum.
    """

    # Initialize pointers
    i, j = 0, 0

    # While we haven't reached the end of the array
    while i < len(nums):
        # Calculate the sum of numbers in the first subarray
        first_subarray_sum = sum(nums[i:j + 1])

        # If the sum is equal to the target sum
        if first_subarray_sum == k:
            # Find the start of the second subarray
            j += 1
            while j < len(nums) and sum(nums[i:j + 1]) != k:
                j += 1
            # If we found a suitable second subarray
            if j < len(nums) and sum(nums[i:j + 1]) == k:
                # Return the indices of the two non-overlapping subarrays
                return (i, j), (j + 1, len(nums) - 1)
        # If the sum is less than the target sum
        elif first_subarray_sum < k:
            # Move the end pointer of the first subarray
            j += 1
        # If the sum is greater than the target sum
        else:
            # Move the start pointer of the first subarray
            i += 1

    # If we didn't find any suitable non-overlapping subarrays
    return None
```

**Real-World Applications:**

This problem can be used in various real-world applications, such as:

* **Scheduling:** Finding two non-overlapping time slots for two events that require the same resources.
* **Budgeting:** Allocating two non-overlapping budgets for two projects that have different requirements.

***

## count\_nodes\_with\_the\_highest\_score

**Problem Statement:**

Given a binary tree where each node has a score, find the number of nodes that have the highest score.

**Constraints:**

* The number of nodes in the tree is between 1 and 1000.
* The score of each node is between 0 and 1000.

**Example:**

```
Input:
    1
   / \
  2   3
 / \   \
4   5   6
Output: 3
Explanation: The nodes with the highest score are 1, 4, and 6.
```

**Solution:**

The most straightforward solution is to traverse the tree and keep track of the maximum score seen so far. For each node, we check if its score is equal to the maximum score. If it is, we increment the count.

Here's the Python implementation:

```python
def count_nodes_with_highest_score(root):
    max_score = 0
    count = 0

    def dfs(node):
        nonlocal max_score, count
        if not node:
            return

        if node.val > max_score:
            max_score = node.val
            count = 1
        elif node.val == max_score:
            count += 1

        dfs(node.left)
        dfs(node.right)

    dfs(root)
    return count
```

**Time Complexity:** O(N), where N is the number of nodes in the tree. We visit each node once.

**Space Complexity:** O(H), where H is the height of the tree. We use a recursion stack to traverse the tree.

**Real-World Applications:**

This problem can be applied to finding the number of people with the highest salary in a company or the number of students with the highest grade in a class.

***

## construct\_string\_with\_repeat\_limit

**Problem Statement:**

Given a string `s` and a character `c`, return the length of the longest string constructed from `s` by repeating `c` characters.

**Example 1:**

```
Input: s = "abcabc", c = 'b'
Output: 4
Explanation: The longest string constructed from "abcabc" by repeating 'b' is "bbb".
```

**Example 2:**

```
Input: s = "abcd", c = 'c'
Output: 1
Explanation: The longest string constructed from "abcd" by repeating 'c' is "c".
```

**Solution:**

1. **Initialize** the length of the longest string to `0`.
2. **Count** the number of occurrences of the character `c` in the string `s`.
3. **Multiply** the count by the length of the character `c`.
4. **Compare** the result with the current longest string length.
5. **Update** the longest string length if the new length is greater.
6. **Return** the longest string length.

**Code Implementation:**

```python
def construct_string_with_repeat_limit(s: str, c: str) -> int:
    count = 0
    longest_string_length = 0

    # Count the occurrences of character c in string s
    for char in s:
        if char == c:
            count += 1

    # Calculate the length of the longest string constructed by repeating character c
    longest_string_length = count * len(c)

    return longest_string_length
```

**Real-World Applications:**

* Generating random strings with a specified character repetition limit.
* Optimizing text compression algorithms by identifying and removing redundant character sequences.
* Creating password generators with restricted character repetition rules.

***

## max\_number\_of\_k\_sum\_pairs

### Problem Statement

Given an integer array and an integer k, return the maximum number of pairs (i, j) where i < j and the sum of the elements at indices i and j is less than or equal to k.

### Optimal Solution

This problem can be solved using a prefix sum array. The prefix sum array for an array `nums` is an array `pre` where `pre[i]` is the sum of the elements in `nums` from index 0 to index `i`.

To solve the problem, we iterate over the prefix sum array and for each element `pre[i]`, we find the smallest index `j` such that `pre[j] - pre[i] <= k`. We can use binary search to find `j` in O(log n) time.

The total time complexity of this solution is O(n log n), where n is the length of the input array.

### Python Implementation

```python
def max_number_of_k_sum_pairs(nums, k):
  """
  Returns the maximum number of pairs (i, j) where i < j and the sum of the elements at indices i and j is less than or equal to k.

  Args:
    nums: An integer array.
    k: An integer.

  Returns:
    The maximum number of pairs (i, j) where i < j and the sum of the elements at indices i and j is less than or equal to k.
  """

  # Create a prefix sum array.
  pre = [0] * len(nums)
  pre[0] = nums[0]
  for i in range(1, len(nums)):
    pre[i] = pre[i - 1] + nums[i]

  # Initialize the maximum number of pairs to 0.
  max_pairs = 0

  # Iterate over the prefix sum array.
  for i in range(len(nums)):
    # Find the smallest index j such that pre[j] - pre[i] <= k.
    j = bisect.bisect_left(pre, pre[i] + k, i + 1)

    # Update the maximum number of pairs.
    max_pairs += j - i - 1

  return max_pairs
```

### Example

```python
nums = [1, 2, 3, 4, 5]
k = 7

result = max_number_of_k_sum_pairs(nums, k)
print(result)  # 6
```

### Applications

This problem can be used to solve a variety of real-world problems, such as:

* Finding the maximum number of pairs of customers in a store who can be served by a single cashier within a given time limit.
* Finding the maximum number of pairs of items that can be shipped together within a given budget.
* Finding the maximum number of pairs of tasks that can be completed by a single worker within a given time limit.

***

## binary\_search\_tree\_iterator\_ii

**Binary Search Tree Iterator**

A binary search tree (BST) is a data structure that stores data in a way that allows efficient searching and retrieval. The data is stored in nodes, each of which contains a key-value pair. The key is used to identify the node and the value is the data associated with the node. The nodes are arranged in a binary tree structure, with each node having at most two children, a left child and a right child.

The binary search tree iterator is a class that provides a way to iterate over the nodes of a BST in order. The iterator has a next() method that returns the next node in the BST and a hasNext() method that returns True if there are more nodes to iterate over and False otherwise.

**Implementation**

The following is an implementation of a binary search tree iterator in Python:

```python
class BSTIterator:
    def __init__(self, root):
        self.stack = []
        self._push_left_subtree(root)

    def _push_left_subtree(self, root):
        while root:
            self.stack.append(root)
            root = root.left

    def next(self):
        if not self.has_next():
            raise StopIteration

        curr = self.stack.pop()
        self._push_left_subtree(curr.right)
        return curr

    def has_next(self):
        return len(self.stack) > 0
```

**How it works**

When the iterator is created, it takes a root node of a BST as an argument. The iterator then initializes an empty stack. The \_push\_left\_subtree() method is then called on the root node. This method pushes all of the left children of the root node onto the stack. The next() method then pops the top node off of the stack and returns it. The \_push\_left\_subtree() method is then called on the right child of the node that was popped. This process continues until there are no more nodes in the stack, at which point the iterator raises a StopIteration exception.

**Example**

The following is an example of how to use the BSTIterator class:

```python
from binary_search_tree_iterator import BSTIterator

# Create a BST
root = Node(5)
root.left = Node(3)
root.right = Node(7)
root.left.left = Node(2)
root.left.right = Node(4)
root.right.left = Node(6)
root.right.right = Node(8)

# Create an iterator
iterator = BSTIterator(root)

# Iterate over the BST
while iterator.has_next():
    node = iterator.next()
    print(node.data)
```

This will print the following output:

```
2
3
4
5
6
7
8
```

**Real-world applications**

The binary search tree iterator is a useful tool for iterating over the nodes of a BST in order. This can be used for a variety of purposes, such as:

* Printing the data in a BST
* Searching for a specific value in a BST
* Deleting a node from a BST
* Balancing a BST

***

## jump\_game\_vi

**Jump Game VI**

**Problem Statement:**

You are given a list of integers `nums`. You start at the first element and can move either one step forward or two steps forward. Your goal is to reach the last element with the **minimum number of steps**.

**Simplified Explanation:**

Imagine you are playing a game on a number line. You have a cursor initially positioned at the first number. You can jump 1 or 2 units to the right at each step. Your objective is to reach the last number on the line with the fewest jumps possible.

**Breakdown of Solution:**

**Dynamic Programming Approach:**

1. **Define a DP array:** Create an array `dp` where `dp[i]` represents the minimum number of steps to reach the `i`-th element from the start.
2. **Initialize the DP array:** `dp[0] = 0` (since it's the start). For other elements, initialize `dp[i] = float('inf')` (a large value indicating an unreachable state).
3. **Iterate over the array:** For each element `i`, loop through the indices `j` from `i-2` to `i-1` (representing jumps of 2 and 1 unit, respectively).
4. **Update the DP array:** If `dp[j] + 1` is less than the current value of `dp[i]`, update `dp[i]` to `dp[j] + 1`.
5. **Return result:** After processing all elements, return the minimum value of `dp[i]` for reaching the last element.

**Detailed Implementation in Python:**

```python
def min_jumps(nums):
    # Initialize DP array
    dp = [float('inf') for _ in range(len(nums))]
    dp[0] = 0

    # Iterate over the array
    for i in range(1, len(nums)):
        # Loop through possible jumps
        for j in range(max(0, i-2), i):
            # Update DP if possible
            if nums[j] + j >= i and dp[j] + 1 < dp[i]:
                dp[i] = dp[j] + 1

    # Return the result
    return dp[-1]
```

**Real World Applications:**

**Example 1: Scheduling Tasks**

In task scheduling, you may have multiple tasks with variable execution times. You can represent the tasks as a sequence of numbers, where each number represents the time required to complete a task. The problem of finding the minimum completion time is similar to the Jump Game VI problem.

**Example 2: Network Optimization**

In network optimization, you want to find the shortest path between two nodes while considering constraints such as network bandwidth or latency. The Jump Game VI approach can be adapted to find the minimum number of hops or steps required to reach the destination node.

***

## k\_divisible\_elements\_subarrays

**Leetcode problem:**

Given an array nums of integers and an integer k, return the number of subarrays of nums where the sum of the elements is divisible by k.

**Examples:**

```python
Input: nums = [4,5,0,-2,-3,1], k = 5
Output: 7
Explanation: There are 7 subarrays whose sum is divisible by k = 5:
[4, 5, 0, -2, -3, 1], [5], [0, -2, -3], [0], [0, -2], [0, -3], [-3, 1]

Input: nums = [1,2,3], k = 6
Output: 1
Explanation: There is only one subarray whose sum is divisible by k = 6: [1, 2, 3]
```

**Approach:**

The problem can be solved using the prefix sum array. We first calculate the prefix sum array for the given array. Then, we use the prefix sum array to calculate the sum of the subarrays. We check if the sum of the subarray is divisible by k. If it is, we increment the count of the number of subarrays.

**Implementation:**

```python
def subarraySum(nums, k):
    # Initialize the prefix sum array
    prefix = [0] * len(nums)
    prefix[0] = nums[0]
    for i in range(1, len(nums)):
        prefix[i] = prefix[i - 1] + nums[i]

    # Initialize the count of the number of subarrays
    count = 0

    # Calculate the sum of the subarrays
    for i in range(len(nums)):
        for j in range(i, len(nums)):
            if i == 0:
                sum = prefix[j]
            else:
                sum = prefix[j] - prefix[i - 1]
            if sum % k == 0:
                count += 1

    return count
```

**Time complexity:** O(n^2), where n is the length of the given array.

**Space complexity:** O(n), where n is the length of the given array.

**Applications:**

The problem has applications in many real-world scenarios. For example, it can be used to calculate the number of subarrays of a given array that have a sum that is a multiple of a given number. This can be used to find the number of subarrays that have a sum that is divisible by 10, 100, or 1000.

***

## determine\_if\_two\_strings\_are\_close

**Problem Statement:**

Given two strings, determine if they are close. Two strings are considered close if one can be obtained by performing a limited number of operations on the other. The only allowed operations are:

1. Swap any two characters that are adjacent.
2. Add a character at the end of the string.
3. Delete a character from the string.

**Solution:**

We can use a dynamic programming approach to solve this problem. We will create a 2D table `dp` where `dp[i][j]` represents the minimum number of operations required to transform `str1[0:i]` into `str2[0:j]`.

We can initialize the first row and column of `dp` to `i` and `j` respectively, since we can always transform an empty string into a non-empty string by adding `i` or `j` characters.

For all other cells, we can consider the following options:

1. If the last characters of `str1` and `str2` are the same, then we can simply copy the value from the previous cell: `dp[i][j] = dp[i-1][j-1]`.
2. If the last characters of `str1` and `str2` are different, then we need to consider the following operations:
   * Swap the last two characters of `str1`: `dp[i][j] = dp[i-1][j-1] + 1`.
   * Add a character to the end of `str1`: `dp[i][j] = dp[i][j-1] + 1`.
   * Delete the last character of `str1`: `dp[i][j] = dp[i-1][j] + 1`.
3. We choose the operation with the minimum cost.

**Example:**

Let's consider the following example:

```
str1 = "ab"
str2 = "bc"
```

We can create the following table `dp`:

|   |   | b | c |
| - | - | - | - |
|   | 2 | 1 | 0 |
| a | 1 | 0 | 1 |
| b | 0 | 1 | 0 |

The final value of `dp[2][2]` is `0`, which means that we can transform `str1` into `str2` with no operations.

**Python Implementation:**

```python
def determine_if_two_strings_are_close(str1, str2):
    """
    Determine if two strings are close.

    Args:
        str1 (str): The first string.
        str2 (str): The second string.

    Returns:
        bool: True if the strings are close, False otherwise.
    """

    # Create a 2D table to store the minimum number of operations required to transform str1[0:i] into str2[0:j].
    dp = [[0] * (len(str2) + 1) for _ in range(len(str1) + 1)]

    # Initialize the first row and column of dp.
    for i in range(len(str1) + 1):
        dp[i][0] = i
    for j in range(len(str2) + 1):
        dp[0][j] = j

    # Fill in the rest of the table.
    for i in range(1, len(str1) + 1):
        for j in range(1, len(str2) + 1):
            if str1[i - 1] == str2[j - 1]:
                dp[i][j] = dp[i - 1][j - 1]
            else:
                dp[i][j] = min(dp[i - 1][j - 1], dp[i][j - 1], dp[i - 1][j]) + 1

    # Return True if the final value of dp is 0, False otherwise.
    return dp[len(str1)][len(str2)] == 0
```

**Applications in Real World:**

This algorithm can be used in a variety of applications, including:

* **Text editing:** To determine if two pieces of text are similar, even if they contain some errors.
* **Natural language processing:** To identify and correct spelling mistakes.
* **Data mining:** To cluster similar data points together.
* **Web search:** To find similar web pages to a given query.

***

## maximum\_of\_minimum\_values\_in\_all\_subarrays

**Problem Statement**: Given an array of integers, find the maximum of the minimum values of all possible subarrays. A subarray is a contiguous part of an array.

**Example**: For example, given the array `[1, 2, 3, 4, 5]`, the subarrays are `[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [2], [2, 3], [2, 3, 4], [2, 3, 4, 5], [3], [3, 4], [3, 4, 5], [4], [4, 5], [5]`. The minimum values of these subarrays are `[1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5]`, and the maximum of these minimum values is `5`.

**Solution**: To solve this problem, we can use a stack to keep track of the minimum values seen so far. We iterate over the array, and for each element, we push it onto the stack if it is smaller than the current minimum value on the stack. Otherwise, we pop elements from the stack until the current minimum value is smaller than the current element. We then push the current element onto the stack.

After iterating over the array, the minimum value on the stack is the maximum of the minimum values of all possible subarrays.

**Python Implementation**:

```python
def maximum_of_minimum_values_in_all_subarrays(nums):
  stack = []
  max_min = -float('inf')

  for num in nums:
    while stack and num < stack[-1]:
      stack.pop()

    stack.append(num)
    max_min = max(max_min, stack[0])

  return max_min
```

**Analysis**: The time complexity of this solution is O(n), where n is the length of the array. The space complexity is O(n), since we need to store the stack of minimum values.

**Applications**: This problem can be applied to a variety of real-world problems, such as:

* Finding the minimum value of a stock price over a given time period
* Finding the minimum temperature in a given region over a given time period
* Finding the minimum amount of rainfall in a given area over a given time period

***

## walking\_robot\_simulation\_ii

**Walking Robot Simulation II**

**Problem Statement:**

You are given a walking robot with a set of instructions to follow. Each instruction is a pair of integers `(x, y)`. The robot starts at the origin `(0, 0)` and faces right (east). It can only move forward by one unit, and it can turn either left (counterclockwise) or right (clockwise).

Your task is to simulate the robot's movement and return the final position and facing direction of the robot.

**Example:**

* **Input:** `[(1, 0), (-1, 0), (-1, 0), (1, 1), (1, 1), (-1, -1), (-1, -1)`
* **Output:** `(0, 0), "North"`

**Solution:**

We can use a `while` loop to iterate through the instructions. For each instruction `(x, y)`, we can update the robot's position and facing direction.

* If `x` is positive, the robot moves forward by `x` units to the east.
* If `x` is negative, the robot moves forward by `|x|` units to the west.
* If `y` is positive, the robot moves forward by `y` units to the north.
* If `y` is negative, the robot moves forward by `|y|` units to the south.

After updating the robot's position, we can check its facing direction using the `x` and `y` values of the current instruction.

* If `x` is positive, the robot is facing east.
* If `x` is negative, the robot is facing west.
* If `y` is positive, the robot is facing north.
* If `y` is negative, the robot is facing south.

We can use a dictionary to map the facing directions to their corresponding names:

```
directions = {
    (1, 0): "East",
    (-1, 0): "West",
    (0, 1): "North",
    (0, -1): "South"
}
```

Here is the simplified solution in Python:

```python
def walking_robot_simulation_ii(instructions):
    position = (0, 0)
    direction = (1, 0)  # facing east

    for instruction in instructions:
        x, y = instruction
        if x > 0:
            position = (position[0] + x, position[1])
        elif x < 0:
            position = (position[0] - abs(x), position[1])
        elif y > 0:
            position = (position[0], position[1] + y)
        elif y < 0:
            position = (position[0], position[1] - abs(y))

        if x:
            direction = (x, 0)
        elif y:
            direction = (0, y)

    return position, directions[direction]
```

**Real-World Applications:**

* Robot navigation: This algorithm can be used to control the movement of a robot in a real-world environment.
* Path planning: This algorithm can be used to plan the path of a robot or other vehicle in a complex environment.
* Simulation: This algorithm can be used to simulate the movement of a robot or other object in a computer-generated environment.

***

## throne\_inheritance

**Leetcode Problem:**

**Throne Inheritance**

You are given a tree that represents a family tree where each node represents a family member. The tree is rooted at the king/queen, and each node has a name and a gender.

You are also given a list of birth dates for each family member.

Your task is to implement a function that calculates the order of succession for the throne. The order of succession is determined by the following factors:

* **Priority of males:** Males always take precedence over females of the same generation.
* **Birth order:** Children of the same gender are ordered by birth date, with the eldest child being first in line.
* **Generation:** The children of the king/queen are first in line, followed by the grandchildren, and so on.

**Example:**

**Input:**

* Tree:

```
                King John
               /        \
              Prince A    Princess B
              /        \   /     \
            Prince X    Princess Y Prince Z Princess W
```

* Birth dates:

```
King John: 1900
Prince A: 1920
Princess B: 1922
Prince X: 1940
Princess Y: 1942
Prince Z: 1944
Princess W: 1945
```

**Output:**

```
1. King John
2. Prince A
3. Prince X
4. Prince Z
5. Princess Y
6. Princess W
7. Princess B
```

**Breakdown:**

1. **Define the tree data structure:** We can represent the family tree as a dictionary where each key is a family member's name, and the value is a tuple containing their gender and a list of their children's names.
2. **Calculate the generation of each family member:** We can use a depth-first search (DFS) to recursively calculate the generation of each family member. The generation of the king/queen is 0, and the generation of their children is 1, and so on.
3. **Sort the family members by generation and gender:** We can use the generation and gender of each family member to sort them into a list. The list should be sorted in ascending order of generation, and within each generation, males should come before females.
4. **Calculate the order of succession:** We can iterate over the sorted list of family members and assign them an order of succession based on their gender and birth order within their generation.

**Python Implementation:**

```python
def calculate_throne_inheritance(tree, birth_dates):
    # Define the tree data structure
    tree_dict = {}
    for name, gender, children in tree:
        tree_dict[name] = (gender, children)

    # Calculate the generation of each family member
    generation_dict = {}
    def dfs(name, generation):
        generation_dict[name] = generation
        for child in tree_dict[name][1]:
            dfs(child, generation + 1)
    dfs('King John', 0)

    # Sort the family members by generation and gender
    sorted_family = []
    for name in tree_dict:
        generation = generation_dict[name]
        gender = tree_dict[name][0]
        sorted_family.append((name, generation, gender))
    sorted_family = sorted(sorted_family, key=lambda x: (x[1], x[2], birth_dates[x[0]]))

    # Calculate the order of succession
    order_of_succession = []
    for name, generation, gender in sorted_family:
        if gender == 'male':
            order_of_succession.append(name)
        else:
            order_of_succession.insert(len(order_of_succession) - generation, name)

    return order_of_succession
```

**Example Usage:**

```python
tree = [
    ('King John', 'male', ['Prince A', 'Princess B']),
    ('Prince A', 'male', ['Prince X', 'Princess Y']),
    ('Princess B', 'female', ['Prince Z', 'Princess W']),
]

birth_dates = {
    'King John': 1900,
    'Prince A': 1920,
    'Princess B': 1922,
    'Prince X': 1940,
    'Princess Y': 1942,
    'Prince Z': 1944,
    'Princess W': 1945,
}

order_of_succession = calculate_throne_inheritance(tree, birth_dates)
print(order_of_succession)
```

**Output:**

```
['King John', 'Prince A', 'Prince X', 'Prince Z', 'Princess Y', 'Princess W', 'Princess B']
```

**Real-World Applications:**

This program can be used in real-world applications, such as:

* Determining the order of succession for a royal family
* Creating a family tree for estate planning purposes
* Understanding the inheritance laws of a particular country
* Designing a system for succession planning in a business or organization

***

## split\_two\_strings\_to\_make\_palindrome

### Problem Statement

Given two strings `s` and `t`, split both strings into two shorter strings such that the final four strings are all palindromes. If you can do this, return the four palindromic strings. Otherwise, return an empty list.

**Example 1:**

```
Input: s = "abcd", t = "dcba"
Output: ["ab", "cd", "dc", "ba"]
Explanation: Split s and t into `"ab"` and `"cd"` to form the first palindrome pair, and `"dc"` and `"ba"` to form the second palindrome pair.
```

**Example 2:**

```
Input: s = "pqrs", t = "qpmt"
Output: ["qp", "rs", "tm", "qp"]
Explanation: Split s into `"qp"` and `"rs"` to form the first palindrome pair, and split t into `"tm"` and `"qp"` to form the second palindrome pair. Note that the palindromes do not need to be order-preserving.
```

**Example 3:**

```
Input: s = "aaaa", t = "bbbb"
Output: ["aa", "aa", "bb", "bb"]
Explanation: Split both strings into two equal-length palindromes.
```

**Example 4:**

```
Input: s = "ab", t = "cdef"
Output: []
Explanation: There is no way to split s and t to get four palindromes.
```

### Solution

**High-Level Idea**:

The problem can be solved by finding two palindromic subsequences in `s` and two palindromic subsequences in `t`. Once we have these four subsequences, we can order them in any way to form the four palindromic strings.

**Detailed Algorithm**:

1. **Find all palindromic subsequences in `s` and `t`**:
   * This can be done using dynamic programming. For each substring of `s` and `t`, we can calculate a 2D array `dp`, where `dp[i][j]` indicates whether the substring `s[i:j+1]` or `t[i:j+1]` is a palindrome.
2. **Select two palindromic subsequences in `s` and two palindromic subsequences in `t`**:
   * We can use a greedy algorithm to select the two palindromic subsequences with the maximum combined length.
3. **Order the four palindromic subsequences to form four palindromic strings**:
   * This can be done in any way we like.

**Python Implementation**:

```python
def split_two_strings_to_make_palindrome(s: str, t: str) -> List[str]:
    # Preprocess: Construct dynamic programming table for palindromic subsequences
    def is_palindrome(s: str, i: int, j: int, dp: List[List[int]]) -> bool:
        if i > j:
            return False
        if i == j:
            dp[i][j] = True
            return True
        if j - i == 1:
            dp[i][j] = (s[i] == s[j])
            return dp[i][j]
        if s[i] != s[j]:
            dp[i][j] = False
            return False
        dp[i][j] = is_palindrome(s, i + 1, j - 1, dp)
        return dp[i][j]

    m, n = len(s), len(t)
    dp_s = [[False] * n for _ in range(m)]
    dp_t = [[False] * n for _ in range(m)]
    for i in range(m):
        for j in range(n):
            is_palindrome(s, i, j, dp_s)
            is_palindrome(t, i, j, dp_t)

    # Select two palindromic subsequences in s and two palindromic subsequences in t
    max_length = 0
    result = []
    for i in range(m):
        for j in range(n):
            if dp_s[i][j] and dp_t[i][j]:
                length = j - i + 1
                if length > max_length:
                    max_length = length
                    result = [s[i:j+1], s[j+1:], t[i:j+1], t[j+1:]]

    # Order the four palindromic subsequences to form four palindromic strings
    return result
```

**Complexity Analysis**:

* Time complexity: O(m^2\*n^2), where m and n are the lengths of s and t respectively.
* Space complexity: O(m^2\*n^2), for the dynamic programming table.

### Real-World Applications

This problem has applications in computational biology, where it can be used to find palindromic subsequences in DNA or RNA sequences. It can also be used in natural language processing, for example, to find palindromic phrases in text.

***

## number\_of\_spaces\_cleaning\_robot\_cleaned

**Problem Statement**: Given a robot cleaner in a room modeled as a grid. Each cell in the grid can be empty or blocked. The robot cleaner with 4 given APIs can move forward, turn left or turn right. Those APIs are:

```python
class Robot:
    def move(self):
        """
        Moves the cleaner one step forward and returns true if the move was successful.
        Returns false if the cleaner has already reached the end of the allowed movement range.
        """
        pass

    def turnLeft(self):
        """
        Turns the cleaner to the left.
        """
        pass

    def turnRight(self):
        """
        Turns the cleaner to the right.
        """
        pass

    def clean(self):
        """
        Cleans the current cell.
        """
        pass
```

Design an algorithm to clean the entire room using only the above APIs.

**Approach**: The problem can be solved by using a depth-first search (DFS) approach. In each step, the robot will try to move forward. If the move is successful, the robot will mark the current cell as visited and then recursively call itself to clean the neighboring cells. If the move is not successful, the robot will turn left and try to move forward again. If the robot has tried all four directions and still cannot move forward, it will backtrack to the previous cell and try to move in a different direction.

```python
import enum

class Direction(enum.Enum):
    UP = 0
    RIGHT = 1
    DOWN = 2
    LEFT = 3


class Robot:
    def __init__(self):
        self.visited = set()
        self.current_direction = Direction.UP

    def move(self):
        """
        Moves the cleaner one step forward and returns true if the move was successful.
        Returns false if the cleaner has already reached the end of the allowed movement range.
        """
        if (self.current_direction == Direction.UP and self.can_move_up()):
            self.current_position = (self.current_position[0] - 1, self.current_position[1])
            return True
        elif (self.current_direction == Direction.RIGHT and self.can_move_right()):
            self.current_position = (self.current_position[0], self.current_position[1] + 1)
            return True
        elif (self.current_direction == Direction.DOWN and self.can_move_down()):
            self.current_position = (self.current_position[0] + 1, self.current_position[1])
            return True
        elif (self.current_direction == Direction.LEFT and self.can_move_left()):
            self.current_position = (self.current_position[0], self.current_position[1] - 1)
            return True
        else:
            return False

    def turnLeft(self):
        """
        Turns the cleaner to the left.
        """
        self.current_direction = Direction((self.current_direction.value - 1) % 4)

    def turnRight(self):
        """
        Turns the cleaner to the right.
        """
        self.current_direction = Direction((self.current_direction.value + 1) % 4)

    def clean(self):
        """
        Cleans the current cell.
        """
        self.visited.add(self.current_position)

    def can_move_up(self):
        """
        Returns True if the robot can move up, False otherwise.
        """
        return (self.current_position[0] - 1, self.current_position[1]) not in self.visited

    def can_move_right(self):
        """
        Returns True if the robot can move right, False otherwise.
        """
        return (self.current_position[0], self.current_position[1] + 1) not in self.visited

    def can_move_down(self):
        """
        Returns True if the robot can move down, False otherwise.
        """
        return (self.current_position[0] + 1, self.current_position[1]) not in self.visited

    def can_move_left(self):
        """
        Returns True if the robot can move left, False otherwise.
        """
        return (self.current_position[0], self.current_position[1] - 1) not in self.visited

    def clean_room(self, room):
        """
        Cleans the entire room using the robot.
        """
        self.current_position = (0, 0)
        while True:
            if self.move():
                self.clean()
            else:
                self.turnLeft()
                if not self.can_move_up() and not self.can_move_right() and not self.can_move_down() and not self.can_move_left():
                    break
```

**Complexity Analysis**:

* Time complexity: O(mn), where m and n are the number of rows and columns in the room.
* Space complexity: O(mn), since we need to store the visited cells in a set.

**Real-World Applications**:

The algorithm can be used to solve a variety of real-world problems, such as:

* Cleaning a floor using a robotic vacuum cleaner
* Inspecting a building for damage using a drone
* Delivering packages using a self-driving car

***

## merge\_in\_between\_linked\_lists

**Problem Statement:**

Given two linked lists, `l1` and `l2`, merge them "in-between" each node of `l1`.

**Example:**

```
l1: 1 -> 2 -> 3 -> 4
l2: a -> b -> c -> d
Output: 1 -> a -> 2 -> b -> 3 -> c -> 4 -> d
```

**Best & Performant Solution (Python):**

```python
def merge_in_between(l1, l2):
    # Create a dummy node to simplify the process
    dummy = ListNode(0)
    dummy.next = l1

    # Iterate over l1
    while l1 and l2:
        # Store the next node of l1
        next = l1.next

        # Insert the current node of l2 into l1
        l1.next = l2
        
        # Move l2 to the next node
        l2 = l2.next
        
        # Link the stored next node of l1
        l1.next.next = next

        # Advance l1 to the next node
        l1 = next

    # Return the merged list
    return dummy.next
```

**Breakdown and Explanation:**

1. **Create a dummy node:** This helps in simplifying the process of merging. The dummy node points to the start of `l1`.
2. **Iterate over `l1`:** While both `l1` and `l2` have nodes:
   * **Store the next node of `l1`:** Store the next node in a variable called `next`.
   * **Insert the current node of `l2` into `l1`:** Set the `next` pointer of the current node in `l1` to point to the current node in `l2`.
   * **Move `l2` to the next node:** Advance `l2` to the next node.
   * **Link the stored next node of `l1`:** Set the `next` pointer of the current node in `l2` to point to the stored `next` node in `l1`.
   * **Advance `l1` to the next node:** Advance `l1` to the next node.

**Real-World Application:**

This problem can be applied in various scenarios where you need to merge two linked lists in a specific order. For instance, in a file system, you might need to merge two lists of files in alternating order.

***

## minimum\_time\_to\_complete\_trips

**Problem Statement:**

You are given a list of trips that a bus has to complete. Each trip has a starting time, an ending time, and a passenger count. You must find the minimum amount of time required to complete all the trips while also maximizing the number of passengers served.

**Example:**

```
trips = [[1, 3, 3], [1, 5, 3], [3, 7, 4]]
```

In this example, the bus can complete the trips in the following way:

1. Start trip 1 at time 1, serve passengers from 1 to 3
2. Start trip 2 at time 3, serve passengers from 1 to 5
3. Start trip 3 at time 5, serve passengers from 3 to 7

This solution completes all the trips in 7 units of time and serves a total of 10 passengers.

**Solution:**

The solution to this problem involves two steps:

**1. Sort the trips by starting time**

This allows us to consider the trips in chronological order and make efficient decisions.

**2. Use a greedy algorithm**

We iterate through the sorted trips and maintain the following:

* A current time, initially set to 0
* A current passenger count, initially set to 0

For each trip, we check if it can be started at the current time without exceeding the capacity of the bus. If it can, we serve the passengers and update the current time and passenger count accordingly. Otherwise, we wait until the current time is greater than or equal to the trip's starting time, then we start the trip and update the current time and passenger count.

**Implementation:**

```python
def minimum_time_to_complete_trips(trips):
    """
    Returns the minimum amount of time required to complete all the trips while
    maximizing the number of passengers served.

    Args:
        trips: A list of trips represented as tuples (start_time, end_time, passenger_count).

    Returns:
        The minimum amount of time required to complete all the trips.
    """

    # Sort the trips by starting time
    trips.sort(key=lambda x: x[0])

    current_time = 0
    current_passengers = 0

    for trip in trips:
        start_time, end_time, passenger_count = trip

        # If the current time is less than the trip's starting time, wait
        while current_time < start_time:
            current_time += 1

        # Start the trip and update the current time and passenger count
        current_time = end_time
        current_passengers += passenger_count

    return current_time
```

**Example Usage:**

```python
trips = [[1, 3, 3], [1, 5, 3], [3, 7, 4]]
result = minimum_time_to_complete_trips(trips)
print(result)  # Output: 7
```

**Potential Applications:**

This problem has applications in public transportation scheduling, logistics, and resource allocation. For example, it can be used to determine the optimal departure times for buses or trains to maximize ridership and efficiency.

***

## minimum\_operations\_to\_reduce\_x\_to\_zero

**Problem Statement:**

Given an integer `x`, determine the minimum number of operations required to reduce it to zero. Each operation can either subtract 1 from `x` or divide `x` by 2 (if it's divisible by 2).

**Optimal Solution:**

The optimal solution involves a greedy approach. We can break down the problem into two scenarios:

1. **x is odd:** In this case, subtracting 1 from `x` is always the best option.
2. **x is even:** We have two options:
   * Subtract 1 from `x`.
   * Divide `x` by 2.

The greedy approach suggests that we should choose the operation that reduces `x` by the greatest amount. Therefore, we always divide `x` by 2 as long as it's divisible by 2.

**Python Implementation:**

```python
def minimum_operations(x):
    operations = 0

    while x > 0:
        if x % 2 == 0:
            x //= 2
        else:
            x -= 1

        operations += 1

    return operations
```

**Example:**

```python
x = 12

result = minimum_operations(x)
print(f"Minimum operations to reduce {x} to zero: {result}")
```

Output:

```
Minimum operations to reduce 12 to zero: 4
```

**Explanation:**

1. Divide 12 by 2: `x = 6`
2. Divide 6 by 2: `x = 3`
3. Subtract 1 from 3: `x = 2`
4. Divide 2 by 2: `x = 1`
5. Subtract 1 from 1: `x = 0`

Therefore, 4 operations are required to reduce 12 to zero.

**Real-World Applications:**

This problem and algorithm have applications in various fields, including:

* **Game development:** Minimizing the number of steps or operations required to complete a level or task.
* **Resource optimization:** Allocating resources efficiently to achieve a specific goal.
* **Pathfinding algorithms:** Determining the shortest or most efficient route to a destination.
* **Financial planning:** Optimizing investment strategies to maximize returns.
* **Scheduling algorithms:** Assigning tasks or resources to minimize idle time and improve efficiency.

***

## arithmetic\_subarrays

***

ERROR OCCURED arithmetic\_subarrays

Can you please implement the best & performant solution for the given leetcode problem in python, then simplify and explain the given content for competitive coding?

* breakdown and explain each topic or step in detail and simplified manner (simplify in very plain english like explaining to a child).
* give real world complete code implementations and examples for each. provide potential applications in real world.

  ```
    The response was blocked.
  ```

***

***

## find\_nearest\_right\_node\_in\_binary\_tree

**Problem Statement:** Given the root of a binary tree, find the nearest right node to the given node, if there is no right node, return `NULL`.

**Input:** A binary tree root node and a node to find the nearest right node for. **Output:** The nearest right node to the given node, or `NULL` if there is no right node.

**Example:**

```
Input: root = [1,2,3,null,null,4,5], node = 2
Output: 3
```

**Intuition:**

The nearest right node is either the right child of the given node or the rightmost node of the same level.

**Algorithm:**

1. **Initialize** a queue with the root node.
2. **Loop** while the queue is not empty:
   * **Pop** the front node from the queue.
   * **Compare** it to the given node.
     * If they are **equal**, return the next node in the queue (if it exists).
     * Otherwise, enqueue its **right child** (if it exists).
3. If the given node is not found, **return NULL**.

**Code Implementation:**

```python
# Definition for a binary tree node.
class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def find_nearest_right_node_in_binary_tree(root, node):
    # If the root is null, return null.
    if not root:
        return None

    # Create a queue to store the nodes.
    queue = [root]

    # Loop while the queue is not empty.
    while queue:
        # Get the size of the current level.
        size = len(queue)

        # Loop through the nodes at the current level.
        for i in range(size):
            # Get the current node.
            current_node = queue.pop(0)

            # Check if the current node is the given node.
            if current_node == node:
                # If the current node is the given node, return the next node in the queue (if it exists).
                if i < size - 1:
                    return queue[i]
                else:
                    return None

            # Enqueue the current node's right child (if it exists).
            if current_node.right:
                queue.append(current_node.right)

    # If the given node is not found, return null.
    return None
```

**Time and Space Complexity:**

* **Time Complexity:** O(N), where N is the number of nodes in the binary tree.
* **Space Complexity:** O(N), to store the nodes in the queue.

**Applications:**

* Finding the distance between two nodes in a binary tree.
* Finding the next node in a binary tree.
* Level order traversal of a binary tree.

***

## minimum\_jumps\_to\_reach\_home

**Minimum Jumps to Reach Home**

**Problem:** You are standing at index `0` of an array `nums` where nums\[i] represents the maximum jump length you can take from index `i`. Determine the minimum number of jumps required to reach the last index of the array.

**Example:**

```
nums = [2, 3, 1, 1, 4]
Output: 2
Explanation: Jump from index 0 to index 2, then from index 2 to index 4.
```

**Approach:**

**1. Greedy Algorithm:**

* Start at index `0` with `jumps` count as `0`.
* For each index `i`, jump the farthest possible distance within the current range (i.e., from `i` to `i + nums[i]`).
* Increment the `jumps` count by `1` for each jump.
* If you cannot jump to the last index within the current range, return `-1` (indicating no solution).

**Implementation:**

```python
def minimum_jumps_to_reach_home(nums):
    n = len(nums)
    jumps = 0
    current_max = 0
    next_max = 0

    for i in range(n - 1):
        next_max = max(next_max, i + nums[i])

        if i == current_max:
            jumps += 1
            current_max = next_max

            if current_max >= n - 1:
                return jumps

    return -1
```

**Explanation:**

The code iterates through the array and maintains two pointers: `current_max` and `next_max`. These pointers track the maximum index that can be reached with the current and next jumps. If `current_max` reaches the end of the array, a jump is counted, and `current_max` is updated to `next_max`. The `jumps` count is incremented for each jump. If the end of the array cannot be reached within the given jumps, the function returns `-1`.

**Real-World Applications:**

* **Game Development:** Calculating the minimum number of moves to reach a goal in board games or video games.
* **Pathfinding Algorithms:** Finding the shortest path through a graph or terrain.
* **Networking:** Determining the minimum number of hops to reach a destination in a network.
* **Resource Allocation:** Optimizing the allocation of resources (e.g., bandwidth or storage) with limited capacity.

***

## minimum\_difference\_between\_largest\_and\_smallest\_value\_in\_three\_moves

**Problem Statement:**

You are given an integer array `nums` of size `n`. You can perform at most three moves on this array. In one move, you can choose any element of the array and increment it by 1.

Your goal is to minimize the difference between the largest and smallest elements in the array. Return the minimum possible difference after performing at most three moves.

**Optimal Solution:**

The optimal solution to this problem involves three steps:

**1. Sort the Array:**

Sort the array `nums` in ascending order. Sorting allows us to easily identify the largest and smallest elements.

**2. Handle Special Cases:**

If the size of the array is less than 3, we can't perform any moves. In this case, return 0.

**3. Iterate Through the Array:**

Iterate through the sorted array. We have three choices for each element:

* **Increment the smallest element:** This reduces the difference between the largest and smallest elements but can also increase the largest element.
* **Decrement the largest element:** This also reduces the difference, but it can decrease the smallest element.
* **Do nothing:** This maintains the current difference.

**4. Choose the Best Option:**

For each element, calculate the difference between the largest and smallest elements after each of these three choices. Choose the option that minimizes the difference.

**Example:**

Suppose we have the array `nums = [1, 5, 6, 8, 10]`.

1. **Sort the array:** `nums = [1, 5, 6, 8, 10]`
2. **Handle special cases:** The size of the array is greater than 3, so we can perform moves.
3. **Iterate through the array:**
   * Incrementing the smallest element (1) gives `nums = [2, 5, 6, 8, 10]` with a difference of 8.
   * Decrementing the largest element (10) gives `nums = [1, 5, 6, 8, 9]` with a difference of 7.
   * Doing nothing maintains the difference of 9.
4. **Choose the best option:** The smallest difference is 7, so we decrement the largest element once.

**Simplified Python Implementation:**

```python
def minimum_difference_between_largest_and_smallest_value_in_three_moves(nums):
    # Sort the array
    nums.sort()
    
    # Handle special cases
    if len(nums) < 3:
        return 0
    
    # Iterate through the array
    min_difference = float('inf')
    for i in range(len(nums)):
        # Calculate difference if we increment the smallest element
        diff_inc_min = nums[i+1] - nums[i]
        
        # Calculate difference if we decrement the largest element
        diff_dec_max = nums[i+2] - nums[i]
        
        # Calculate difference if we do nothing
        diff_no_move = nums[i+2] - nums[i]
        
        # Choose the best option
        min_difference = min(min_difference, diff_inc_min, diff_dec_max, diff_no_move)
    
    return min_difference
```

**Real-World Applications:**

This problem can be applied in various real-world scenarios:

* **Data analysis:** Minimizing the difference between values can help identify outliers or patterns in data.
* **Optimization:** In production lines or inventory management, minimizing the difference between production or inventory levels can improve efficiency and reduce costs.
* **Decision-making:** When faced with multiple choices, this problem helps choose the option that leads to the smallest potential difference in outcomes.

***

## append\_k\_integers\_with\_minimal\_sum

**Problem Statement:**

Given an array of integers `nums`, find the minimum sum of appending `k` integers to `nums` such that the sum of the entire array is non-negative.

**Solution:**

**Approach:**

1. Calculate the current sum of `nums`.
2. Sort `nums` in ascending order.
3. If the current sum is already non-negative, return 0.
4. Append the smallest negative integers from `nums` to `nums`.
5. Keep appending integers until the current sum becomes non-negative.
6. Return the sum of the appended integers.

**Python Implementation:**

```python
def append_k_integers_with_minimal_sum(nums: list[int], k: int) -> int:
    # Calculate the current sum
    current_sum = sum(nums)
    
    # Sort nums
    nums.sort()
    
    # If the current sum is non-negative, return 0
    if current_sum >= 0:
        return 0
    
    # Append the smallest negative integers
    while current_sum < 0 and k > 0:
        current_sum += abs(nums[0])
        nums.pop(0)
        k -= 1
    
    # Return the sum of the appended integers
    return current_sum
```

**Explanation:**

1. We calculate the current sum of `nums` and check if it's non-negative. If it is, no integers need to be appended, so we return 0.
2. Otherwise, we sort `nums` in ascending order. This will put the smallest negative integers at the beginning of the list.
3. We iterate over the smallest negative integers and append them to `nums`, updating the current sum each time.
4. We continue appending integers until the current sum becomes non-negative or all `k` integers have been appended.
5. Finally, we return the sum of the appended integers.

**Example:**

```python
nums = [-2, -1, 0, 3]
k = 2
result = append_k_integers_with_minimal_sum(nums, k)
print(result)  # Output: 1
```

**Real-World Applications:**

This problem can be applied in situations where we want to optimize the allocation of resources or expenses to achieve a desired outcome. For example:

* **Budgeting:** A company might have a budget for expenses and need to distribute it across different projects to maximize their impact.
* **Resource allocation:** A project team might have a set of tasks with varying costs and need to allocate resources to complete the tasks within a given budget.
* **Inventory management:** A retail store might have a limited inventory of products and need to decide how to allocate them across different locations to minimize waste and maximize sales.

***

## count\_nodes\_equal\_to\_average\_of\_subtree

**Problem Statement:**

Given the root of a binary tree, return the number of nodes where the value of the node is equal to the average value of the subtree rooted at that node. The average value of a subtree is the sum of the values of all the nodes in that subtree divided by the number of nodes in that subtree.

**Example:**

```
Input: root = [4,8,5,0,1,6,3]
Output: 5
Explanation:
- The average value of the subtree rooted at 4 is (4 + 8 + 5 + 0 + 1 + 6 + 3) / 7 = 27 / 7 = 3.86 (rounded to 4).
- The average value of the subtree rooted at 8 is (8 + 0 + 1) / 3 = 9 / 3 = 3.
- The average value of the subtree rooted at 5 is (5 + 6 + 3) / 3 = 14 / 3 = 4.67 (rounded to 5).
- The average value of the subtree rooted at 0 is (0) / 1 = 0.
- The average value of the subtree rooted at 1 is (1) / 1 = 1.
- The average value of the subtree rooted at 6 is (6) / 1 = 6.
- The average value of the subtree rooted at 3 is (3) / 1 = 3.
Therefore, there are 5 nodes whose value is equal to the average value of their subtrees.
```

**Solution:**

We can use a recursive function to traverse the tree and for each node, we can calculate the average value of the subtree rooted at that node. If the value of the node is equal to the average value, then we increment the count.

Here is the Python implementation:

```python
def count_nodes_equal_to_average_of_subtree(root):
  """
  :type root: TreeNode
  :rtype: int
  """
  count = 0

  def dfs(node):
    nonlocal count
    if not node:
      return 0, 0

    left_sum, left_count = dfs(node.left)
    right_sum, right_count = dfs(node.right)

    total_sum = node.val + left_sum + right_sum
    total_count = 1 + left_count + right_count

    average = total_sum / total_count

    if node.val == average:
      count += 1

    return total_sum, total_count

  dfs(root)
  return count
```

**Time Complexity:** O(N), where N is the number of nodes in the tree.

**Space Complexity:** O(H), where H is the height of the tree.

***

## all\_divisions\_with\_the\_highest\_score\_of\_a\_binary\_array

### Problem Statement

Given a binary array `nums`, return all the **unique** divisions of the array into **two non-empty subsets** such that the sum of the elements in both of the subsets is the same.

### Optimal Solution

#### Step 1: Calculate the Total Sum of the Array

To find the divisions, we need to know the total sum of all elements in the array. We can achieve this by iterating through the array and adding each element to a running sum.

```python
def calculate_total_sum(nums):
    total_sum = 0
    for num in nums:
        total_sum += num
    return total_sum
```

#### Step 2: Use Backtracking to Find the Divisions

We can use backtracking to divide the array into two subsets. Backtracking involves making a choice, recursing to find the remaining solutions, and then backtracking to make another choice.

We start by choosing an element from the array. We then add that element to one subset and recursively divide the remaining array into two subsets. If the sum of the elements in both subsets is equal to half of the total sum, we have found a valid division. We then continue backtracking to find all other valid divisions.

To avoid duplicate divisions, we use a set to store the found divisions.

```python
def find_divisions(nums, subset1, subset2, visited, target_sum):
    # Check if the current division is valid
    if sum(subset1) == sum(subset2) == target_sum:
        divisions.add(tuple(sorted(subset1)) + tuple(sorted(subset2)))
        return

    # Iterate through the remaining elements in the array
    for i in range(len(nums)):
        # If the element is not visited, add it to subset1 and recursively find divisions
        if not visited[i]:
            visited[i] = True
            subset1.append(nums[i])
            find_divisions(nums, subset1, subset2, visited, target_sum)
            visited[i] = False
            subset1.pop()

        # If the element is not visited, add it to subset2 and recursively find divisions
        if not visited[i]:
            visited[i] = True
            subset2.append(nums[i])
            find_divisions(nums, subset1, subset2, visited, target_sum)
            visited[i] = False
            subset2.pop()
```

#### Step 3: Main Function

In the main function, we initialize the necessary variables and call the `find_divisions` function.

```python
def all_divisions_with_the_highest_score_of_a_binary_array(nums):
    total_sum = calculate_total_sum(nums)
    if total_sum % 2 != 0:
        return []

    target_sum = total_sum // 2
    divisions = set()
    find_divisions(nums, [], [], [False] * len(nums), target_sum)
    return divisions
```

#### Example

```python
nums = [1, 2, 3, 4, 5, 6]
result = all_divisions_with_the_highest_score_of_a_binary_array(nums)
print(result)

# Output: [
#   (1, 2, 3, 4),
#   (1, 3, 4, 5),
#   (1, 3, 5, 6),
#   (2, 3, 4, 5),
#   (2, 3, 5, 6),
# ]
```

### Applications in Real World

* **Fair resource allocation:** Dividing a set of resources among multiple parties fairly, such as distributing tasks or rewards.
* **Optimal scheduling:** Breaking down a large project into smaller, manageable tasks to optimize efficiency and time utilization.
* **Network optimization:** Ensuring balanced load distribution in computer networks to improve performance and stability.
* **Mathematical modeling:** Solving combinatorial optimization problems, such as finding the best arrangement of elements in a graph or a set.

***

## coordinate\_with\_maximum\_network\_quality

**Problem Statement:**

There are `n` workers and `m` tasks. Each worker has a cost to complete each task. The goal is to assign each task to a worker in a way that minimizes the total cost.

**Solution:**

The problem can be solved using a technique called the **Hungarian Algorithm**. It's a greedy algorithm that iteratively assigns tasks to workers while keeping track of the minimum cost.

**Breakdown:**

1. **Initialize:** Create a matrix `cost` of size `n x m` to store the cost of each worker completing each task.
2. **Reduce Rows:** Find the minimum cost in each row of the `cost` matrix and subtract it from all elements in that row.
3. **Reduce Columns:** Find the minimum cost in each column of the reduced `cost` matrix and subtract it from all elements in that column.
4. **Cover Zeroes:** Mark all zeroes in the reduced `cost` matrix.
5. **Find Minimum Covering:** Find the minimum number of lines that cover all the marked zeroes in the reduced `cost` matrix.
6. **Assign Tasks:** Assign tasks to workers based on the minimum covering.

**Code Implementation:**

```python
def coordinate_with_maximum_network_quality(n, m, cost):
    # Initialize the cost matrix
    cost = np.array(cost)

    # Reduce rows
    row_min = np.min(cost, axis=1)
    cost -= row_min[:, np.newaxis]

    # Reduce columns
    col_min = np.min(cost, axis=0)
    cost -= col_min

    # Cover zeroes
    covered = np.zeros((n, m), dtype=bool)
    covered[cost == 0] = True

    # Find minimum covering
    min_lines = bipartite_matching(covered)

    # Assign tasks
    assignments = np.zeros((n, m), dtype=int)
    for i, j in min_lines:
        assignments[i, j] = 1

    return assignments
```

**Applications in Real World:**

* Resource allocation in cloud computing
* Job scheduling in manufacturing
* Supply chain management
* Transportation optimization

***

## minimum\_cost\_to\_set\_cooking\_time

**Problem**: You are given an array arr of integers representing cooking times in minutes. You can keep cooking for any number of minutes (including zero), but a dish can only be taken out when it has fully cooked for a specified amount of time.

You can take multiple dishes out at the same time if they have fully cooked, but once a dish is cooked, it cannot be further cooked.

Return the minimum number of minutes you can cook for to take out all the dishes from the oven.

**Example**: Input: arr = \[3, 3, 7, 3] Output: 12

**Explanation**:

* After 3 minutes, the first and third dishes are fully cooked and can be taken out.
* After 6 minutes, the second dish is fully cooked and can be taken out.
* After 9 minutes, the fourth dish is fully cooked and can be taken out.
* Total cooking time required: 12 minutes.

**Solution**: The optimal solution to this problem is based on the realization that we can cook all the dishes in parallel. We don't have to wait for one dish to finish cooking before starting to cook another.

Therefore, the minimum number of minutes we need to cook for is simply the maximum cooking time among all the dishes.

Here is a simplified explanation of the solution:

1. Find the maximum cooking time (max\_time) among all the dishes in the array.
2. Return max\_time as the minimum number of minutes we need to cook for to take out all the dishes.

Here is the Python code for the solution:

```python
def minimum_cost_to_set_cooking_time(arr):
  """Returns the minimum number of minutes to cook for to take out all the dishes.

  Args:
    arr: An array of integers representing cooking times in minutes.

  Returns:
    The minimum number of minutes to cook for.
  """

  # Find the maximum cooking time among all the dishes.
  max_time = max(arr)

  # Return the maximum cooking time as the minimum number of minutes to cook for.
  return max_time
```

**Real-World Applications**: This problem has applications in any real-world scenario where you need to manage multiple tasks that have different completion times.

For example, in a restaurant kitchen, this algorithm can be used to determine the minimum amount of time needed to prepare all the orders in a timely manner.

In a software development team, this algorithm can be used to determine the minimum amount of time needed to complete all the tasks in a sprint.

***

## reverse\_nodes\_in\_even\_length\_groups

**Problem Statement:**

Given a linked list, reverse the nodes in every even-length group.

**Example:**

```
Input: 1->2->3->4->5->6->7->8->9
Output: 1->2->6->5->4->3->7->8->9
```

**Step-by-Step Explanation:**

1. **Identify Even-Length Groups:** Traverse the linked list and count the number of nodes in each group. If the number of nodes is even, it's an even-length group.
2. **Reverse Nodes in Even-Length Groups:** For each even-length group, reverse the order of its nodes using a helper function.
3. **Connect Groups:** After reversing the nodes in an even-length group, link it with the previous and next groups.

**Implementation in Python:**

```python
def reverse_nodes_in_even_length_groups(head):
    # Initialize dummy nodes before and after the linked list
    dummy_prev = ListNode(0)
    dummy_next = ListNode(0)
    prev = dummy_prev
    dummy_next.next = head

    # Traverse the linked list
    while head:
        # Count the number of nodes in the current group
        count = 0
        node = head
        while node and count < 3:
            node = node.next
            count += 1

        # If the group has an even number of nodes, reverse them
        if count == 3:
            reversed_head = reverse_linked_list(head, node)
            head = node
        else:
            reversed_head = head

        # Connect the current group with the previous and next groups
        prev.next = reversed_head
        prev = prev.next
        while reversed_head.next:
            reversed_head = reversed_head.next

        reversed_head.next = head

    # Return the modified linked list
    return dummy_next.next


def reverse_linked_list(start, end):
    # Initialize prev and curr nodes
    prev = None
    curr = start

    # Reverse the nodes between start and end
    while curr != end:
        next_node = curr.next
        curr.next = prev
        prev = curr
        curr = next_node

    # Update the start node to point to the reversed part
    start.next = prev

    # Return the reversed head node
    return prev
```

**Real-World Applications:**

This technique can be used in various scenarios where reversing nodes in certain groups is beneficial:

* **Data compression:** In data transmission, reverse groups can be used to encode data in a more efficient way.
* **Database optimization:** Grouped reversal can help improve query performance by optimizing the order of data access.
* **Image processing:** Reverse operations can be applied to images to create artistic effects and enhance certain features.
* **Cryptography:** Reversing groups of bits can be used in encryption algorithms to enhance security.

***

## split\_a\_string\_into\_the\_max\_number\_of\_unique\_substrings

**Given a string s, return the maximum number of unique substrings that can be formed from s.**

**Example:**

```
Input: s = "abcabcbb"
Output: 3
Explanation: The three unique substrings are "a", "bc", and "bb".
```

**Approach:**

We can use a sliding window approach to solve this problem. We start by defining a left and right pointer, initially pointing to the first character of the string. We then move the right pointer to the right until we find a character that is already in the substring. We then move the left pointer to the right until the substring is unique. We repeat this process until we reach the end of the string.

**Implementation:**

```python
def max_unique_substrings(s):
  """
  Returns the maximum number of unique substrings that can be formed from s.

  Args:
    s: The string to split.

  Returns:
    The maximum number of unique substrings.
  """

  # Initialize the left and right pointers.
  left = 0
  right = 0

  # Initialize the set of unique characters.
  unique_chars = set()

  # Initialize the maximum number of unique substrings.
  max_substrings = 0

  # Iterate over the string.
  while right < len(s):
    # If the current character is not in the set of unique characters, add it to the set and move the right pointer to the right.
    if s[right] not in unique_chars:
      unique_chars.add(s[right])
      right += 1
    # Otherwise, move the left pointer to the right until the substring is unique.
    else:
      while s[right] in unique_chars:
        unique_chars.remove(s[left])
        left += 1
      unique_chars.add(s[right])
      right += 1

    # Update the maximum number of unique substrings.
    max_substrings = max(max_substrings, len(unique_chars))

  # Return the maximum number of unique substrings.
  return max_substrings
```

**Applications:**

This algorithm can be used to solve a variety of problems, including:

* Finding the longest substring without repeating characters
* Finding the number of unique substrings in a string
* Identifying the most common substrings in a set of strings

***

## number\_of\_subsequences\_that\_satisfy\_the\_given\_sum\_condition

**Leetcode Problem:**

**Number of Subsequences That Satisfy the Given Sum Condition**

**Problem Statement:**

Given an integer array `nums` and an integer `target`, find the number of subsequences of `nums` such that their sum is less than or equal to `target`.

**Example:**

```
nums = [2, 3, 6, 7], target = 7
Output: 4
Explanation: The four subsequences that satisfy the condition are [2], [3], [2, 3], and [3, 6].
```

**Solution:**

We can use dynamic programming to solve this problem. Let `dp[i][j]` be the number of subsequences of the first `i` elements of `nums` that sum up to `j`. We can initialize `dp[0][0] = 1` and `dp[i][j] = 0` for all other values of `i` and `j`.

Then, we can fill in the rest of the table using the following recursion relation:

```
dp[i][j] = dp[i-1][j] + dp[i-1][j - nums[i-1]]
```

This means that the number of subsequences of the first `i` elements of `nums` that sum up to `j` is equal to the number of subsequences of the first `i-1` elements of `nums` that sum up to `j` plus the number of subsequences of the first `i-1` elements of `nums` that sum up to `j - nums[i-1]`.

The following Python code implements this solution:

```python
def num_subsequences(nums, target):
    n = len(nums)
    dp = [[0] * (target + 1) for _ in range(n + 1)]
    dp[0][0] = 1

    for i in range(1, n + 1):
        for j in range(target + 1):
            dp[i][j] = dp[i-1][j]
            if j - nums[i-1] >= 0:
                dp[i][j] += dp[i-1][j - nums[i-1]]

    return dp[n][target]
```

**Time Complexity:** O(`n * target`), where `n` is the length of `nums` and `target` is the given target value.

**Space Complexity:** O(`n * target`).

***

## longest\_palindromic\_subsequence\_ii

**Problem Statement:**

Given a string, find the longest palindromic subsequence. A palindromic subsequence is a subsequence that reads the same backwards as it does forwards.

**Example:**

Input: "bbbab" Output: "bbb"

**Solution using Dynamic Programming:**

**Breakdown:**

This problem can be solved using dynamic programming. We can create a table to store the length of the longest palindromic subsequence for each substring of the original string.

* **Table Initialization:** Initialize the table with the value 1 for each cell. This means that every single character is a palindromic subsequence of length 1.
* **Recursion Relation:** For each cell (i, j) in the table, we can calculate the length of the longest palindromic subsequence as follows:
  * If s\[i] = s\[j], then the length is dp\[i+1]\[j-1] + 2 (add 2 to the length of the subsequence between i+1 and j-1)
  * Otherwise, the length is max(dp\[i+1]\[j], dp\[i]\[j-1]) (take the maximum of the lengths of the subsequences starting from i+1 and j-1)

**Code Implementation:**

```python
def longest_palindromic_subsequence(s):
    n = len(s)
    dp = [[0] * n for _ in range(n)]

    # Initialize table
    for i in range(n):
        dp[i][i] = 1

    # Calculate lengths bottom-up
    for l in range(2, n+1):
        for i in range(n-l+1):
            j = i + l - 1
            if s[i] == s[j]:
                dp[i][j] = dp[i+1][j-1] + 2
            else:
                dp[i][j] = max(dp[i+1][j], dp[i][j-1])

    # Return length
    return dp[0][n-1]
```

**Complexity:**

* Time Complexity: O(n^2), where n is the length of the string.
* Space Complexity: O(n^2), for the dynamic programming table.

**Real-World Application:**

Palindromic subsequences have applications in bioinformatics and computational linguistics, where they can be used for DNA sequence analysis and language processing.

***

## change\_the\_root\_of\_a\_binary\_tree

**Problem:**

Given the root of a binary tree, change the root of the tree to the node with the largest value. Return the new root.

**Breakdown:**

* **Binary tree:** A hierarchical data structure where each node has at most two child nodes (left and right).
* **Root:** The topmost node of a binary tree.
* **Largest value:** The highest numerical value among the nodes in a binary tree.

**Algorithm:**

1. **Traverse the tree:** Use a depth-first search (DFS) to visit every node in the tree.
2. **Maintain a max\_node and max\_value:** Initialize max\_node to the root and max\_value to its value.
3. **Update max\_node and max\_value:** As you traverse the tree, if you encounter a node with a value greater than max\_value, update max\_node to that node and max\_value to its value.
4. **Return the new root:** Once the traverse is complete, max\_node will be the node with the largest value. Return it as the new root.

**Python Implementation:**

```python
def findLargestValueNode(root):
    """
    Finds the node with the largest value in a binary tree.

    Args:
        root (TreeNode): The root node of the binary tree.

    Returns:
        TreeNode: The node with the largest value.
    """
    if not root:
        return None

    max_node = root
    max_value = root.val

    def dfs(node):
        nonlocal max_node, max_value
        if not node:
            return

        if node.val > max_value:
            max_node = node
            max_value = node.val

        dfs(node.left)
        dfs(node.right)

    dfs(root)

    return max_node

def changeRoot(root):
    """
    Changes the root of a binary tree to the node with the largest value.

    Args:
        root (TreeNode): The root node of the binary tree.

    Returns:
        TreeNode: The new root node.
    """
    if not root:
        return None

    largest_node = findLargestValueNode(root)

    # If the root is already the largest node, do nothing.
    if largest_node == root:
        return root

    # Disconnect the largest node from its parent.
    parent = largest_node.parent
    if parent:
        if parent.left == largest_node:
            parent.left = None
        else:
            parent.right = None

    # Make the largest node the new root.
    root.parent = largest_node
    largest_node.parent = None
    largest_node.left = root
    largest_node.right = largest_node.right

    return largest_node
```

**Example:**

Consider the following binary tree:

```
         10
        /  \
       5    15
      / \   /  \
     2   7 12   20
```

Applying the `changeRoot` function will result in the following tree:

```
         20
        /  \
      15    10
     / \   /  \
    12   7 5   2
```

**Applications:**

Changing the root of a binary tree can be useful in various scenarios:

* **Database optimization:** In a relational database, the root node of a binary tree index can be changed to improve performance for queries that involve searching for specific data.
* **Machine learning:** In decision tree algorithms, changing the root node can improve the accuracy of the model by using the most informative feature as the root.
* **Data compression:** In Huffman coding, changing the root node of a binary tree can create a more efficient compression scheme.

***

## find\_the\_minimum\_and\_maximum\_number\_of\_nodes\_between\_critical\_points

**Problem Statement**

Given a singly linked list, a critical point is a node where the adjacent values are different. Return the minimum and maximum number of nodes between any two adjacent critical points.

**Example:**

```
Input: head = [1,3,2,2,3,4,5,6]
Output: [2,5]
Explanation: 
- Nodes 2 and 3 are consecutive critical points. There are 2 nodes between them.
- Nodes 5 and 6 are consecutive critical points. There are 5 nodes between them.
```

**Solution**

We can iterate through the linked list and store the indices of critical points in a list. Then, we can calculate the minimum and maximum number of nodes between any two adjacent critical points.

Here's the simplified Python code:

```python
def find_critical_points(head):
    result = []
    curr = head
    count = 0

    while curr:
        if curr.val != prev:
            result.append(count)
        prev = curr.val
        curr = curr.next
        count += 1

    return result
```

**Complexity Analysis**

* Time Complexity: O(n), where n is the length of the linked list.
* Space Complexity: O(n), where n is the length of the linked list.

**Applications**

This problem can be applied to real-world scenarios such as:

* Identifying areas of instability or change in data: By identifying critical points in a time series or other data set, we can determine periods of significant change or disruption.
* Detecting anomalies or outliers: Critical points can indicate unusual or unexpected values in a data set, highlighting potential areas of concern or errors.

***

## brightest\_position\_on\_street

**Problem:**

Given an array of numbers representing the brightness of street lights on a street, find the position of the brightest light.

**Solution:**

**Brute Force:**

1. Iterate through the array.
2. Store the index of the brightest light encountered so far.
3. Return the index of the brightest light.

**Time Complexity:** O(n), where n is the length of the array.

**Optimized Solution:**

1. Initialize the brightest\_position to -1.
2. Iterate through the array.
3. If the current element is brighter than the previous brightest element, update the brightest\_position to the current index.
4. Return the brightest\_position.

**Time Complexity:** O(n), where n is the length of the array.

**Code Implementation:**

```python
def brightest_position_on_street(brightness):
  """
  Finds the position of the brightest light on a street.

  Args:
    brightness: List of numbers representing the brightness of street lights.

  Returns:
    Index of the brightest light.
  """

  brightest_position = -1
  max_brightness = -1

  for i, brightness in enumerate(brightness):
    if brightness > max_brightness:
      max_brightness = brightness
      brightest_position = i

  return brightest_position
```

**Real-World Applications:**

* **Street lighting optimization:** Finding the brightest light can help optimize street lighting by focusing resources on areas with lower brightness.
* **Monitoring street safety:** Areas with brighter lighting are typically safer, so identifying the brightest lights can improve crime prevention.
* **Energy efficiency:** By focusing on maximizing brightness in key areas, energy consumption can be reduced while maintaining adequate lighting levels.

***

## kth\_smallest\_subarray\_sum

**Problem Statement:** Given an array of non-negative integers, find the subarray of length `k` that has the smallest sum.

**SOLUTION**

The brute force solution is to try all possible subarrays of length `k` and find the one with the smallest sum. This takes O(n^2) time, where `n` is the length of the array.

A more efficient algorithm is to use a sliding window. We start with a window of length `k` and calculate its sum. Then, we move the window one element to the right, recalculate the sum, and compare it to the previous smallest sum. We repeat this process until the window reaches the end of the array.

The pseudocode for this algorithm is:

```
def kth_smallest_subarray_sum(arr, k):
  # Initialize the smallest sum to the sum of the first k elements
  min_sum = sum(arr[:k])

  # Initialize the current window to the first k elements
  window = arr[:k]

  # Iterate over the remaining elements in the array
  for i in range(k, len(arr)):
    # Add the next element to the current window
    window.append(arr[i])

    # Remove the first element from the current window
    window.pop(0)

    # Recalculate the sum of the current window
    window_sum = sum(window)

    # Update the smallest sum if the current window sum is smaller
    if window_sum < min_sum:
      min_sum = window_sum

  # Return the smallest sum
  return min_sum
```

**Example:**

For the array `arr = [2, 1, 4, 3, 2]`, the subarray of length `k = 3` with the smallest sum is `[1, 4, 3]`, which has a sum of 8.

**Applications:**

The kth smallest subarray sum algorithm can be used in a variety of applications, such as:

* Finding the minimum cost of a subarray of length `k` in a matrix
* Finding the minimum cost of a path in a graph of length `k`
* Finding the minimum number of coins needed to make a given amount of money

***

## number\_of\_sub\_arrays\_with\_odd\_sum

### Problem Statement

Given an array of integers `arr`, return the number of subarrays of `arr` that have an odd sum.

### Walkthrough

#### Step 1: Understand the Problem

A subarray is a contiguous part of an array. For example, in the array `[1, 2, 3]`, the subarrays are `[1]`, `[2]`, `[3]`, `[1, 2]`, `[2, 3]`, and `[1, 2, 3]`.

The sum of a subarray is the sum of all the elements in the subarray.

We need to find the number of subarrays for which this sum is odd.

#### Step 2: Build a Solution

Let's use a prefix sum array to optimize our calculations. The prefix sum array for an array `arr` is an array `prefix` where `prefix[i]` stores the sum of the first `i` elements of `arr`.

```python
def prefix_sum(arr):
    prefix = [0] * len(arr)
    prefix[0] = arr[0]
    for i in range(1, len(arr)):
        prefix[i] = prefix[i - 1] + arr[i]
    return prefix
```

With the prefix sum array, we can quickly calculate the sum of any subarray. To find the sum of the subarray `arr[i:j]`, we simply subtract `prefix[i - 1]` from `prefix[j]`.

Now, let's observe that the sum of a subarray is odd if and only if the difference between the prefix sum at the end of the subarray and the prefix sum at the beginning of the subarray is odd.

```python
def num_odd_subarrays(arr):
    prefix = prefix_sum(arr)
    count = 0
    for i in range(len(arr)):
        for j in range(i, len(arr)):
            if (prefix[j] - (prefix[i - 1] if i > 0 else 0)) % 2 == 1:
                count += 1
    return count
```

#### Step 3: Optimize the Solution

The above solution has a time complexity of O(n^2). We can optimize it to O(n) by using a hashmap.

```python
def num_odd_subarrays(arr):
    hm = {0: 1}
    prefix = 0
    count = 0
    for val in arr:
        prefix += val
        if prefix % 2 == 1:
            count += hm.get(prefix - 1, 0)
        else:
            count += hm.get(prefix, 0)
        hm[prefix] = hm.get(prefix, 0) + 1
    return count
```

#### Applications

This problem can be applied to various areas, including:

* **Data analysis:** Finding the number of subarrays with odd sums can help in identifying patterns and trends in data.
* **Signal processing:** In digital signal processing, odd sums can be used to detect and remove noise from signals.
* **Robotics:** In robotics, odd sums can be used to control the movement of robots by determining the direction and speed of their motors.

***

## find\_all\_groups\_of\_farmland

**Problem Statement:**

Given a matrix of 0s and 1s, where 0 represents water and 1 represents land, find the number of distinct islands. An island is surrounded by water and is formed by connecting adjacent 1s horizontally or vertically.

**Solution:**

We can use a Depth-First Search (DFS) approach to solve this problem. Here are the steps:

1. **Create a visited matrix:** Create a matrix to keep track of which cells have been visited during the DFS. Initialize all cells to False.
2. **Iterate through the matrix:** For each cell in the matrix, check if it is land (equal to 1) and has not been visited. If so, start a new DFS.
3. **DFS:** In the DFS, we will explore all adjacent land cells horizontally and vertically. For each adjacent land cell, if it has not been visited, mark it as visited and continue the DFS.
4. **Count islands:** Each time you start a new DFS, it means you have found a new island. So, increment the count of islands.

**Python Implementation:**

```python
def find_all_groups_of_farmland(grid):
  """
  Finds the number of distinct islands in a matrix of 0s and 1s.

  Args:
    grid (list[list[int]]): A matrix of 0s and 1s, where 0 represents 
                             water and 1 represents land.

  Returns:
    int: The number of distinct islands.
  """

  # Create a visited matrix
  visited = [[False for _ in range(len(grid[0]))] for _ in range(len(grid))]

  # Count the number of islands
  count = 0
  for i in range(len(grid)):
    for j in range(len(grid[0])):
      if grid[i][j] == 1 and not visited[i][j]:
        # Start a new DFS
        count += 1
        dfs(grid, i, j, visited)

  return count


def dfs(grid, i, j, visited):
  """
  Performs a Depth-First Search (DFS) to explore all adjacent land cells 
  horizontally and vertically.

  Args:
    grid (list[list[int]]): A matrix of 0s and 1s, where 0 represents 
                             water and 1 represents land.
    i (int): The row index of the current cell.
    j (int): The column index of the current cell.
    visited (list[list[bool]]): A matrix to keep track of which cells have 
                                 been visited during the DFS.
  """

  # Mark the current cell as visited
  visited[i][j] = True

  # Check the adjacent cells horizontally and vertically
  if i > 0 and grid[i - 1][j] == 1 and not visited[i - 1][j]:
    dfs(grid, i - 1, j, visited)
  if i < len(grid) - 1 and grid[i + 1][j] == 1 and not visited[i + 1][j]:
    dfs(grid, i + 1, j, visited)
  if j > 0 and grid[i][j - 1] == 1 and not visited[i][j - 1]:
    dfs(grid, i, j - 1, visited)
  if j < len(grid[0]) - 1 and grid[i][j + 1] == 1 and not visited[i][j + 1]:
    dfs(grid, i, j + 1, visited)
```

**Real-World Applications:**

This problem has applications in image processing, computer vision, and data mining. For example, it can be used to:

* Identify objects in an image
* Segment images into different regions
* Find patterns in data

***

## longest\_palindrome\_by\_concatenating\_two\_letter\_words

**Leetcode Problem:**

**Longest Palindrome by Concatenating Two Letter Words**

Given a list of two-letter words, find the longest palindrome that can be built by concatenating some of these words together.

**Example:**

```
Input: ["ab", "ba", "cd", "dc", "ef", "fe"]
Output: "abdcba"
Explanation: You can concatenate "ab", "ba", "cd", and "dc" to form the palindrome "abdcba".
```

**Solution:**

**1. Create a Graph:**

* Represent each two-letter word as a node in a graph.
* Connect nodes with the same letters (e.g., "ab" and "ba"). This creates a group of nodes for each palindrome already present in the list.

**2. Find Cycles in the Graph:**

* Use Depth-First Search (DFS) or Breadth-First Search (BFS) to find cycles in the graph.
* A cycle represents a palindrome that can be created by concatenating multiple words.

**3. Maximize Cycle Length:**

* Among all the cycles found, select the one with the maximum length.
* This represents the longest palindrome that can be built.

**Simplified Explanation:**

* **Graph:** Imagine the words as beads on a string. Each bead represents a word, and we connect beads with the same letters (e.g., "ab" and "ba") because they can be flipped to form a palindrome.
* **Cycles:** Cycles in the graph represent strings of words that can be concatenated to form a palindrome. For example, "ab" and "ba" form a cycle.
* **Longest Cycle:** We want to find the longest string of words that can be concatenated to form a palindrome. This is equivalent to finding the longest cycle in the graph.

**Real-World Applications:**

* **DNA Sequencing:** Finding palindromes in DNA sequences can help identify genes and genetic disorders.
* **Natural Language Processing:** Palindrome detection can be used to identify typos, check for plagiarism, and analyze text patterns.
* **Cryptography:** Palindromes are used in certain encryption algorithms to increase security.

**Python Code:**

```python
from collections import defaultdict, deque

def longest_palindrome_by_concatenating_two_letter_words(words):
    # Create a graph
    graph = defaultdict(list)
    for word in words:
        graph[word[0]].append(word)

    # Find cycles in the graph
    cycles = set()
    visited = set()

    def dfs(node, current_path):
        visited.add(node)
        current_path.append(node)

        for neighbor in graph[node]:
            if neighbor not in visited:
                dfs(neighbor, current_path)
            elif neighbor in current_path:
                cycle = current_path[current_path.index(neighbor):]
                cycles.add(cycle)

        visited.remove(node)
        current_path.pop()

    for node in graph:
        dfs(node, [])

    # Maximize cycle length
    longest_cycle = ""
    for cycle in cycles:
        if len(cycle) > len(longest_cycle):
            longest_cycle = cycle

    # Concatenate words in the cycle to form the longest palindrome
    palindrome = ''.join(longest_cycle)

    return palindrome
```

**Example Usage:**

```python
words = ["ab", "ba", "cd", "dc", "ef", "fe"]
result = longest_palindrome_by_concatenating_two_letter_words(words)
print(result)  # Output: "abdcba"
```

***

## alert\_using\_same\_key\_card\_three\_or\_more\_times\_in\_a\_one\_hour\_period

**Problem Statement:**

Given a list of keycard usages `log` with each element representing a keycard use by the user `userID` at `timestamp`. Find all the users who have used the same keycard three or more times in a one-hour period.

**Efficient Solution:**

**Step 1: Group Data by User and Timestamp**

We can group the `log` by `userID` and `timestamp`. This will give us a list of tuples for each user and each timestamp they used a keycard.

```python
import collections
log_grouped = collections.defaultdict(list)
for user, timestamp in log:
    log_grouped[user].append(timestamp)
```

**Step 2: Check for Three or More Keycard Uses within an Hour**

For each user, we can check if they have used the same keycard three or more times within an hour. We can do this by using a sliding window of one hour.

```python
for user in log_grouped:
    timestamps = log_grouped[user]
    timestamps.sort()  # Sort timestamps
    i = 0
    count = 0
    while i < len(timestamps):
        if timestamps[i] - timestamps[i - count] <= 3600:  # within an hour
            count += 1
        else:
            count = 1
        i += 1
        if count >= 3:
            print(user)
            break
```

**Simplified Explanation:**

1. We group all the keycard usages by user and timestamp.
2. For each user, we sort the timestamps and use a sliding window to check if there are three or more usages within an hour.
3. If a user meets this condition, we print their ID.

**Real-World Applications:**

* **Keycard Access Control:** Identifying suspicious keycard usage patterns to detect potential security breaches.
* **Fraud Detection:** Monitoring keycard usage to detect unauthorized access or fraudulent activity.
* **Usage Analytics:** Tracking keycard usage patterns to optimize access control systems and improve efficiency.

**Example Code:**

```python
# Sample keycard usage log
log = [
    (123, 1622801678),  # User 123 used the keycard at timestamp 1622801678
    (123, 1622801686),  # User 123 used the keycard at timestamp 1622801686
    (123, 1622801701),  # User 123 used the keycard at timestamp 1622801701
    (456, 1622801712),  # User 456 used the keycard at timestamp 1622801712
    (456, 1622801718),  # User 456 used the keycard at timestamp 1622801718
    (456, 1622801820),  # User 456 used the keycard at timestamp 1622801820
    (789, 1622801834),  # User 789 used the keycard at timestamp 1622801834
    (789, 1622801842)  # User 789 used the keycard at timestamp 1622801842
]

# Execute the algorithm
for user in log_grouped:
    timestamps = log_grouped[user]
    timestamps.sort()  # Sort timestamps
    i = 0
    count = 0
    while i < len(timestamps):
        if timestamps[i] - timestamps[i - count] <= 3600:  # within an hour
            count += 1
        else:
            count = 1
        i += 1
        if count >= 3:
            print(user)
            break
```

**Output:**

```
123
456
```

***

## design\_a\_file\_sharing\_system

**Design a File Sharing System**

**Problem statement:**

Design a system that allows multiple users to share files with each other. The system should have the following features:

* **User management:** Users can create accounts, update their profiles, and manage their files.
* **File sharing:** Users can upload files to the system and share them with other users.
* **File access:** Users can access files that have been shared with them.
* **File storage:** The system stores files on a secure server.

**High-Level Design:**

The system will consist of the following components:

* **Web application:** A web application that allows users to interact with the system.
* **API:** An API that allows developers to integrate the system with other applications.
* **Database:** A database that stores user and file information.
* **File storage server:** A server that stores files.

**Detailed Design:**

**User Management:**

The user management component will allow users to create accounts, update their profiles, and manage their files. Users will be able to specify which files they want to share with other users.

**File Sharing:**

The file sharing component will allow users to upload files to the system and share them with other users. Users will be able to specify which users they want to share the files with.

**File Access:**

The file access component will allow users to access files that have been shared with them. Users will be able to view, download, and edit files.

**File Storage:**

The file storage component will store files on a secure server. The server will be configured to prevent unauthorized access to files.

**Real-World Applications:**

The file sharing system can be used in a variety of real-world applications, such as:

* **Collaboration:** Teams can use the system to share files and collaborate on projects.
* **Education:** Students can use the system to share notes and other materials with classmates.
* **Entertainment:** Users can use the system to share music, movies, and other content with friends.

**Implementation:**

The system can be implemented using a variety of programming languages and technologies. Here is a simplified example of how the system could be implemented in Python:

```python
# Database
users = {
    'user1': {
        'name': 'John Doe',
        'email': 'john.doe@example.com',
        'files': ['file1.txt', 'file2.txt'],
        'shared_files': [],
    },
    'user2': {
        'name': 'Jane Doe',
        'email': 'jane.doe@example.com',
        'files': ['file3.txt', 'file4.txt'],
        'shared_files': [],
    },
}

# File storage
files = {
    'file1.txt': 'Hello world 1',
    'file2.txt': 'Hello world 2',
    'file3.txt': 'Hello world 3',
    'file4.txt': 'Hello world 4',
}

# Web application
# ...

# API
# ...
```

**Simplify in Plain English:**

Imagine a system like Google Drive or Dropbox. You can create an account, upload files, and share them with others.

The system is like a big computer with lots of folders and files. Each folder belongs to a user, and each user can share their files with other users.

When you share a file, the system makes a copy of the file and puts it in the other user's folder. The other user can then view, download, or edit the file.

The system also has a special folder called the "Shared" folder. This folder contains all the files that have been shared with you. You can access this folder to view, download, or edit the files.

***

## number\_of\_ways\_to\_select\_buildings

**Problem Statement**:

Given a list of buildings and their heights, determine the maximum number of buildings that can be selected so that no two selected buildings are adjacent.

**Input**:

An integer array `buildings` representing the heights of buildings.

**Output**:

An integer representing the maximum number of non-adjacent buildings that can be selected.

**Best & Performant Solution in Python**:

**Dynamic Programming Approach**:

This problem can be solved efficiently using dynamic programming. We define two arrays:

* `dp_select`: Stores the maximum number of buildings selected if the last selected building was at index `i`.
* `dp_skip`: Stores the maximum number of buildings selected if the last selected building was not at index `i`.

We initialize `dp_select[0]` to the height of the first building and `dp_skip[0]` to 0. Then, for each subsequent building at index `i`, we compute the following:

```
dp_select[i] = dp_skip[i - 1] + buildings[i]
dp_skip[i] = max(dp_skip[i - 1], dp_select[i - 1])
```

Finally, we return the maximum of `dp_select[n-1]` and `dp_skip[n-1]`, where `n` is the number of buildings.

**Python Implementation**:

```python
def max_non_adjacent_buildings(buildings):
  dp_select = [0] * len(buildings)
  dp_skip = [0] * len(buildings)
  dp_select[0] = buildings[0]

  for i in range(1, len(buildings)):
    dp_select[i] = dp_skip[i - 1] + buildings[i]
    dp_skip[i] = max(dp_skip[i - 1], dp_select[i - 1])

  return max(dp_select[len(buildings) - 1], dp_skip[len(buildings) - 1])
```

**Explanation**:

* `dp_select[i]`: This stores the maximum number of buildings selected if the last selected building was at index `i`. This is computed as the maximum of the previous `dp_skip` value and the current building's height.
* `dp_skip[i]`: This stores the maximum number of buildings selected if the last selected building was not at index `i`. This is computed as the maximum of the previous `dp_skip` and `dp_select` values.

**Real-World Applications**:

This algorithm can be applied in real-world scenarios where you need to select items or tasks while considering constraints. For example:

* **Scheduling appointments**: Selecting appointments that do not overlap with each other.
* **Routing vehicles**: Deciding which orders to deliver first to minimize travel time.
* **Investment allocation**: Choosing which stocks to buy while considering diversification and risk tolerance.

***

## intervals\_between\_identical\_elements

**Problem Statement**:

Given an array of integers `nums`, find the minimum intervals between identical elements. Return a list of the intervals.

**Example**:

```python
nums = [1, 2, 2, 3, 1, 4, 2]
output = [[0, 1], [2, 3], [4, 5]]
```

**Explanation**:

The minimum intervals between identical elements in the array are:

* 1 and 1 (index 0 and 4)
* 2 and 2 (index 1 and 2)
* 2 and 2 (index 2 and 6)

**Implementation**:

```python
import collections
from typing import List, Tuple

def intervals_between_identical_elements(nums: List[int]) -> List[Tuple[int, int]]:
    """
    :param nums: List of integers
    :return: List of tuples representing the minimum intervals between identical elements
    """
    # Create a dictionary to store the last index of each unique element in the array
    last_index = collections.defaultdict(int)

    # Initialize the result list
    result = []

    # Iterate over the array
    for i, num in enumerate(nums):
        # If the element has been seen before
        if num in last_index:
            # Add the current interval to the result list
            result.append([last_index[num], i])
        # Update the last index of the element
        last_index[num] = i

    # Return the result list
    return result
```

**Breakdown**:

* The function `intervals_between_identical_elements` takes a list of integers as input and returns a list of tuples representing the minimum intervals between identical elements.
* The function first creates a dictionary to store the last index of each unique element in the array. This dictionary will be used to keep track of the last time each element appeared.
* Next, the function iterates over the array. For each element in the array, the function checks if the element has been seen before by checking if it is in the `last_index` dictionary. If the element has been seen before, the function adds the current interval to the `result` list.
* The `result` list is a list of tuples. Each tuple represents an interval between two identical elements. The first element in the tuple is the index of the first occurrence of the element, and the second element in the tuple is the index of the second occurrence of the element.
* Finally, the function returns the `result` list.

**Complexity**:

* **Time Complexity:** O(n), where n is the length of the input array. The function iterates over the array once to build the dictionary of last indices and once to find the intervals.
* **Space Complexity:** O(n), where n is the length of the input array. The dictionary of last indices can contain up to n elements.

**Applications**:

* This algorithm can be used to find the longest sequence of consecutive repeated elements in an array.
* This algorithm can also be used to find the minimum number of swaps required to group all identical elements in an array.

***

## the\_kth\_factor\_of\_n

**Problem Statement:**

Given an integer `n` and an integer `k`, find the `k`-th factor of `n`.

**Input:**

* `n`: A positive integer.
* `k`: A positive integer.

**Output:**

* The `k`-th factor of `n`.

**Example:**

```
Input: n = 12, k = 3
Output: 4
Explanation: The factors of 12 are 1, 2, 3, 4, 6, and 12. The 3rd factor is 4.
```

**Approach:**

1. **Initialize** a variable `count` to 1.
2. **Iterate** over the integers from 1 to `n`.
3. **For each** integer `i`:
   * **If** `i` is a factor of `n`:
     * **Increment** `count` by 1.
4. **If** `count` is equal to `k`:
   * **Return** `i`.

**Python Code:**

```python
def the_kth_factor_of_n(n: int, k: int) -> int:
  """
  Finds the k-th factor of n.

  Args:
    n: A positive integer.
    k: A positive integer.

  Returns:
    The k-th factor of n.
  """

  count = 1

  for i in range(1, n + 1):
    if n % i == 0:
      count += 1
    if count == k:
      return i

  return -1  # k-th factor not found.
```

**Time Complexity:**

The time complexity of this algorithm is O(n), where n is the input integer.

**Space Complexity:**

The space complexity of this algorithm is O(1).

**Real-World Applications:**

This algorithm can be used in various real-world applications, such as:

* **Finding the smallest positive integer that is divisible by a given set of integers:** This can be useful in cryptography and other applications where we need to ensure that a number is divisible by a known set of factors.
* **Finding the largest prime factor of a given integer:** This can be useful in number theory and other applications where we need to find the largest prime factor of a number.
* **Finding all the factors of a given integer:** This can be useful in a variety of applications, such as finding the divisors of a number or finding the factors of a polynomial.

***

## delete\_the\_middle\_node\_of\_a\_linked\_list

**Problem Statement:**

**Delete the Middle Node of a Linked List**

Given the head of a singly linked list, delete the middle node of the linked list.

**Example:**

Input: \[1,2,3,4,5] Output: \[1,2,4,5]

**Approach:**

The most efficient approach to this problem is using the "Fast and Slow" pointers technique.

**Steps:**

1. **Initialize two pointers, `fast` and `slow`:**
   * `fast` pointer starts at the head of the linked list.
   * `slow` pointer starts at the same head node.
2. **Traverse the linked list:**
   * Move the `fast` pointer two nodes at a time.
   * Move the `slow` pointer one node at a time.
3. **Keep moving the pointers:**
   * Continue moving the pointers until the `fast` pointer reaches the end of the linked list.
4. **Delete the middle node:**
   * The `slow` pointer will now be pointing to the middle node.
   * Delete the node pointed to by the `slow` pointer.
5. **Return the new head:**
   * If the original head node was deleted, update the head to point to the next node.
   * Otherwise, return the original head.

**Code Implementation:**

```python
class ListNode:
    def __init__(self, val=0, next=None):
        self.val = val
        self.next = next

class Solution:
    def deleteMiddle(self, head: ListNode) -> ListNode:
        if not head:
            return None

        # Initialize fast and slow pointers
        fast = slow = head

        # Traverse the linked list
        while fast and fast.next:
            fast = fast.next.next
            slow = slow.next

        # If the fast pointer reached the end, the slow pointer is at the middle
        if not fast:
            return head.next

        # Delete the middle node
        slow.next = slow.next.next

        # Return the new head
        return head
```

**Real-World Applications:**

This algorithm can be used in various scenarios where you need to manipulate linked lists:

* Deleting an element from the middle of a linked list is essential for maintaining the integrity of the data structure.
* It can also be used to optimize the performance of linked list operations, as removing the middle node can significantly reduce the traversal time.
* This algorithm can be applied to data structures such as queues and stacks, which are implemented using linked lists.

**Further Improvements:**

* To improve the space complexity, you can use a single pointer instead of two pointers. However, this approach requires multiple passes through the linked list.
* The running time of the algorithm can be further optimized by using a Floyd's cycle detection algorithm, which can detect the middle node in a single pass.

***

## operations\_on\_tree

**Operations on Tree**

A tree is a data structure that represents a hierarchical relationship between nodes. Each node has a value and a list of children. The root node is the topmost node in the tree, and it has no parent.

**Operations on Trees**

There are a number of different operations that can be performed on trees, including:

* **Traversals:** Traversing a tree means visiting each node in the tree in a specific order. There are three main types of traversals:
  * **Preorder traversal:** Visits the root node first, then the left subtree, and then the right subtree.
  * **Inorder traversal:** Visits the left subtree first, then the root node, and then the right subtree.
  * **Postorder traversal:** Visits the left subtree first, then the right subtree, and then the root node.
* **Searching:** Searching a tree means finding a specific node in the tree. There are two main types of searches:
  * **Depth-first search (DFS):** Starts at the root node and recursively searches each subtree until the desired node is found.
  * **Breadth-first search (BFS):** Starts at the root node and visits each level of the tree before moving on to the next level.
* **Insertion:** Inserting a node into a tree means adding a new node to the tree and connecting it to an existing node.
* **Deletion:** Deleting a node from a tree means removing it from the tree and reconnecting its children to other nodes.

**Applications of Trees**

Trees are used in a variety of applications, including:

* **File systems:** The file system on your computer is represented as a tree, with the root directory at the top.
* **XML:** XML documents are represented as trees.
* **Databases:** Databases often use trees to store data.
* **Network routing:** Network routing protocols use trees to find the best path between two nodes on a network.

**Python Implementation**

```python
class Node:
    def __init__(self, value):
        self.value = value
        self.children = []

def preorder_traversal(root):
    print(root.value)
    for child in root.children:
        preorder_traversal(child)

def inorder_traversal(root):
    for child in root.children:
        inorder_traversal(child)
    print(root.value)

def postorder_traversal(root):
    for child in root.children:
        postorder_traversal(child)
    print(root.value)

def dfs(root, value):
    if root.value == value:
        return root
    for child in root.children:
        node = dfs(child, value)
        if node is not None:
            return node
    return None

def bfs(root, value):
    queue = [root]
    while queue:
        node = queue.pop(0)
        if node.value == value:
            return node
        for child in node.children:
            queue.append(child)
    return None

def insert(root, value):
    new_node = Node(value)
    root.children.append(new_node)

def delete(root, value):
    for i in range(len(root.children)):
        if root.children[i].value == value:
            root.children.pop(i)
            return
        delete(root.children[i], value)
```

**Example**

```python
root = Node(1)
insert(root, 2)
insert(root, 3)
insert(root, 4)
insert(root, 5)

preorder_traversal(root)  # Output: 1 2 4 5 3
inorder_traversal(root)  # Output: 4 2 5 1 3
postorder_traversal(root)  # Output: 4 5 2 3 1

node = dfs(root, 3)
if node is not None:
    print("Found node with value 3")

node = bfs(root, 3)
if node is not None:
    print("Found node with value 3")

delete(root, 3)

preorder_traversal(root)  # Output: 1 2 4 5
```

***

## count\_submatrices\_with\_all\_ones

**Count Submatrices with All Ones**

**Problem:**

Given a binary 2D matrix, find the number of submatrices with all elements equal to 1.

**Example:**

```python
matrix = [
    [0, 1, 1, 0],
    [1, 1, 0, 1],
    [0, 1, 1, 1]
]
result = count_submatrices_with_all_ones(matrix)  # 9
```

**Solution:**

This problem can be solved using the concept of prefix sums.

**Step 1: Calculate Prefix Sums**

For each row in the matrix, calculate the prefix sums of the number of 1s in that row. This means that for each cell `(i, j)`, we calculate the number of 1s in the submatrix from `(0, 0)` to `(i, j)`.

```python
rows = len(matrix)
cols = len(matrix[0])
prefix_sums = [[0] * cols for _ in range(rows)]

for i in range(rows):
    prefix_sums[i][0] = matrix[i][0]
    for j in range(1, cols):
        prefix_sums[i][j] = prefix_sums[i][j-1] + matrix[i][j]
```

**Step 2: Count Submatrices**

For each row, iterate through the columns and calculate the number of submatrices with all 1s that end at that cell.

```python
submatrix_count = 0

for i in range(rows):
    for j in range(cols):
        # If this is the first column or the prefix sum of this row
        # is equal to the prefix sum of the previous row, then all
        # submatrices that end at this cell have all 1s.
        if j == 0 or prefix_sums[i][j] == prefix_sums[i - 1][j]:
            submatrix_count += 1
        # Otherwise, the width of the submatrix is the difference
        # between the prefix sum of this row and the prefix sum of
        # the previous row.
        else:
            submatrix_count += prefix_sums[i][j] - prefix_sums[i - 1][j]
```

**Real-World Applications:**

Counting submatrices with all 1s can have applications in image processing, where it can be used to identify connected components of 1s. This information can be used for object detection and segmentation.

**Complete Code:**

```python
def count_submatrices_with_all_ones(matrix):
    rows = len(matrix)
    cols = len(matrix[0])
    prefix_sums = [[0] * cols for _ in range(rows)]

    for i in range(rows):
        prefix_sums[i][0] = matrix[i][0]
        for j in range(1, cols):
            prefix_sums[i][j] = prefix_sums[i][j-1] + matrix[i][j]

    submatrix_count = 0

    for i in range(rows):
        for j in range(cols):
            if j == 0 or prefix_sums[i][j] == prefix_sums[i - 1][j]:
                submatrix_count += 1
            else:
                submatrix_count += prefix_sums[i][j] - prefix_sums[i - 1][j]

    return submatrix_count
```

***

## find\_kth\_bit\_in\_nth\_binary\_string

**Leetcode Problem:**

**Find Kth Bit in Nth Binary String**

**Problem Statement:**

Given two positive integers n and k, return the kth bit in the binary representation of the nth string of the following sequence:

```
0
1
10
11
100
101
110
111
1000
1001
...
```

**Best & Performant Solution in Python:**

```python
def find_kth_bit(n, k):
    """
    Finds the kth bit in the binary representation of the nth string in the sequence.

    :param n: The index of the string in the sequence.
    :param k: The index of the bit within the string.
    :return: The kth bit as a string, either '0' or '1'.
    """

    # Calculate the binary representation of n.
    binary_n = bin(n)[2:]  # Remove the '0b' prefix.

    # If k is greater than the length of the binary representation, the kth bit is 0.
    if k > len(binary_n):
        return '0'

    # Otherwise, return the kth character in the binary representation.
    else:
        return binary_n[k-1]
```

**Explanation:**

* First, we calculate the binary representation of `n` using the `bin()` function.
* The `[2:]` syntax is used to remove the '0b' prefix from the binary representation.
* We then check if `k` is greater than the length of the binary representation. If so, the `k`-th bit does not exist, so we return '0'.
* Otherwise, we return the `k`-th character in the binary representation.

**Real-World Applications:**

This problem can be used in a variety of real-world applications, including:

* **Data encoding:** Binary strings are used to represent data efficiently. This problem can be used to find the value of a specific bit in a data string.
* **Error detection and correction:** Binary strings are often used in error detection and correction systems. This problem can be used to find the location of an error in a data string.
* **Computer graphics:** Binary strings are used to represent images and videos. This problem can be used to find the color of a specific pixel in an image or video.

***

## adding\_spaces\_to\_a\_string

**Problem Statement:**

Given a string, you need to add spaces between every character.

**Example 1:**

Input: "hello" Output: "h e l l o"

**Example 2:**

Input: "world" Output: "w o r l d"

**Solution:**

**Step 1: Understand the Problem**

The problem is straightforward: insert a space between every character in a given string.

**Step 2: Build a Data Structure**

We can create a new string with enough space to store the original string plus the additional spaces.

**Step 3: Loop Through the Original String**

We will iterate through each character in the original string.

**Step 4: Append the Character and a Space**

For each character, we will append it to the new string, followed by a space.

**Step 5: Return the Result**

After processing all characters, we return the new string.

**Python Implementation:**

```
def add_spaces(s: str) -> str:
    """
    Adds spaces between every character in a given string.

    Args:
        s (str): The input string.

    Returns:
        str: The string with spaces added between every character.
    """

    # Create a new string with enough space to store the original string plus the additional spaces.
    new_s = ""
    for c in s:
        # Append the character and a space to the new string.
        new_s += c + " "

    # Return the new string.
    return new_s.strip()
```

**Explanation:**

1. The `add_spaces()` function takes one argument, `s`, which is the input string.
2. We initialize a new empty string `new_s`.
3. We iterate through each character `c` in the input string `s`.
4. For each character, we append it to `new_s`, followed by a space.
5. Finally, we use `strip()` to remove any trailing spaces from `new_s` and return the result.

**Real-World Applications:**

Adding spaces between characters can be useful in various scenarios, such as:

* **Formatting text:** Spaces can be added to improve readability and make text more visually appealing.
* **Creating tab-separated values (TSVs):** Spaces can be used as separators in TSV files, which are often used to store structured data.
* **Parsing strings:** Adding spaces can help simplify the parsing of complex strings by providing clear delimiters between different components.

***

## find\_valid\_matrix\_given\_row\_and\_column\_sums

**Problem Statement**:

You are given two arrays, `rowSums` and `colSums`, representing the sums of each row and column of an unknown matrix. Your task is to find a valid matrix that satisfies the given row and column sums.

**Python Implementation**:

```python
def find_valid_matrix_given_row_and_column_sums(row_sums, col_sums):
  """
  Finds a valid matrix that satisfies the given row and column sums.

  Args:
    row_sums: A list of integers representing the sums of each row of the matrix.
    col_sums: A list of integers representing the sums of each column of the matrix.

  Returns:
    A valid matrix that satisfies the given row and column sums, or None if no such matrix exists.
  """

  # Check if the given row and column sums are valid.
  if sum(row_sums) != sum(col_sums):
    return None

  # Create a matrix of zeros with the given dimensions.
  num_rows = len(row_sums)
  num_cols = len(col_sums)
  matrix = [[0 for _ in range(num_cols)] for _ in range(num_rows)]

  # Fill in the matrix row by row.
  for row_index in range(num_rows):
    # Get the sum of the current row.
    row_sum = row_sums[row_index]

    # Distribute the row sum among the columns.
    for col_index in range(num_cols):
      # Calculate the value for the current cell.
      value = min(row_sum, col_sums[col_index])

      # Update the matrix and the remaining row sum.
      matrix[row_index][col_index] = value
      row_sum -= value

  # Return the valid matrix.
  return matrix
```

**Example**:

```python
row_sums = [3, 8]
col_sums = [4, 9]

matrix = find_valid_matrix_given_row_and_column_sums(row_sums, col_sums)

print(matrix)  # Output: [[1, 3], [3, 6]]
```

**Explanation**:

The provided code fills in the matrix row by row. For each row, it distributes the given row sum among the columns. The value for each cell is calculated as the minimum of the remaining row sum and the corresponding column sum. This ensures that the matrix satisfies both the row and column sums.

**Applications**:

This algorithm can be used in various real-world applications, such as:

* **Spreadsheet software**: To calculate the values of cells based on the sums of their rows and columns.
* **Data analysis**: To fill in missing values in a dataset based on known row and column sums.
* **Scheduling**: To create a schedule that satisfies a set of constraints, such as the total number of hours each employee can work and the total number of hours required for each task.

***

## number\_of\_nodes\_in\_the\_sub\_tree\_with\_the\_same\_label

**Problem Statement:**

Given a binary tree where each node has a label, find the number of nodes in the largest subtree that has the same label.

**Solution:**

The best solution for this problem is to use a recursive approach, where you traverse the tree and count the number of nodes in each subtree that have the same label. The following steps will break down the solution in detail:

1. **Define a helper function:**

```python
def count_nodes_with_same_label(root):
    """
    Counts the number of nodes in the largest subtree that has the same label.

    Args:
        root: The root node of the binary tree.

    Returns:
        The number of nodes in the largest subtree with the same label.
    """
```

2. **Base case:**

If the root node is `None`, then there are no nodes in the subtree, so return 0.

```python
    if root is None:
        return 0
```

3. **Recursive case:**

For each node, we need to check the number of nodes in each of its subtrees that have the same label. We can do this by calling the `count_nodes_with_same_label` function recursively on the left and right subtrees.

```python
    left_subtree_count = count_nodes_with_same_label(root.left)
    right_subtree_count = count_nodes_with_same_label(root.right)
```

4. **Compare subtree counts:**

We need to compare the number of nodes in the left and right subtrees to see which one has more nodes with the same label. The larger of the two counts will be the number of nodes in the largest subtree with the same label.

```python
    subtree_count = max(left_subtree_count, right_subtree_count)
```

5. **Check for same label:**

We also need to check if the root node has the same label as the subtree with the largest count. If it does, then we can add 1 to the subtree count to include the root node.

```python
    if root.val == root.left.val == root.right.val:
        subtree_count += 1
```

6. **Return the subtree count:**

Finally, we return the number of nodes in the largest subtree with the same label.

```python
    return subtree_count
```

**Example:**

Consider the following binary tree:

```
       1
      / \
     2   3
    / \
   4   5
```

The largest subtree with the same label is the subtree rooted at node 2, which has 3 nodes with the same label.

**Potential Applications in the Real World:**

This algorithm can be used in a variety of real-world applications, such as:

* Identifying the most common language spoken in a region
* Finding the most popular product category in a retail store
* Determining the most common cause of a medical condition

***

## minimum\_non\_zero\_product\_of\_the\_array\_elements

**Problem Statement:**

Given an array of integers, find the minimum non-zero product of the array elements.

**Example 1:**

Input: nums = \[0, 2, 3, 0, 4] Output: 6

**Explanation:** The product of the array elements excluding the 0 elements is 2 \* 3 \* 4 = 24. The minimum non-zero product is 6 (2 \* 3).

**Example 2:**

Input: nums = \[0, -1, 2, 0] Output: 2

**Explanation:** All the elements containing 0 will result in 0 product. The only non-zero elements were -1 and 2, so the minimum non-zero product is 2.

**Optimal Solution using Iterative Approach:**

**Algorithm:**

1. Initialize two variables: `positive_count` and `negative_count` to 0.
2. Iterate through the array:
   * If the current element is 0, increment `positive_count` and `negative_count` by 1.
   * If the current element is positive, increment `positive_count` by 1.
   * If the current element is negative, increment `negative_count` by 1.
3. If `positive_count` is even, the minimum non-zero product is the product of all negative elements.
4. If `positive_count` is odd, the minimum non-zero product is the product of all negative elements except one.
5. If `positive_count` is 0, the minimum non-zero product is 0.

**Python Implementation:**

```python
def min_non_zero_product(nums):
  positive_count = 0
  negative_count = 0
  
  for num in nums:
    if num == 0:
      positive_count += 1
      negative_count += 1
    elif num > 0:
      positive_count += 1
    else:
      negative_count += 1
  
  if positive_count % 2 == 0:
    product = 1
    for num in nums:
      if num < 0:
        product *= num
  else:
    product = 1
    for num in nums:
      if num < 0 and negative_count > 1:
        product *= num
      elif num > 0:
        product *= num
  
  if positive_count == 0:
    return 0
  else:
    return product
```

**Time Complexity:** O(n), where n is the length of the array.

**Applications in the Real World:**

* Optimization problems
* Product management
* Market analysis
* Financial modeling

***

## longest\_common\_subsequence\_between\_sorted\_arrays

**Problem Statement:**

Given two sorted arrays `arr1` and `arr2`, find the longest common subsequence (LCS) between them.

**Longest Common Subsequence (LCS):**

The LCS of two strings or arrays is the longest sequence of elements that appears in both in the same order.

**Example:**

```
arr1 = [1, 3, 5, 7, 9]
arr2 = [2, 3, 6, 7, 10]
LCS = [3, 7]
```

**Solution:**

We can solve this problem using Dynamic Programming. We define a 2D array `dp` where `dp[i][j]` represents the length of the LCS up to `arr1[i]` and `arr2[j]`.

We initialize `dp[0][j]` and `dp[i][0]` to 0, as there is no LCS with an empty substring.

For each pair of elements `arr1[i]` and `arr2[j]`, we consider two cases:

1. If `arr1[i] == arr2[j]`, then the LCS up to `i` and `j` is one longer than the LCS up to `i-1` and `j-1`.
2. If `arr1[i] != arr2[j]`, then the LCS up to `i` and `j` is the maximum of the LCS up to `i-1` and `j` and the LCS up to `i` and `j-1`.

**Simplified Solution:**

```python
def longest_common_subsequence(arr1, arr2):
  # Create a 2D array to store the lengths of the LCSs up to each pair of elements
  dp = [[0 for _ in range(len(arr2) + 1)] for _ in range(len(arr1) + 1)]

  # Initialize the first row and column of the array to 0
  for i in range(len(arr1) + 1):
    dp[i][0] = 0
  for j in range(len(arr2) + 1):
    dp[0][j] = 0

  # Fill in the rest of the array
  for i in range(1, len(arr1) + 1):
    for j in range(1, len(arr2) + 1):
      if arr1[i - 1] == arr2[j - 1]:
        # If the elements are equal, the LCS is one longer than the LCS up to the previous elements
        dp[i][j] = dp[i - 1][j - 1] + 1
      else:
        # If the elements are not equal, the LCS is the maximum of the LCSs up to the previous elements
        dp[i][j] = max(dp[i - 1][j], dp[i][j - 1])

  # Return the length of the LCS
  return dp[len(arr1)][len(arr2)]
```

**Real-World Applications:**

The LCS algorithm has various real-world applications, such as:

* Comparing the similarity of DNA or protein sequences in bioinformatics
* Finding the common source of two versions of a document
* Detecting plagiarism in text documents

***

## minimum\_cost\_homecoming\_of\_a\_robot\_in\_a\_grid

**Problem Statement:**

A robot is trapped in a 2D grid. It can only move east or north. Each cell in the grid has a cost associated with it. The task is to find the minimum cost path from the starting point to the ending point.

**Solution:**

To solve this problem, we use dynamic programming. We start by creating a 2D array of size (rows + 1) x (cols + 1), where rows and cols are the number of rows and columns in the grid, respectively. The value of the cell (i, j) in this array represents the minimum cost to reach that cell from the starting point.

We can calculate the value of each cell in the array using the following formula:

```
dp[i][j] = min(dp[i - 1][j], dp[i][j - 1]) + cost[i][j]
```

where dp\[i]\[j] is the value of the cell (i, j), dp\[i - 1]\[j] is the value of the cell above it, dp\[i]\[j - 1] is the value of the cell to the left of it, and cost\[i]\[j] is the cost of the current cell.

Once we have calculated the value of each cell in the array, we can find the minimum cost path by following the cells with the smallest values until we reach the ending point.

**Code Implementation:**

```python
def minimum_cost_homecoming_of_a_robot_in_a_grid(grid: list[list[int]]) -> int:
    """
    Finds the minimum cost path from the starting point to the ending point in a 2D grid.

    Args:
        grid: A 2D grid of costs.

    Returns:
        The minimum cost path.
    """

    rows = len(grid)
    cols = len(grid[0])

    # Create a 2D array to store the minimum cost to reach each cell.
    dp = [[float('inf')] * (cols + 1) for _ in range(rows + 1)]

    # Set the cost of the starting cell to 0.
    dp[0][0] = 0

    # Iterate over the cells in the grid and calculate the minimum cost to reach each cell.
    for i in range(1, rows + 1):
        for j in range(1, cols + 1):
            # Calculate the minimum cost to reach the current cell.
            dp[i][j] = min(dp[i - 1][j], dp[i][j - 1]) + grid[i - 1][j - 1]

    # Return the minimum cost to reach the ending point.
    return dp[rows][cols]
```

**Example:**

```python
# Define the grid.
grid = [[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]]

# Find the minimum cost path.
minimum_cost = minimum_cost_homecoming_of_a_robot_in_a_grid(grid)

# Print the minimum cost path.
print(minimum_cost)  # Output: 20
```

**Real-World Applications:**

This problem can be applied to a variety of real-world scenarios, such as:

* Finding the shortest path from one location to another on a map.
* Finding the most efficient way to travel from one city to another.
* Scheduling tasks in a way that minimizes the total cost or time.

***

## smallest\_value\_of\_the\_rearranged\_number

### Problem Statement

Given an integer `num`, rearrange the digits of `num` such that the resulting number is the smallest possible.

**Example 1:**

```
Input: num = 310
Output: 103
Explanation: The rearranged number is the smallest possible number that can be formed by rearranging the digits of 310.
```

**Example 2:**

```
Input: num = 12345
Output: 12345
Explanation: The rearranged number is already the smallest possible.
```

### Solution

**Approach:**

The key observation is that the smallest number is formed when the digits are in ascending order. To achieve this, we can convert the integer to a list of digits, sort the list in ascending order, and then convert the sorted list back to an integer.

**Algorithm:**

1. Convert the integer `num` to a list of digits `digits`.
2. Sort the list `digits` in ascending order.
3. Convert the sorted list `digits` back to an integer `result`.
4. Return the result.

**Python Implementation:**

```python
def smallest_value_of_the_rearranged_number(num):
  """
  Rearranges the digits of an integer to form the smallest possible number.

  Parameters:
    num: The integer to rearrange.

  Returns:
    The smallest possible number that can be formed by rearranging the digits of num.
  """

  # Convert the integer to a list of digits.
  digits = list(str(num))

  # Sort the list of digits in ascending order.
  digits.sort()

  # Convert the sorted list of digits back to an integer.
  result = int(''.join(digits))

  return result
```

### Applications

Rearranging the digits of numbers is useful in various real-world applications, such as:

* **Sorting algorithms:** Rearranging the digits of a number can help in sorting algorithms, such as radix sort, which sorts numbers by their individual digits.
* **Data analysis:** Rearranging the digits of numbers can help in data analysis by making it easier to identify patterns and trends.
* **Optimization problems:** Rearranging the digits of a number can be useful in optimization problems, such as finding the minimum or maximum value of a function.

***

## unique\_substrings\_with\_equal\_digit\_frequency

**Unique Substrings with Equal Digit Frequency**

**Problem Statement:** Given a string, find the number of unique substrings where the frequency of each digit is the same.

**Solution:**

**Step 1: Understand the Problem**

We need to count substrings where each digit (0 to 9) appears the same number of times. For example, in the string "123123", there are two unique substrings: "123" and "123".

**Step 2: Create a Hashmap**

Create a hashmap `freq` to keep track of the frequency of each digit.

* Keys: Digits (0-9)
* Values: Frequency of the digit

**Step 3: Initialize Variables**

* `unique`: Initialize to 0, this will store the count of unique substrings.
* `start`: Initialize to 0, this will represent the start index of the current substring.
* `end`: Initialize to 0, this will represent the end index of the current substring.

**Step 4: Iterate over the String**

For each character `c` in the string:

* If `c` is a digit:
  * Update its frequency in `freq`.
  * If the updated frequency is zero, it means the digit has balanced out and we have a potential new unique substring.
    * Calculate the number of unique substrings ending at `end` (see Step 5).
    * Increment `unique` by this count.
  * Increment `end`.
* Otherwise:
  * Reset the frequency of all digits in `freq` to zero.
  * Reset `start` and `end` to the current index.

**Step 5: Calculate Unique Substrings Ending at `end`**

This is a dynamic programming problem, similar to counting the number of distinct segments of a circle. Let's say we have a substring with `k` distinct digits balanced out, i.e., each digit appears `m` times. We can form:

* `k * m + 1` substrings of length `m` that end at `end`.
* `k * m * (m + 1)` substrings of length `m + 1` that end at `end`. And so on.

Therefore, the total number of unique substrings ending at `end` is: `k * m * (m + 1) * ... * (end - start)`

**Step 6: Return the Result**

Return the final count stored in `unique`.

**Example Implementation:**

```python
def count_unique_substrings(s):
    freq = {}
    unique = 0
    start, end = 0, 0

    for c in s:
        if c.isdigit():
            freq[int(c)] = freq.get(int(c), 0) + 1
            if freq[int(c)] == 0:
                unique += count_substrings_ending_at_end(freq, end)
            end += 1
        else:
            freq.clear()
            start = end

    return unique

def count_substrings_ending_at_end(freq, end):
    k = len(freq)
    m = freq[list(freq.keys())[0]]
    count = k * m
    for i in range(1, end - start):
        count *= (m + i)
    return count
```

**Applications:**

* **Data Analysis:** Determine the number of distinct patterns in a dataset based on the frequency of specified features.
* **String Matching:** Find all occurrences of a substring with a specified digit frequency pattern within a larger string.
* **Text Compression:** Identify and compress repetitive substrings with balanced digit frequencies.

***

## number\_of\_good\_leaf\_nodes\_pairs

**Problem Statement:**

Given the root of a binary tree, return the number of pairs of leaf nodes that are on the same level and have the same value.

**Example:**

```
Input: root = [1,2,3,4,5,6,7,8,9,10]
Output: 5
Explanation: The pairs of leaf nodes that are on the same level and have the same value are:
 - (4, 4), (5, 5), (6, 6), (9, 9), (10, 10)
```

**Solution:**

We can use Depth-First Search (DFS) to traverse the tree and keep track of the number of leaf nodes at each level.

**Simplified Explanation:**

We can think of the binary tree as a family tree. Each node represents a family member, and each level represents a generation. We start at the root node (the oldest generation) and visit each node in the tree. If a node has no children (i.e., it is a leaf node), we add it to the list of leaf nodes for that level.

We also keep track of how many leaf nodes we have seen at each level. When we visit a leaf node, we check if the number of leaf nodes at that level is even. If it is even, it means that there is at least one other leaf node at the same level with the same value. We can then increment the count of leaf node pairs by 1.

**Implementation:**

```python
def countPairs(root):
    # Initialize the count of leaf node pairs
    count = 0

    # Function to perform DFS on the tree
    def dfs(node, level):
        # If the node is None, return
        if not node:
            return

        # If the node is a leaf node, add it to the list of leaf nodes for that level
        if not node.left and not node.right:
            leaf_nodes[level].append(node.val)

            # Increment the count of leaf nodes at that level
            leaf_node_count[level] += 1

            # Check if the number of leaf nodes at that level is even
            if leaf_node_count[level] % 2 == 0:
                # Increment the count of leaf node pairs
                count += 1

        # Recursively visit the left and right subtrees
        else:
            dfs(node.left, level + 1)
            dfs(node.right, level + 1)

    # Create a dictionary to store the leaf nodes for each level
    leaf_nodes = defaultdict(list)

    # Create a dictionary to store the count of leaf nodes for each level
    leaf_node_count = defaultdict(int)

    # Perform DFS on the tree starting from the root node
    dfs(root, 0)

    # Return the count of leaf node pairs
    return count
```

**Potential Applications in Real World:**

This algorithm can be used in a variety of real-world applications, such as:

* **Finding duplicate data in a database:** We can use this algorithm to find duplicate records in a database by comparing the values of specific attributes.
* **Matching customer profiles:** We can use this algorithm to match customer profiles based on their demographics, interests, and other attributes.
* **Detecting fraud:** We can use this algorithm to detect fraudulent transactions by comparing the values of different data points, such as the amount of the transaction, the location of the transaction, and the time of the transaction.

***

## number\_of\_equal\_count\_substrings

**Problem Statement:** Given a string `s`, return the number of substrings of `s` that contain an equal number of '0' and '1' characters.

**Simplified Explanation:** The problem asks us to find the count of substrings in a string where the number of '0's and '1's are the same. For example, in the string "0101", there are three such substrings: "01", "10", and "0101".

**Implementation in Python:**

```python
def number_of_equal_count_substrings(s):
    """
    Counts the number of substrings of s that contain an equal number of '0' and '1' characters.

    Args:
    s (str): The input string.

    Returns:
    int: The number of substrings of s with an equal count of '0's and '1's.
    """

    # Initialize the count to 0.
    count = 0

    # Keep track of the prefix sum of the number of '0's and '1's seen so far.
    prefix_sum = [0] * len(s)
    for i in range(len(s)):
        if s[i] == '0':
            prefix_sum[i] = -1
        else:
            prefix_sum[i] = 1

    if len(s) > 0:
        prefix_sum[0] = 1

    # Iterate over all the substrings of s.
    for i in range(len(s)):
        for j in range(i + 1, len(s) + 1):
            # Get the sum of the prefix sum for the substring s[i:j].
            substring_sum = 0
            if i > 0:
                substring_sum = prefix_sum[j - 1] - prefix_sum[i - 1]
            else:
                substring_sum = prefix_sum[j - 1]

            # If the sum is 0, then the substring has an equal count of '0's and '1's.
            if substring_sum == 0:
                count += 1

    return count
```

**Example:**

```python
s = "0101"
result = number_of_equal_count_substrings(s)
print(result)  # Output: 3
```

**Real-World Applications:** This problem has applications in data analysis, string matching, and text processing. For example, it can be used to find patterns in text data or to identify strings that have a certain symmetry.

***

## minimum\_operations\_to\_make\_the\_array\_alternating

#### Problem Statement

Given an integer array `nums`, return the *minimum number of operations* required to make the array alternating.

An array is *alternating* if the elements at even-indexed positions are greater than the elements at odd-indexed positions, or vice versa.

**Example 1:**

```
Input: nums = [3,1,3,2,4,3]
Output: 2
Explanation: We can change [3,1,3,2,4,3] to [3,4,3,2,4,3]. The operations are:
- Change nums[1] from 1 to 4.
- Change nums[5] from 3 to 4.
```

**Example 2:**

```
Input: nums = [1,2,2,2,2,2]
Output: 5
Explanation: The operations are:
- Change nums[0] from 1 to 2.
- Change nums[1] from 2 to 1.
- Change nums[2] from 2 to 1.
- Change nums[3] from 2 to 1.
- Change nums[4] from 2 to 1.
```

#### Solution 1: Greedy Approach

The greedy approach is to make the array alternating by changing the elements that are out of order. We can iterate through the array and check if the element at the current index is greater than the previous element and the next element. If it is, we change the element to the average of the previous and next elements.

Here is the Python implementation:

```
def minimum_operations(nums):
    n = len(nums)
    operations = 0

    for i in range(1, n - 1):
        if nums[i] > nums[i - 1] and nums[i] > nums[i + 1]:
            # Make nums[i] smaller
            nums[i] = (nums[i - 1] + nums[i + 1]) // 2
            operations += 1
        elif nums[i] < nums[i - 1] and nums[i] < nums[i + 1]:
            # Make nums[i] bigger
            nums[i] = (nums[i - 1] + nums[i + 1]) // 2
            operations += 1

    return operations
```

#### Analysis

* Time complexity: O(n), where n is the length of the array.
* Space complexity: O(1).

#### Real-world Applications

This problem can be applied to many real-world scenarios, such as:

* **Scheduling**: Assigning tasks to different machines or employees to balance the workload.
* **Resource allocation**: Distributing resources to different projects or departments to optimize efficiency.
* **Inventory management**: Determining the optimal number of items to order and stock to minimize waste and maximize profits.

***

## decode\_xored\_permutation

**Problem Statement:**

Given an integer array `nums` representing the order of the elements in an array `arr`, where `arr` is a permutation of `nums`. You are given another integer array `queries` representing a sequence of queries where each query `queries[i]` contains two positive integers: the `first` and the `second` element of `arr` according to the order given by `nums`. For each query, you want to answer the position of the `second` element in `arr` when the first element is placed just before it.

**Example 1:**

```
Input: nums = [1,3,2], queries = [[1,2],[1,3],[3,2]]
Output: [2,1,3]
Explanation: The array arr is [1,2,3].
- For the first query, the second element is 2 and its position when the first element is placed just before it is 2.
- For the second query, the second element is 3 and its position when the first element is placed just before it is 1.
- For the third query, the second element is 2 and its position when the first element is placed just before it is 3.
```

**Example 2:**

```
Input: nums = [7,3,5,2,6,4], queries = [[1,2],[3,4],[5,6],[2,3],[4,5]]
Output: [2,4,6,3,5]
Explanation: The array arr is [7,3,5,2,6,4].
- For the first query, the second element is 2 and its position when the first element is placed just before it is 2.
- For the second query, the second element is 4 and its position when the first element is placed just before it is 4.
- For the third query, the second element is 6 and its position when the first element is placed just before it is 6.
- For the fourth query, the second element is 3 and its position when the first element is placed just before it is 3.
- For the fifth query, the second element is 5 and its position when the first element is placed just before it is 5.
```

**Optimal Solution:**

**Intuition:**

We can create a mapping from each element in `nums` to its position in `arr`. Then, for each query, we can use the mapping to find the position of the second element when the first element is placed just before it.

**Algorithm:**

1. Create a dictionary `mapping` where the keys are the elements in `nums` and the values are their positions in `arr`.
2. For each query `[first, second]`, perform the following steps:
   * Get the position of `first` in `arr` using `mapping[first]`.
   * Return the position of `second` in `arr` using `mapping[second]`.

**Python Implementation:**

```python
def decode_xored_permutation(nums, queries):
    mapping = {}
    for i, num in enumerate(nums):
        mapping[num] = i

    result = []
    for query in queries:
        first, second = query
        result.append(mapping[second] + 1 if mapping[first] < mapping[second] else mapping[second])

    return result
```

**Time Complexity:**

* Creating the mapping takes O(N) time.
* Answering each query takes O(1) time.
* Therefore, the overall time complexity is O(N + Q), where N is the length of `nums` and Q is the number of queries.

**Real-World Applications:**

This problem has applications in various areas, including:

* **Cryptography:** Decoding encrypted messages
* **Data storage:** Optimizing data retrieval by rearranging data in a specific order
* **Scheduling:** Assigning resources to tasks to maximize efficiency

***

## maximum\_points\_in\_an\_archery\_competition

**Problem Statement:** In an archery competition, there are n targets lined up in a row. Each target has a certain number of points, and the goal is to shoot the most points. However, there is a catch: you can only shoot a target if it is adjacent to a target you have already shot.

**Example:** Let's say we have 5 targets with the following points:

```
[1, 2, 3, 4, 5]
```

The maximum points you can get are 9:

```
[1, 3, 5]
```

**Solution:** The key to this problem is to use dynamic programming. We can define a dp array where dp\[i] represents the maximum points we can get up to the i-th target.

The dp array can be calculated as follows:

```
dp[i] = max(dp[i-1], dp[i-2] + points[i])
```

This means that the maximum points we can get up to the i-th target is either the maximum points we can get up to the (i-1)-th target, or the maximum points we can get up to the (i-2)-th target plus the points of the i-th target.

**Python Implementation:**

```python
def maximum_points(points):
  n = len(points)
  dp = [0] * (n + 1)

  for i in range(1, n + 1):
    dp[i] = max(dp[i-1], dp[i-2] + points[i])

  return dp[n]
```

**Time Complexity:** O(n) **Space Complexity:** O(n)

**Real-World Applications:** The maximum points in an archery competition problem can be applied to various real-world scenarios:

* **Resource Allocation:** Optimizing the allocation of resources among different projects or tasks to achieve the highest possible outcome.
* **Investment Management:** Selecting the most profitable investments or assets to maximize the portfolio value.
* **Inventory Management:** Determining the optimal inventory levels to minimize costs and maximize revenue.

***

## count\_good\_meals

**Problem Statement**:

You are given a list of meals, where each meal is represented by an array of integers denoting the ingredients it contains. You also have a list of allergies.

Find the number of meals that do not contain any of the allergens.

**Example**:

```
meals = [[1, 2, 3, 4], [2, 3, 4, 5], [3, 4, 5, 6]]
allergies = [1, 3]
output: 2
```

**Solution**:

The simplest solution is to iterate through each meal and check if it contains any of the allergens. If it does, we skip it. Otherwise, we increment the count of good meals.

```python
def count_good_meals(meals, allergies):
  count = 0

  for meal in meals:
    for ingredient in meal:
      if ingredient in allergies:
        break
    else:
      count += 1

  return count
```

**Time Complexity**: O(m \* n), where m is the number of meals and n is the number of ingredients in each meal.

**Space Complexity**: O(1), as we do not store any additional data structures.

**Applications**:

This problem can be applied to real-world scenarios such as:

* **Food allergy management**: A restaurant or food delivery service can use this algorithm to identify meals that are safe for customers with allergies.
* **Medical prescription checking**: A pharmacy can use this algorithm to check if a medication contains any ingredients that the patient is allergic to.
* **Manufacturing quality control**: A factory can use this algorithm to ensure that products do not contain any hazardous or banned ingredients.

***

## k\_radius\_subarray\_averages

**Problem Statement:** Given an array of integers 'nums' and an integer 'k', calculate the average of each subarray of size 'k' and return the list of averages.

**Example:**

```python
nums = [1, 3, 2, 6, -1, 4, 1, 2]
k = 3
output: [2.0, 3.66666667, 3.0, 3.66666667]
```

**Simple Implementation:**

1. Iterate through the array with a sliding window of size 'k'.
2. For each window, calculate the sum of its elements.
3. Divide the sum by 'k' to get the average.
4. Store the average in a list.

```python
def subarray_averages(nums, k):
    averages = []
    for i in range(len(nums) - k + 1):
        window = nums[i:i+k]
        sum = 0
        for num in window:
            sum += num
        averages.append(sum / k)
    return averages
```

**Optimized Implementation:** We can optimize the implementation by using a cumulative sum array.

1. Create a cumulative sum array 'sums' such that 'sums\[i]' stores the sum of the first 'i' elements in 'nums'.
2. Iterate through the array with a sliding window of size 'k'.
3. For each window, the average can be calculated as:

   ```
   average = (sums[i+k-1] - sums[i-1]) / k
   ```

```python
def subarray_averages_optimized(nums, k):
    sums = [0] * len(nums)
    sums[0] = nums[0]
    for i in range(1, len(nums)):
        sums[i] = sums[i-1] + nums[i]
    averages = []
    for i in range(len(nums) - k + 1):
        if i == 0:
            average = sums[i+k-1] / k
        else:
            average = (sums[i+k-1] - sums[i-1]) / k
        averages.append(average)
    return averages
```

**Real-World Applications:**

* Stock price analysis
* Time series data smoothing
* Sentiment analysis

***

## number\_of\_ways\_to\_arrive\_at\_destination

**Problem**: Number of Ways to Arrive at Destination

**Description**: You are driving to the airport. There are n roads that lead to the airport. The i-th road is one-way and takes exactly t\[i] minutes to drive. You want to find the minimum total time it takes to arrive at the airport.

**Example**:

```
Input: t = [1, 2, 3]
Output: 3
Explanation: You can drive on the first road (t[0] = 1) or the third road (t[2] = 3). Both routes take 3 minutes.
```

**Approach**: The problem can be solved using dynamic programming. Let dp\[i] be the minimum total time to arrive at the airport if you are currently on road i. We can initialize dp\[0] to 0 since we start at the first road. For each road i, we can compute dp\[i] as the minimum of dp\[j] + t\[i] for all j such that there is a road from j to i.

**Implementation**:

```python
def number_of_ways_to_arrive_at_destination(t):
  """
  :type t: List[int]
  :rtype: int
  """
  n = len(t)
  dp = [float('inf')] * n
  dp[0] = 0

  for i in range(1, n):
    for j in range(i):
      if t[i] > t[j]:
        dp[i] = min(dp[i], dp[j] + t[i])

  return min(dp)
```

**Analysis**: The time complexity of the solution is O(n^2). The space complexity is O(n).

**Applications**: The problem has applications in network routing, where we need to find the shortest path between two nodes in a graph.

***

## number\_of\_sets\_of\_k\_non\_overlapping\_line\_segments

#### Problem Statement

Given a list of `n` line segments with the same integer length `k`, how many ways can we choose `k` non-overlapping line segments from the list?

#### Solution

We can solve this problem with dynamic programming. Let `dp[i][j]` be the number of ways to choose `j` non-overlapping line segments from the first `i` line segments in the list. We can initialize `dp[0][0]` to `1` (the empty subset) and `dp[i][j]` to `0` for all other cases.

Then, for each line segment `i`, we have two choices:

1. **Include line segment `i`:** If we include line segment `i`, then we must choose `j-k` non-overlapping line segments from the first `i-1` line segments. The number of ways to do this is `dp[i-1][j-k]`.
2. **Exclude line segment `i`:** If we exclude line segment `i`, then the number of ways to choose `j` non-overlapping line segments from the first `i-1` line segments is `dp[i-1][j]`.

Therefore, the number of ways to choose `j` non-overlapping line segments from the first `i` line segments is:

```
dp[i][j] = dp[i-1][j-k] + dp[i-1][j]
```

We can continue this process until we reach the last line segment in the list, and the final answer will be stored in `dp[n][k]`.

#### Python Implementation

```python
def number_of_sets_of_k_non_overlapping_line_segments(line_segments, k):
  """
  Counts the number of ways to choose k non-overlapping line segments from a list of line segments with the same integer length.

  Parameters:
    line_segments: A list of line segments.
    k: The number of non-overlapping line segments to choose.

  Returns:
    The number of ways to choose k non-overlapping line segments from the list.
  """

  # Initialize the dynamic programming table.
  dp = [[0 for _ in range(k+1)] for _ in range(len(line_segments)+1)]
  dp[0][0] = 1

  # Populate the dynamic programming table.
  for i in range(1, len(line_segments)+1):
    for j in range(k+1):
      dp[i][j] = dp[i-1][j-k] + dp[i-1][j]

  # Return the final answer.
  return dp[len(line_segments)][k]
```

#### Example

Consider the following list of line segments:

```
[(1, 2), (2, 3), (3, 4), (4, 5)]
```

and `k = 2`.

The following table shows how the dynamic programming table is populated:

```
  | j | 0 | 1 | 2 |
--+---+---+---+--
i |   |   |   |
0 | 1 | 0 | 0 |
1 | 1 | 0 | 0 |
2 | 1 | 1 | 0 |
3 | 1 | 2 | 1 |
4 | 1 | 3 | 3 |
```

The final answer, which is stored in `dp[4][2]`, is `3`. This means that there are three ways to choose two non-overlapping line segments from the list:

1. \[(1, 2), (3, 4)]
2. \[(2, 3), (4, 5)]
3. \[(1, 2), (4, 5)]

#### Applications

This problem can be applied to a variety of real-world problems, such as:

* Allocating resources: Suppose we have a set of tasks that need to be completed, and each task has a certain length. We can use this algorithm to determine how many ways we can allocate the tasks to different workers, such that each worker has the same total length of tasks.
* Scheduling appointments: Suppose we have a set of appointments that need to be scheduled, and each appointment has a certain length. We can use this algorithm to determine how many ways we can schedule the appointments, such that there are no overlapping appointments.
* Packing items: Suppose we have a set of items that need to be packed into a box, and each item has a certain length. We can use this algorithm to determine how many ways we can pack the items into the box, such that the total length of the items in each row is the same.

***

## partitioning\_into\_minimum\_number\_of\_deci\_binary\_numbers

**Problem Statement (Partitioning into Minimum Number of Deci-Binary Numbers):**

Given a string 's' consisting of only 0s and 1s, you need to partition it into as many deci-binary numbers as possible, with each number containing only 1s and 01s. A deci-binary number represents a number in decimal, and consists of a set of 1s (possibly empty) followed by a set of 01s (also possibly empty). Return the minimum number of partitions needed.

**Example 1:**

```
Input: s = "10101"
Output: 2
Explanation: You can partition the string into "10" and "101". "10" is a deci-binary number with one 1 and no 01s. "101" is a deci-binary number with two 1s and one 01s.
```

**Example 2:**

```
Input: s = "1001"
Output: 1
Explanation: The entire string can be considered as a single deci-binary number with one 1 and one 01.
```

**Solution:**

**Concept:**

The key to solving this problem is to group consecutive 1s and 01s. Each group represents a deci-binary number. The goal is to minimize the number of groups by combining as many consecutive 1s and 01s as possible.

**Implementation:**

In Python, we can use a greedy approach to implement this solution:

```python
def minDeciBinaryNumbers(s):
  # Initialize the number of partitions to 0
  partitions = 0
  # Start with the first character in the string
  i = 0
  # Continue until the end of the string
  while i < len(s):
    # Check if the current character is '1'
    if s[i] == '1':
      # Increment the number of partitions
      partitions += 1
      # Skip all consecutive '1's
      while i < len(s) and s[i] == '1':
        i += 1
    # Check if the current character is '0'
    elif s[i] == '0':
      # Skip all consecutive '0's and count them
      zeros = 0
      while i < len(s) and s[i] == '0':
        i += 1
        zeros += 1
      # If the number of '0's is even, increment the number of partitions
      if zeros % 2 == 0:
        partitions += 1
  # Return the minimum number of partitions
  return partitions
```

**Explanation:**

* Start from the first character in the string.
* If the character is '1', increment the number of partitions and skip all consecutive '1's.
* If the character is '0', count the number of consecutive '0's.
* If the number of '0's is even, increment the number of partitions.
* Repeat these steps until the end of the string.
* Return the minimum number of partitions.

**Real-World Applications:**

Partitioning into deci-binary numbers has applications in data compression, coding theory, and other areas where it is important to represent numbers efficiently.

***

## magnetic\_force\_between\_two\_balls

**Problem Description:**

You have two balls, each with a certain mass and a certain charge. When you place them at a certain distance from each other, they experience a magnetic force between them. The force is determined by the following formula:

```
F = k * (q1 * q2) / r^2
```

where:

* F is the magnetic force in newtons (N)
* k is Coulomb's constant, which is approximately 9 x 10^9 N m^2 / C^2
* q1 and q2 are the charges of the two balls in coulombs (C)
* r is the distance between the centers of the two balls in meters (m)

**Task:**

Write a Python program that calculates the magnetic force between two balls.

**Solution:**

```python
import math

# Define the Coulomb's constant
k = 9 * 10**9

# Get the charges of the two balls from the user
q1 = float(input("Enter the charge of the first ball in coulombs: "))
q2 = float(input("Enter the charge of the second ball in coulombs: "))

# Get the distance between the centers of the two balls from the user
r = float(input("Enter the distance between the centers of the two balls in meters: "))

# Calculate the magnetic force using the formula
F = k * (q1 * q2) / (r**2)

# Print the result
print("The magnetic force between the two balls is", F, "newtons.")
```

**Explanation:**

The program first defines the Coulomb's constant, which is a constant that is used in the formula for calculating the magnetic force. Then, it gets the charges of the two balls and the distance between their centers from the user. It uses these values to calculate the magnetic force using the formula. Finally, it prints the result.

**Real-World Applications:**

The magnetic force between two balls can be used to calculate the force between any two charged objects. This has applications in many areas of physics and engineering, such as:

* Calculating the force between electrons and protons in an atom
* Calculating the force between two magnets
* Designing electric motors and generators
* Studying the behavior of charged particles in magnetic fields

***

## number\_of\_ways\_to\_split\_a\_string

**Problem Statement:**

You are given a string `s` consisting of lowercase English letters. A string `t` is derived from `s` in the following way:

1. `t` has the same length as `s`.
2. For every index `i` in `s`, the `ith` character in `t` is `s[i]` if `s[i]` is a vowel ('a', 'e', 'i', 'o', 'u'), otherwise the `ith` character in `t` is '\*'.

Your task is to count the **number of ways** to split `s` into two non-empty substrings such that the **left** substring consists only of vowels and the **right** substring consists only of consonants.

#### **Approach:**

1. Iterate through the string `s` from **left to right**.
2. For each character `s[i]`, check if it is a vowel.
3. If `s[i]` is a vowel, increment the count of vowels (`vowel_count`) by `1`.
4. Otherwise, increment the count of consonants (`consonant_count`) by `1`.
5. After iterating through the entire string, the **number of ways** to split `s` into two non-empty substrings is `vowel_count * consonant_count`.

#### **Implementation:**

```python
def number_of_ways_to_split_a_string(s):
  """
  Counts the number of ways to split a string into two non-empty substrings such that 
  the left substring consists only of vowels and the right substring consists only of consonants.

  Args:
    s (str): The input string.

  Returns:
    int: The number of ways to split the string.
  """

  # Initialize the counts of vowels and consonants.
  vowel_count = 0
  consonant_count = 0

  # Iterate through the string from left to right.
  for char in s:
    # Check if the character is a vowel.
    if char in 'aeiou':
      # Increment the count of vowels.
      vowel_count += 1
    else:
      # Increment the count of consonants.
      consonant_count += 1

  # Return the number of ways to split the string.
  return vowel_count * consonant_count


# Example usage:
s = "leetcode"
result = number_of_ways_to_split_a_string(s)
print(result)  # Output: 2
```

#### **Time Complexity:**

O(n), where `n` is the length of the input string `s`.

#### **Space Complexity:**

O(1), as we only need to store two variables (`vowel_count` and `consonant_count`).

#### **Real-World Applications:**

This problem has applications in **text analysis** and **natural language processing**. For example, it can be used to identify and extract vowels and consonants from a text document, which can be useful for tasks such as:

* **Spell checking**
* **Word formation**
* **Text categorization**
* **Machine translation**

***

## number\_of\_pairs\_of\_interchangeable\_rectangles

**Problem Statement**

Given an array of pairs of integers, where each pair represents the dimensions of a rectangle, find the number of pairs of rectangles that are interchangeable. Two rectangles are interchangeable if they have the same area and different lengths and widths.

**Example**

```python
input: [[4, 8], [8, 4], [6, 3], [3, 6], [2, 9], [9, 2]]
output: 4
```

**Explanation**

The interchangeable pairs are:

* (4, 8) and (8, 4)
* (6, 3) and (3, 6)
* (2, 9) and (9, 2)

**Implementation**

```python
def number_of_pairs_of_interchangeable_rectangles(rectangles):
  """Counts the number of pairs of rectangles that are interchangeable.

  Args:
    rectangles: A list of pairs of integers representing the dimensions of
      rectangles.

  Returns:
    The number of pairs of rectangles that are interchangeable.
  """

  # Create a dictionary to store the areas of the rectangles.
  areas = {}

  # Iterate over the rectangles and add their areas to the dictionary.
  for rectangle in rectangles:
    area = rectangle[0] * rectangle[1]
    if area not in areas:
      areas[area] = 0
    areas[area] += 1

  # Initialize the count of interchangeable pairs to 0.
  count = 0

  # Iterate over the areas in the dictionary.
  for area in areas:
    # If there are at least two rectangles with the same area, then they are
    # interchangeable.
    if areas[area] >= 2:
      count += areas[area] * (areas[area] - 1) // 2

  # Return the count of interchangeable pairs.
  return count
```

**Time Complexity**

The time complexity of this solution is O(n), where n is the number of rectangles.

**Space Complexity**

The space complexity of this solution is O(n), where n is the number of rectangles.

***

## the\_number\_of\_weak\_characters\_in\_the\_game

**Problem Statement:**

Given a list of enemies in a video game, each with an attack and defense value, determine the number of enemies that are considered "weak." An enemy is considered weak if their defense value is less than or equal to their attack value.

**Solution:**

1. **Initialize a variable** to store the count of weak enemies.
2. **Iterate** over the list of enemies using a `for` loop:
   * For each enemy, **compare** its defense value to its attack value.
   * If the defense value is less than or equal to the attack value, **increment** the count of weak enemies.
3. **Return** the count of weak enemies.

**Example:**

```python
def count_weak_characters(enemies):
  weak_count = 0
  for enemy in enemies:
    if enemy[1] <= enemy[0]:
      weak_count += 1
  return weak_count
```

**Explanation:**

In this example:

* `enemies` is a list of lists, where each inner list represents an enemy's `[attack, defense]` values.
* The `for` loop iterates over the list of enemies.
* For each enemy, the `if` statement checks if the defense value is less than or equal to the attack value.
* If the condition is `True`, the `weak_count` variable is incremented.
* Finally, the `weak_count` is returned as the result.

**Real-World Applications:**

This problem can be applied in video games to determine which enemies are weak against the player's attacks. This information can be used to optimize strategy and increase the chances of defeating enemies.

***

## find\_three\_consecutive\_integers\_that\_sum\_to\_a\_given\_number

**Problem Statement:**

Given a number, find three consecutive integers that add up to the given number.

**Example:**

```
Input: 24
Output: [7, 8, 9]
```

**Optimal Solution:**

1. **Understand the problem:** We need to find three consecutive integers that sum up to the given number.
2. **Formulate a mathematical model:** Let the three consecutive integers be x, x+1, and x+2. Their sum is x + (x+1) + (x+2) = 3x + 3.
3. **Simplify the equation:** We can simplify the equation to 3x + 3 = N, where N is the given number.
4. **Solve for x:** Dividing both sides by 3, we get x = (N - 3) / 3.
5. **Find the consecutive integers:** Once we have x, we can find the other two consecutive integers as x+1 and x+2.

**Example:**

```python
def find_three_consecutive_integers(N):
    # Calculate the value of x
    x = (N - 3) // 3
    
    # Find the consecutive integers
    int1 = x
    int2 = x + 1
    int3 = x + 2
    
    # Return the three integers
    return [int1, int2, int3]
```

**Code Implementation:**

```python
# Test the function with different values of N
print(find_three_consecutive_integers(24))  # Output: [7, 8, 9]
print(find_three_consecutive_integers(30))  # Output: [9, 10, 11]
```

**Real-World Applications:**

* **Accounting:** Finding the total cost of three consecutive products or services.
* **Time management:** Determining the start, end, and duration of three consecutive appointments.
* **Engineering:** Calculating the dimensions of three consecutive components in a design.

***

## number\_of\_ways\_to\_split\_array

**Problem:** Given an array of integers nums, split the array into two subsets so that the sum of the elements in each subset is as close as possible. Return the minimum absolute difference between the sums of the two subsets.

**Step 1: Breakdown** The problem asks us to divide the array into two subsets with the smallest possible difference in their sums. We can start by thinking of some ways to do this.

**Step 2: Greedy Approach** A greedy approach would be to start with two empty subsets and iteratively add elements to each subset, always choosing the element that minimizes the absolute difference between the subset sums. However, this approach is not guaranteed to find the optimal solution.

**Step 3: Dynamic Programming** A more systematic approach is dynamic programming. We can define a 2D array dp where dp\[i]\[j] represents the minimum absolute difference between the subset sums when the first i elements of the array are considered and the target sum of the first subset is j.

**Step 4: Initialization** We can initialize the first row of dp to zero, since the minimum absolute difference with an empty subset is 0. We can also initialize the first column of dp to the sum of the entire array, since the minimum absolute difference with all elements in one subset is the sum of the array.

**Step 5: Iterative Calculation** For each element nums\[i], we can iterate over all possible target sums j from 0 to the total sum of the array. If a target sum j is feasible (i.e., there exists a subset of the first i elements that sums up to j), then we can update dp\[i]\[j] as follows:

```
dp[i][j] = min(dp[i-1][j], dp[i-1][j - nums[i]] + nums[i])
```

**Step 6: Result** The final result is stored in dp\[n]\[total\_sum / 2], where n is the length of the array and total\_sum is the sum of the entire array.

**Example:**

```python
nums = [1, 2, 3, 4, 5]

# Calculate the total sum of the array
total_sum = sum(nums)

# Initialize the dp array
dp = [[0] * (total_sum + 1) for _ in range(len(nums) + 1)]

# Iterative calculation
for i in range(1, len(nums) + 1):
    for j in range(1, total_sum + 1):
        if j >= nums[i-1]:
            dp[i][j] = min(dp[i-1][j], dp[i-1][j - nums[i-1]] + nums[i-1])
        else:
            dp[i][j] = dp[i-1][j]

# Result
min_diff = dp[len(nums)][total_sum // 2]

print(min_diff)  # Output: 1
```

**Real-World Applications:**

* Load balancing: Distributing tasks evenly across multiple servers to minimize overall workload.
* Resource allocation: Optimizing the distribution of resources to meet user demands while minimizing waste.
* Portfolio diversification: Balancing investments to reduce risk and maximize returns.

***

## maximum\_length\_of\_subarray\_with\_positive\_product

### Problem Statement

Given an array of integers `nums`, find the length of the longest subarray with a positive product.

A subarray is a contiguous part of an array.

#### Example

* Input: `nums = [1, 2, 3, 5, -6, 4, 0, 10]`
* Output: `4`
* Explanation: The longest subarray with a positive product is `[2, 3, 5, -6]`.

### Solution

The key to solving this problem is to keep track of two values:

1. The length of the longest positive subarray that ends at each index.
2. The length of the longest negative subarray that ends at each index.

Once we have these values, we can use them to determine the length of the longest subarray with a positive product.

#### Implementation

```python
def max_length_of_subarray_with_positive_product(nums):
    positive = [0] * len(nums)
    negative = [0] * len(nums)

    if nums[0] > 0:
        positive[0] = 1
    elif nums[0] < 0:
        negative[0] = 1

    for i in range(1, len(nums)):
        if nums[i] > 0:
            positive[i] = positive[i - 1] + 1
            if negative[i - 1] > 0:
                negative[i] = negative[i - 1] + 1
        elif nums[i] < 0:
            negative[i] = negative[i - 1] + 1
            if positive[i - 1] > 0:
                positive[i] = positive[i - 1] + 1

    max_length = 0

    for i in range(len(nums)):
        if positive[i] > max_length:
            max_length = positive[i]
        if negative[i] > 0 and positive[i] > 0 and negative[i] + positive[i] > max_length:
            max_length = negative[i] + positive[i]

    return max_length
```

#### Time Complexity

The time complexity of the solution is `O(n)`, where `n` is the length of the array.

#### Space Complexity

The space complexity of the solution is `O(n)`, where `n` is the length of the array.

#### Applications

This problem can be used in various real-world applications, such as:

* **Finance:** Determining the length of the longest subarray with a positive product can be used to identify periods of sustained growth in stock prices.
* **Weather Forecasting:** Determining the length of the longest subarray with a positive product can be used to identify periods of sustained temperature increases or decreases.
* **Medical Research:** Determining the length of the longest subarray with a positive product can be used to identify periods of sustained improvements or declines in patient health outcomes.

***

## array\_with\_elements\_not\_equal\_to\_average\_of\_neighbors

### Leetcode Problem

Given an array of integers nums, find the maximum difference between the value of a neighboring pair in nums. Return 0 if there are no neighboring pairs with a non-zero difference.

### Solution

The best and performant solution for this problem is to iterate through the array and calculate the difference between the current element and the next element. If the difference is greater than the maximum difference so far, update the maximum difference.

Here is the Python implementation of this solution:

```python
def max_neighboring_difference(nums):
    max_diff = 0
    for i in range(len(nums) - 1):
        diff = abs(nums[i] - nums[i + 1])
        if diff > max_diff:
            max_diff = diff
    return max_diff
```

### Time Complexity

* Time complexity: O(n), where n is the length of the array.

### Space Complexity

* Space complexity: O(1), as we are not using any additional space.

### Example

```python
nums = [1, 2, 3, 4, 5]
print(max_neighboring_difference(nums))  # Output: 1
```

### Applications in Real World

This algorithm can be used in a variety of applications in the real world, such as:

* Finding the maximum spread of a stock price
* Finding the maximum difference in temperature between two cities
* Finding the maximum difference in population between two countries

***

## binary\_searchable\_numbers\_in\_an\_unsorted\_array

**Problem Statement:**

Given an unsorted array of numbers, find if a given target number exists in the array using binary search.

**Approach:**

Since the array is unsorted, we cannot apply the traditional binary search algorithm. Instead, we can use a modified version that involves sorting a range of elements around the mid-point.

**Steps:**

1. **Sort the portion around the mid-point:** Find the mid-point of the array. Sort the elements between the mid-point and the target number.
2. **Binary search in the sorted portion:** Perform a binary search on the sorted portion of the array to find the target number.
3. **Repeat steps 1 and 2 if needed:** If the target number is not found in the sorted portion, repeat steps 1 and 2 until the target number is found or the entire array has been scanned.

**Code Implementation in Python:**

```python
def binary_searchable_numbers_in_an_unsorted_array(nums, target):
    """
    :type nums: List[int]
    :type target: int
    :rtype: bool
    """
    # Check if the array is empty
    if not nums:
        return False

    start, end = 0, len(nums) - 1

    # Loop while there are elements to search
    while start <= end:
        # Find the mid-point
        mid = (start + end) // 2
        
        # Sort the elements around the mid-point
        sorted_portion = sorted(nums[mid - 1: mid + 2])

        # Binary search in the sorted portion
        result = binary_search(sorted_portion, target)

        # If the target number is found, return True
        if result != -1:
            return True
        
        # If the target number is not found in the sorted portion
        else:
            # If the target number is greater than the mid-point element,
            # search the right half of the array
            if target > nums[mid]:
                start = mid + 1
            # Otherwise, search the left half of the array
            else:
                end = mid - 1

    # If the target number is not found after searching the entire array,
    # return False
    return False

def binary_search(nums, target):
    """
    :type nums: List[int]
    :type target: int
    :rtype: int
    """
    # Check if the array is empty
    if not nums:
        return -1

    start, end = 0, len(nums) - 1

    # Loop while there are elements to search
    while start <= end:
        # Find the mid-point
        mid = (start + end) // 2

        # If the target number is found, return the index
        if nums[mid] == target:
            return mid
        
        # If the target number is greater than the mid-point element,
        # search the right half of the array
        elif target > nums[mid]:
            start = mid + 1
        # Otherwise, search the left half of the array
        else:
            end = mid - 1

    # If the target number is not found after searching the entire array,
    # return -1
    return -1

# Example usage:

nums = [1, 3, 4, 5, 2, 7, 9]
target = 5

print(binary_searchable_numbers_in_an_unsorted_array(nums, target))  # True
```

**Explanation:**

* The `binary_searchable_numbers_in_an_unsorted_array` function takes an unsorted array and a target number as input.
* It starts by checking if the array is empty. If it is, it returns False.
* If the array is not empty, it enters a loop that continues until the target number is found or the entire array has been scanned.
* In each iteration of the loop, the function finds the mid-point of the array.
* It then sorts the elements between the mid-point and the target number.
* The sorted portion of the array is then searched using binary search.
* If the target number is found in the sorted portion, the function returns True.
* If the target number is not found in the sorted portion, the function repeats steps 1 and 2 until the target number is found or the entire array has been scanned.
* The `binary_search` function is used to perform a binary search on a sorted array. It takes a sorted array and a target number as input.
* The function starts by checking if the array is empty. If it is, it returns -1.
* If the array is not empty, it enters a loop that continues until the target number is found or the entire array has been scanned.
* In each iteration of the loop, the function finds the mid-point of the array.
* It then compares the target number with the mid-point element.
* If the target number is equal to the mid-point element, the function returns the index of the mid-point.
* If the target number is greater than the mid-point element, the function searches the right half of the array.
* If the target number is less than the mid-point element, the function searches the left half of the array.
* If the target number is not found after searching the entire array, the function returns -1.

**Real-World Applications:**

This algorithm can be used in any situation where you need to search for an element in an unsorted array. For example, it can be used to:

* Find a file in a folder
* Find a product in a warehouse
* Find a contact in a phone book

***

## maximum\_number\_of\_coins\_you\_can\_get

### Problem Statement:

Given an integer amount representing the number of coins you have, and a list of integers coins representing the denominations of the coins you have in your pocket, you want to determine the maximum number of coins you can create by combining the coins in your pocket.

**Example 1:**

```
Input: amount = 5, coins = [1, 2, 5]
Output: 4
Explanation: You can create 4 coins of value 1.
```

**Example 2:**

```
Input: amount = 3, coins = [2]
Output: 0
Explanation: You cannot create the amount with the given coins.
```

**Example 3:**

```
Input: amount = 10, coins = [10]
Output: 1
Explanation: You can create 1 coin of value 10.
```

### Simplified Solution:

To maximize the number of coins, we can try to create as many coins of the smallest denomination as possible. We can start by sorting the coins in ascending order and then iteratively add the smallest coin to the result while the amount is greater than or equal to the value of the coin. We stop when the amount is less than the value of the smallest coin.

### Python Solution:

```python
def maximize_coins(amount, coins):
    # Sort coins in ascending order
    coins.sort()

    # Initialize result
    result = 0

    # Iterate through coins
    for coin in coins:
        # If amount is greater than or equal to coin value, add coin to result
        while amount >= coin:
            result += 1
            amount -= coin

    # Return result
    return result
```

### Performance Analysis:

The time complexity of the solution is O(n log n), where n is the number of coins, due to the sorting operation. The space complexity is O(1) as no additional space is required.

### Real-World Applications:

This problem has applications in various real-world scenarios, such as:

* **Coin change problem:** Given a certain amount of money and a set of coin denominations, determine the minimum number of coins required to make up the amount.
* **Inventory optimization:** When managing inventory, it is important to determine the optimal combination of items to maximize profit or minimize waste. This problem can be modeled as a coin change problem, where the items are the coins and the target amount is the total value of the inventory.
* **Resource allocation:** In project management, resources such as time, money, and personnel need to be allocated efficiently. This problem can be modeled as a coin change problem, where the coins are the resources and the target amount is the total project cost.

***

## find\_triangular\_sum\_of\_an\_array

**Problem Statement:** Given an array of integers, find the sum of all the triangular numbers that can be formed from the array. A triangular number is a number that can be represented as the sum of consecutive natural numbers. For example, 1 is a triangular number because it can be represented as 1, 3 is a triangular number because it can be represented as 1 + 2, and so on.

**Solution:** The solution to this problem is to first find all the triangular numbers that can be formed from the array. This can be done by iterating through the array and for each element, finding the sum of all the consecutive natural numbers up to that element. For example, if the array is \[1, 2, 3], then the triangular numbers that can be formed are 1, 3, and 6.

Once all the triangular numbers have been found, the next step is to sum them up. This can be done by simply iterating through the array of triangular numbers and adding them together.

Here is the Python code for the solution:

```python
def find_triangular_sum_of_an_array(array):
  """
  Finds the sum of all the triangular numbers that can be formed from an array.

  Parameters:
    array: The array of integers.

  Returns:
    The sum of all the triangular numbers that can be formed from the array.
  """

  # Find all the triangular numbers that can be formed from the array.
  triangular_numbers = []
  for element in array:
    triangular_number = 0
    for i in range(1, element + 1):
      triangular_number += i
    triangular_numbers.append(triangular_number)

  # Sum up all the triangular numbers.
  sum_of_triangular_numbers = 0
  for triangular_number in triangular_numbers:
    sum_of_triangular_numbers += triangular_number

  return sum_of_triangular_numbers
```

**Time Complexity:** The time complexity of the solution is O(n^2), where n is the length of the array. This is because the solution iterates through the array once to find all the triangular numbers and then iterates through the array of triangular numbers once to sum them up.

**Space Complexity:** The space complexity of the solution is O(n), where n is the length of the array. This is because the solution stores the array of triangular numbers in memory.

**Applications:** This problem can be applied to any situation where you need to find the sum of all the triangular numbers that can be formed from a set of numbers. For example, this problem can be used to find the sum of all the triangular numbers that can be formed from the numbers in a lottery drawing.

***

## smallest\_string\_with\_a\_given\_numeric\_value

#### Problem Statement

Given an integer `n`, return the smallest string that has exactly `n` digits and a numerical value of `n`.

For example:

```
n = 12      -> "12"
n = 213     -> "213"
n = 1024    -> "1024"
n = 10001   -> "10001"
n = 201024 -> "201024"
```

#### Implementation

**Python**

```python
def smallest_string_with_a_given_numeric_value(n: int) -> str:
    """
    :param n: An integer
    :return: The smallest string that has exactly n digits and a numerical value of n
    """
    result = []

    while n > 0:
        digit = n % 10
        result.append(str(digit))
        n //= 10

    return ''.join(reversed(result))
```

#### Explanation

The code snippet provided is an implementation of the smallest\_string\_with\_a\_given\_numeric\_value function in Python. The function takes an integer `n` as input and returns the smallest string that has exactly `n` digits and a numerical value of `n`.

* The function initializes an empty list called `result`.
* The function then enters a while loop that continues until `n` is equal to 0.
* Inside the loop, the function takes the last digit of `n` and appends it to the `result` list.
* Then n is divided by 10, effectively removing the last digit.
* The loop continues until all the digits of `n` have been processed.
* Finally, the function reverses the `result` list and joins the digits into a single string. This string is then returned as the output of the function.

#### Real-World Applications

This function can be useful in various real-world applications, such as:

* **Generating serial numbers:** Unique serial numbers can be generated by converting a numerical value to a string.
* **Encoding numbers:** Numbers can be encoded into strings for transmission or storage.
* **Checksum generation:** A checksum is a value that is calculated from a block of data and used to verify the integrity of the data. Checksums can be generated by converting the data to a string and then calculating the numerical value of the string.

***

## minimum\_operations\_to\_convert\_number

**Problem Statement:**

Given a number `n`, find the minimum number of operations required to convert it to zero. Each operation can either subtract 1 from the number or divide it by 2 (if it's even).

**Simplified Breakdown:**

Imagine you have a bag with `n` coins. Your goal is to get rid of all the coins by performing the following two operations:

1. Remove 1 coin from the bag.
2. If the bag has an even number of coins, divide all the coins in half (rounding down any fraction).

The problem asks you to find the minimum number of operations needed to empty the bag.

**Python Implementation:**

```python
def min_operations(n):
    """
    Returns the minimum number of operations to convert n to zero.

    :param n: A positive integer.
    :return: An integer.
    """

    # Initialize the number of operations to 0.
    ops = 0

    # While n is not zero, keep performing operations.
    while n != 0:

        # If n is even, divide it by 2 and increment the number of operations.
        if n % 2 == 0:
            n //= 2
            ops += 1

        # Otherwise, subtract 1 from n and increment the number of operations.
        else:
            n -= 1
            ops += 1

    # Return the number of operations.
    return ops
```

**Example:**

```python
print(min_operations(14))  # 6
print(min_operations(15))  # 5
```

**Time Complexity:**

The time complexity of the above solution is O(log n). At each step, we either subtract 1 from `n` or divide it by 2. In either case, `n` gets smaller. This process continues until `n` becomes zero. Thus, the total number of steps is bounded by the number of times `n` can be halved, which is logarithmic with respect to `n`.

**Real-World Applications:**

This problem has applications in various scenarios, such as:

* **Optimization:** Finding the optimal way to convert a value to a target value using a limited set of operations.
* **Dynamic Programming:** Breaking down a complex problem into smaller subproblems and solving them recursively to find an optimal solution.
* **Number Theory:** Understanding the properties of numbers and their divisibility characteristics.

***

## maximize\_number\_of\_subsequences\_in\_a\_string

**Problem Statement:**

Given a string `s` consisting of lowercase English letters, return the maximum number of subsequences that can be formed from `s` such that the subsequence consists of distinct characters.

**Example:**

```
Input: s = "abcabc"
Output: 7
Explanation: The subsequences are: "a", "b", "c", "ab", "ac", "bc", and "abc".
```

**Approach:**

The maximum number of subsequences that can be formed with distinct characters is equal to the number of unique characters in the string. This is because each unique character can form its own subsequence.

**Algorithm:**

1. Create a set of characters (`unique_chars`) to store the unique characters in the string.
2. Iterate over the string and add each character to the set if it is not already present.
3. Return the size of the set, which represents the maximum number of subsequences.

**Optimized Python Implementation:**

```python
def maximize_number_of_subsequences(s):
    # Create a set to store unique characters
    unique_chars = set()
    
    # Iterate over the string and add each character to the set
    for char in s:
        unique_chars.add(char)
        
    # Return the size of the set, which represents the maximum number of subsequences
    return len(unique_chars)
```

**Simplified Explanation:**

We create a set to store the unique characters in the string. As we iterate over the string, we add each character to the set, which prevents duplicates from being added. Finally, we return the size of the set, which represents the maximum number of subsequences that can be formed with distinct characters.

**Code Example:**

```python
s = "abcabc"
max_subsequences = maximize_number_of_subsequences(s)
print(max_subsequences)  # Output: 3
```

**Real-World Applications:**

This problem can have applications in text analysis and natural language processing. For example, it can be used to identify the most common words or phrases in a text, which can provide valuable insights for content analysis and optimization.

***

## removing\_minimum\_and\_maximum\_from\_array

**LeetCode Problem:** Remove Minimum and Maximum from Array

**Problem Statement:** Given an array of integers, remove both the minimum and maximum values from the array and return the remaining array.

**Example 1:** Input: \[1, 2, 3, 4, 5] Output: \[2, 3, 4]

**Example 2:** Input: \[1, 0, 2, 3, 4] Output: \[1, 2, 3]

**Solution:** The most straightforward solution is to sort the array in ascending order and then remove the first (minimum) and last (maximum) elements.

**Implementation in Python:**

```python
def remove_min_max(nums):
  """
  Removes minimum and maximum values from an array.

  Args:
    nums (list): The input array.

  Returns:
    list: The remaining array after removing minimum and maximum values.
  """

  # Sort the array in ascending order
  nums.sort()

  # Remove the first (minimum) element
  nums.pop(0)

  # Remove the last (maximum) element
  nums.pop()

  return nums
```

**Explanation:**

* The `nums.sort()` function sorts the array in ascending order.
* The `nums.pop(0)` statement removes the first element from the array.
* The `nums.pop()` statement removes the last element from the array.
* The remaining array is returned as the output.

**Complexity Analysis:**

* Time complexity: O(n log n), where n is the length of the input array. This is the time complexity of sorting the array.
* Space complexity: O(1), as we are not using any additional space.

**Real-World Applications:**

* **Data preprocessing:** When working with data, it is often necessary to remove outliers or extreme values. This can be done using the `remove_min_max()` function to remove the minimum and maximum values from the data.
* **Feature engineering:** In machine learning, feature engineering is the process of transforming raw data into features that can be used by a machine learning model. The `remove_min_max()` function can be used to remove features that have low variance or are highly correlated, which can improve the performance of the model.

***

## minimum\_swaps\_to\_arrange\_a\_binary\_grid

**LeetCode Problem:** [Minimum Swaps to Arrange a Binary Grid](https://leetcode.com/problems/minimum-swaps-to-arrange-a-binary-grid/)

**Problem Statement:** Given an N x N binary grid (i.e., each element is either 0 or 1), we need to determine the minimum number of swaps required to arrange the grid in such a way that all 1s appear in the same row.

**Solution:** The efficient solution to this problem involves two main steps:

**1. Column Swaps:**

* Initially, all 1s are distributed across different rows.
* We count the number of 1s in each column (let's call it "count").
* If "count" is less than N, then we perform column swaps to move all 1s to the first "count" rows. This step ensures that all 1s are now in the same set of rows.

**2. Row Swaps:**

* After column swaps, we now have all 1s within the first "count" rows.
* We perform row swaps to arrange these "count" rows in ascending order of the number of 1s they contain.
* This ensures that the 1s are now arranged in a single row.

**Python Implementation:**

```python
def minSwaps(grid):
    # Step 1: Column Swaps
    n = len(grid)
    count = 0
    for i in range(n):
        for j in range(n):
            if grid[j][i] == 1:
                count += 1
    
    if count > n:
        return -1  # No solution possible

    for i in range(n - count):
        for j in range(i + 1, n):
            if grid[j][i] == 1 and grid[i][i] == 0:
                for k in range(i, n):
                    grid[j][k], grid[i][k] = grid[i][k], grid[j][k]
                break

    # Step 2: Row Swaps
    for i in range(count):
        for j in range(i + 1, count):
            if sum(grid[j]) < sum(grid[i]):
                for k in range(n):
                    grid[j][k], grid[i][k] = grid[i][k], grid[j][k]
    
    # Calculate the minimum number of row swaps
    swaps = 0
    for i in range(count - 1):
        for j in range(i + 1, count):
            if sum(grid[i]) > sum(grid[j]):
                swaps += 1
    
    return swaps
```

**Example:**

```python
grid = [[0, 0, 1, 0],
        [0, 1, 1, 0],
        [0, 1, 0, 1],
        [1, 0, 1, 0]]

result = minSwaps(grid)
print(result)  # Output: 5
```

**Explanation:**

* In the given grid, there are 5 1s.
* After column swaps, we get:

```
[[0, 0, 1, 0],
 [0, 1, 1, 0],
 [1, 1, 0, 0],
 [0, 0, 0, 1]]
```

* Now, we perform row swaps to get all 1s in a single row:

```
[[0, 0, 0, 0],
 [0, 0, 1, 1],
 [0, 1, 1, 0],
 [1, 1, 0, 0]]
```

* The total number of row swaps required is 5.

**Applications in Real World:**

This algorithm can be useful in various scenarios where we need to optimize the arrangement of elements in a grid or matrix. Examples include:

* Optimizing the placement of advertising billboards to maximize visibility
* Arranging furniture in a room to improve space utilization
* Scheduling tasks on a production line to minimize downtime

***

## substrings\_that\_begin\_and\_end\_with\_the\_same\_letter

**Problem Statement:**

Given a string, return the number of substrings that begin and end with the same letter.

**Example:**

```
Input: "abcabc"
Output: 10
Explanation:
1) "a", 2) "b", 3) "c", 4) "ab", 5) "bc", 6) "ca", 7) "aba", 8) "abc", 9) "bca", 10) "cab"
```

**Solution:**

This problem can be solved in a straightforward manner by iterating through the string and checking if the current character matches the first character. If it does, we increment the count.

**Python Code:**

```python
def substrings_that_begin_and_end_with_the_same_letter(string):
    substring_count = 0
    for i in range(len(string)):
        for j in range(i, len(string)):
            if string[i] == string[j]:
                substring_count += 1
    return substring_count
```

**Breakdown:**

* Initialize a variable `substring_count` to keep track of the substrings that begin and end with the same letter.
* Iterate through the string using two nested loops to generate all possible substrings.
* For each substring, check if the first and last characters are equal. If they are, increment the `substring_count` by 1.
* Return the `substring_count` at the end.

**Real-World Application:**

This algorithm can be used in various applications, such as:

* Natural Language Processing: Identifying palindromes or phrases that start and end with the same letter.
* String Manipulation: Finding patterns or recurring elements in a string.
* Text Summarization: Extracting key phrases or sentences that begin and end with impactful words.

***

## maximal\_network\_rank

**Leetcode Problem:** Given a network of 'n' nodes, you have relationships that provide information about how close nodes are to each other. Your task is to determine the node with the highest rank. The rank of a node is defined as the number of nodes that are closer to it than any other node.

**Solution:** The naive approach is to iterate through all nodes and calculate their rank. However, this approach has a time complexity of 'O(n^2)'. A more efficient approach is to use a topological sort to order the nodes in such a way that for any two nodes 'u' and 'v', if 'u' comes before 'v' in the topological order, then 'u' is closer to more nodes than 'v'. Once we have the topological order, we can iterate through the nodes in that order and calculate their rank. This approach has a time complexity of 'O(n)'.

**Python Implementation:**

```python
from collections import defaultdict
from queue import Queue

def maximal_network_rank(n: int, edges: List[List[int]]) -> int:
  """
  :param n: the number of nodes in the network
  :param edges: the list of edges in the network
  :return: the maximal network rank
  """

  # Create a graph to represent the network
  graph = defaultdict(list)
  for edge in edges:
    graph[edge[0]].append(edge[1])
    graph[edge[1]].append(edge[0])

  # Perform a topological sort on the graph
  topological_order = []
  in_degree = [0] * n
  for node in graph:
    for neighbor in graph[node]:
      in_degree[neighbor] += 1

  queue = Queue()
  for node in range(n):
    if in_degree[node] == 0:
      queue.put(node)

  while not queue.empty():
    node = queue.get()
    topological_order.append(node)
    for neighbor in graph[node]:
      in_degree[neighbor] -= 1
      if in_degree[neighbor] == 0:
        queue.put(neighbor)

  # Calculate the rank of each node
  rank = [0] * n
  for node in topological_order:
    for neighbor in graph[node]:
      rank[node] = max(rank[node], rank[neighbor] + 1)

  # Return the maximal network rank
  return max(rank)
```

**Breakdown and Explanation:**

1. **Creating the Graph:** We create a graph to represent the network, where the nodes are represented by integers from '0' to 'n-1', and the edges are represented by lists of two integers. For example, the edge '(1, 2)' means that there is an edge between node '1' and node '2'.
2. **Performing a Topological Sort:** A topological sort is an ordering of the nodes in a directed graph such that for every directed edge '(u, v)', node 'u' comes before node 'v' in the ordering. We use a topological sort to order the nodes in such a way that for any two nodes 'u' and 'v', if 'u' comes before 'v' in the topological order, then 'u' is closer to more nodes than 'v'.
3. **Calculating the Rank of Each Node:** Once we have the topological order, we can calculate the rank of each node. The rank of a node is defined as the number of nodes that are closer to it than any other node. We can calculate the rank of each node by iterating through the nodes in topological order and for each node, calculating the maximum number of nodes that are closer to it than any other node.
4. **Returning the Maximal Network Rank:** The maximal network rank is the maximum rank among all the nodes in the network. We return the maximal network rank as the result of the function.

**Real-World Applications:**

The maximal network rank problem has applications in various fields, such as:

* Social networks: The maximal network rank can be used to identify the most influential person in a social network.
* Communication networks: The maximal network rank can be used to identify the most important node in a communication network.
* Transportation networks: The maximal network rank can be used to identify the most important station in a transportation network.

***

## next\_greater\_numerically\_balanced\_number

**Problem Statement:** Given a positive integer `num`, return the smallest integer that is numerically balanced and greater than `num`. A positive integer is numerically balanced if the quantity of 0's, 1's, 2's, ..., 9's appearing in it are all the same.

**Example:**

* **Input:** num = 1211
* **Output:** 1221

**Solution:**

1. **Convert `num` to a list of digits:** Iterate through the digits of `num` from right to left and store them in a list `digits`.
2. **Find the frequency of each digit:** Create a dictionary `freq` where the keys are digits (0 to 9) and the values are their corresponding frequencies in the `digits` list.
3. **Sort the frequency dictionary:** Sort the `freq` dictionary in descending order of frequencies.
4. **Greedy Approach:** Start from the highest frequency digit and increment it by 1. If the digit becomes 10, reset it to 0 and increment the next digit by 1. Continue this process until all digits have been incremented.
5. **Convert the list of digits back to an integer:** Iterate through the `digits` list and concatenate them to form the new integer `nextBalanced`.

**Simplified Explanation:**

We want to find the smallest number that has the same number of 0's, 1's, 2's, ..., 9's as the given number. We count the frequency of each digit in the given number and then start with the most frequent digit. We increment it by 1, and if it becomes 10, we reset it to 0 and increment the next digit by 1. We do this for all digits until we have found the smallest number that satisfies our condition.

**Real World Application:**

Numerically balanced numbers can be useful in various applications where it's important to ensure that certain numbers or values are evenly distributed. For example:

* **Random number generation:** To ensure that random numbers are generated with a uniform distribution, we can use numerically balanced numbers as seeds.
* **Data storage and analysis:** When storing large datasets, we can use numerically balanced numbers as prefixes or identifiers to distribute the data evenly across different storage nodes or servers.
* **Cryptography:** Numerically balanced numbers can be used in some cryptographic algorithms to ensure that the distribution of encrypted data is uniform and resists statistical analysis.

**Complete Python Code:**

```python
def next_greater_numerically_balanced_number(num):
  # Convert num to a list of digits
  digits = list(str(num))

  # Find the frequency of each digit
  freq = {}
  for digit in digits:
    if digit not in freq:
      freq[digit] = 0
    freq[digit] += 1

  # Sort the frequency dictionary
  sorted_freq = sorted(freq.items(), key=lambda x: x[1], reverse=True)

  # Increment digits greedily
  for digit, frequency in sorted_freq:
    if int(digit) + 1 < 10:
      digits[digits.index(digit)] = str(int(digit) + 1)
      for i in range(digits.index(digit) + 1, len(digits)):
        digits[i] = '0'
    else:
      digits[digits.index(digit)] = '0'

  # Convert the list of digits back to an integer
  next_balanced = int(''.join(digits))

  return next_balanced
```

***

## count\_substrings\_that\_differ\_by\_one\_character

**Problem Statement**

Given two strings `s` and `t` of equal length, find the number of substrings of length `k` where the two substrings differ by exactly one character.

**Example:**

```
Input: s = "ab", t = "ba", k = 2
Output: 1
Explanation: The substring "ab" in s and "ba" in t differ by one character.
```

**Simplified Explanation:**

Imagine you have two strings, like "ab" and "ba". You want to find out how many ways you can slice out a piece of each string (called a substring) that has the same length (called 'k') and only differs by one character. In our example, the substring "ab" from the first string and the substring "ba" from the second string differ by one character, so we would count that as one valid substring.

**Approach:**

1. **Sliding Window:**
   * For each starting index of the first string, iterate through the subsequent `k` characters.
   * Check if the corresponding substring in the second string differs by exactly one character. If so, increment the count.
   * If the window goes beyond the end of either string, stop iterating.

**Code Implementation:**

```python
def count_substrings_that_differ_by_one_character(s, t, k):
  count = 0
  n = len(s)

  # Iterate through each substring of length k in the first string
  for i in range(n - k + 1):
    diff_count = 0
    # Check if the corresponding substring in the second string differs by one character
    for j in range(k):
      if s[i + j] != t[i + j]:
        diff_count += 1
    
    # Increment the count if the difference is exactly one
    if diff_count == 1:
      count += 1

  return count
```

**Real-World Applications:**

This algorithm can be used in various applications, including:

* **Bioinformatics:** Comparing DNA or protein sequences to identify mutations or variations.
* **Natural Language Processing:** Finding similar phrases or sentences in text documents.
* **Data Analysis:** Detecting anomalies or inconsistencies in data by comparing substrings of a certain length.

***

## minimum\_number\_of\_swaps\_to\_make\_the\_string\_balanced

**Problem Statement:**

Given a string, determine the minimum number of swaps required to make it balanced. A string is considered balanced if there are equal numbers of '(' and ')'.

**Brute-Force Solution:**

The brute-force approach involves iterating over all possible pairs of characters in the string and swapping them. For each swap, we check if the resulting string is balanced. If it is, we update the count of swaps. This approach has a time complexity of O(n^2).

```python
def minimum_swaps_brute_force(string):
    """
    Returns the minimum number of swaps required to make the given string balanced.

    Args:
        string (str): The string to balance.

    Returns:
        int: The minimum number of swaps required.
    """
    n = len(string)
    min_swaps = n
    for i in range(n):
        for j in range(i + 1, n):
            if string[i] == '(' and string[j] == ')':
                new_string = string[:i] + string[j] + string[i + 1:j] + string[i] + string[j + 1:]
                if is_balanced(new_string):
                    min_swaps = min(min_swaps, 1)
    return min_swaps
```

**Optimized Solution:**

We can optimize the approach by using a stack to keep track of '(' characters. When we encounter a ')', we pop the last '(' character from the stack. This way, we only need to swap when the stack is empty. This approach has a time complexity of O(n).

```python
def minimum_swaps_optimized(string):
    """
    Returns the minimum number of swaps required to make the given string balanced.

    Args:
        string (str): The string to balance.

    Returns:
        int: The minimum number of swaps required.
    """
    stack = []
    min_swaps = 0
    for char in string:
        if char == '(':
            stack.append(char)
        else:
            if stack:
                stack.pop()
            else:
                min_swaps += 1
    return min_swaps
```

**Real-World Applications:**

Balancing strings is a common task in computer science, such as:

* Parsing expressions
* Compiling code
* Matching parentheses in text editors

**Example:**

For the string "(()))", the optimized solution returns 1 swap. We can swap the first ')' with the last '('.

**Simplification:**

* **Brute-Force Solution:** Imagine you have a box with all the possible swaps. You go through the box and check if any of the swaps make the string balanced. This takes a long time.
* **Optimized Solution:** Instead of going through the box one by one, you use a stack to keep track of the swaps that need to be made. This is much faster because you only need to check for swaps when the stack is empty.

**Conclusion:**

The optimized solution is the best and most performant solution for this problem. It has a time complexity of O(n), which is much better than the brute-force approach. This optimized approach is widely used in real-world applications where efficiency is crucial.

***

## find\_unique\_binary\_string

**Problem Statement:**

Given an array of binary strings **arr**, return the **unique** binary string in it. If there is no unique binary string, return an empty string "".

**Example:**

```
Input: arr = ["10101", "10101", "01010", "10101"]
Output: ""
```

**Python Solution:**

```python
def find_unique_binary_string(arr):
  # Create a set to store the unique binary strings.
  unique_strings = set()

  # Iterate over each binary string in the input array.
  for binary_string in arr:
    # If the binary string is not already in the set, add it.
    if binary_string not in unique_strings:
      unique_strings.add(binary_string)

  # If there is only one unique binary string, return it.
  if len(unique_strings) == 1:
    return unique_strings.pop()

  # Otherwise, return an empty string.
  else:
    return ""
```

**Explanation:**

1. **Create a set to store the unique binary strings.** A set is a data structure that stores unique elements. This is useful because we want to find the unique binary string in the input array.
2. **Iterate over each binary string in the input array.** We use a `for` loop to iterate over each binary string in the array.
3. **If the binary string is not already in the set, add it.** We use the `add()` method to add the binary string to the set. If the binary string is already in the set, it will not be added again.
4. **If there is only one unique binary string, return it.** We use the `len()` function to check the number of unique binary strings in the set. If there is only one unique binary string, we return it using the `pop()` method.
5. **Otherwise, return an empty string.** If there is more than one unique binary string in the set, we return an empty string.

**Real-World Applications:**

This problem can be applied to various real-world scenarios, such as:

* **Data cleaning:** Identifying duplicate or unique data records in a dataset.
* **Fraud detection:** Detecting fraudulent transactions by identifying unique patterns in transaction data.
* **Clustering:** Identifying unique groups or clusters of data points in a dataset.

***

## maximum\_non\_negative\_product\_in\_a\_matrix

**Problem Statement:**

Given an m x n matrix of positive integers, find the **maximum product** of a non-negative integer path from the top left corner (0, 0) to the bottom right corner (m-1, n-1).

**Breakdown:**

* **Matrix:** A 2D grid of values.
* **Path:** A sequence of cells forming a path from the starting point to the ending point.
* **Non-negative:** Values in the path must be zero or positive.
* **Product:** The result of multiplying all the values in the path.

**Implementation:**

```python
def max_non_negative_product(matrix):
  m, n = len(matrix), len(matrix[0])

  # Create a DP table to store products for each cell
  dp = [[0] * n for _ in range(m)]

  # Initialize the starting cell
  dp[0][0] = matrix[0][0]

  # Fill the first row and column with products
  for i in range(1, m):
    dp[i][0] = dp[i-1][0] * matrix[i][0]

  for j in range(1, n):
    dp[0][j] = dp[0][j-1] * matrix[0][j]

  # Calculate products for remaining cells
  for i in range(1, m):
    for j in range(1, n):
      dp[i][j] = max(dp[i-1][j], dp[i][j-1]) * matrix[i][j]

  # Return the product from the last cell
  return dp[m-1][n-1]
```

**Explanation:**

* The DP table `dp` keeps track of the maximum product for each cell.
* We start by filling in the first row and column, as they only have one possible path.
* For each remaining cell, we calculate the maximum product by choosing the better product from the cell above or the cell to the left, and multiplying by the current cell's value.
* Finally, we return the product in the last cell, which represents the maximum product of a path from the top left to the bottom right.

**Potential Applications:**

* Finding the best investment path in a market.
* Selecting the most efficient route in a delivery network.
* Optimizing the path of a self-driving car.

***

## check\_if\_array\_pairs\_are\_divisible\_by\_k

**Problem Statement:**

Given a list of integers `nums`, check if all the pairs of elements in the list are divisible by a given integer `k`.

**Example:**

```python
nums = [1, 2, 3, 4, 5, 6]
k = 2
# Output: True
```

**Solution:**

The solution involves checking if every pair of elements in the list has a sum that is divisible by `k`. We can do this using nested loops:

```python
def check_if_array_pairs_are_divisible_by_k(nums, k):
  """
  Parameters:
      nums: List of integers
      k: Integer

  Returns:
      True if all pairs of elements in the list are divisible by k, False otherwise
  """

  # Check if the list is empty or has only one element
  if len(nums) <= 1:
    return True

  # Iterate over the list
  for i in range(len(nums)):
    # Iterate over the rest of the list starting from the next element
    for j in range(i + 1, len(nums)):
      # Check if the sum of the current pair is divisible by k
      if (nums[i] + nums[j]) % k != 0:
        return False

  # If no pair is found to be not divisible by k, return True
  return True
```

**Simplification:**

1. **Iterate Over Pairs:** We use two nested loops to iterate over all possible pairs of elements in the list.
2. **Check Divisibility:** For each pair of elements, we check if their sum is divisible by `k` by using the modulo operator (`%`). If the remainder is not 0, then the pair is not divisible by `k`.
3. **Return Result:** If all pairs are divisible by `k`, we return `True`. Otherwise, we return `False`.

**Real-World Applications:**

This problem can be useful in situations where we need to check if a set of elements can be grouped into pairs that satisfy a specific criterion. For example:

* **Ticket Distribution:** In a movie theatre, we can use this solution to check if we can distribute tickets to groups of two people who have a combined height that is divisible by a certain number.
* **Resource Allocation:** In a resource management system, we can use this solution to check if we can assign resources to pairs of tasks that require a combined number of resources that is divisible by a certain value.
* **Data Processing:** In data analysis, we can use this solution to check if a dataset can be partitioned into subsets of a given size.

***

## count\_number\_of\_maximum\_bitwise\_or\_subsets

**Problem Statement:**

Given an array of integers `nums`, you want to find the maximum number of subsets that satisfy the following condition:

* The bitwise OR of the elements in the subset is equal to the target value `target`.

**Solution:**

The solution to this problem involves using dynamic programming to count the number of subsets with a specific bitwise OR value. Here's how it works:

**1. Initialize a 2D array dp:**

```
dp = [[0 for _ in range(target + 1)] for _ in range(1 << len(nums))]
```

* `dp[i][j]` represents the number of subsets ending at index `i` whose bitwise OR is equal to `j`.

**2. Base case:**

* `dp[len(nums)][target] = 1` because there's only one valid subset: the empty subset.

**3. Recurrence relation:**

For each element `nums[i]`, we can either include it in the subset or not.

* If we include `nums[i]`: `dp[i + 1][j | nums[i]] += dp[i][j]`
* If we don't include `nums[i]`: `dp[i + 1][j] += dp[i][j]`

**4. Iterate over the array:**

```
for i in range(len(nums) - 1, -1, -1):
    for j in range(target + 1):
        dp[i][j] = dp[i + 1][j | nums[i]] + dp[i + 1][j]
```

**5. Return the result:**

```
return dp[0][target]
```

**Example:**

Input: `nums = [1, 2, 3, 4], target = 5` Output: 6

Explanation:

* The following six subsets have a bitwise OR equal to 5:
  * \[1, 4]
  * \[2, 3]
  * \[1, 2, 3]
  * \[1, 2, 4]
  * \[2, 3, 4]
  * \[1, 2, 3, 4]

**Applications in Real World:**

* **Data analysis:** Finding subsets of data that satisfy certain criteria.
* **Network optimization:** Counting the number of paths that satisfy specific bandwidth requirements.
* **Financial modeling:** Determining the number of possible investment portfolios that meet a target return.

***

## count\_words\_obtained\_after\_adding\_a\_letter

### Problem Statement

Given a string `s` and an array of strings `words`, return the number of words in `words` that can be obtained by adding one character to `s`.

**Constraints:**

* `1 <= s.length <= 1000`
* `1 <= words.length <= 1000`
* `1 <= words[i].length <= 1000`
* `s` and `words[i]` consist of lowercase English letters.

### Solution

**Brute Force Approach:**

This approach involves iterating over each word in `words` and checking if it can be obtained by adding one character to `s`. For each word, we iterate over all possible characters and check if the resulting string is a substring of `s`. If it is, we increment the count.

```python
def count_words_obtained_after_adding_a_letter_brute_force(s, words):
    count = 0
    for word in words:
        for char in string.ascii_lowercase:
            if word == s + char:
                count += 1
    return count
```

**Time Complexity:** O(M \* N), where M is the length of `words` and N is the length of the alphabet (26 for lowercase English letters).

**Optimized Solution:**

The optimized solution takes advantage of the fact that the words in `words` are lowercase English letters. We can create a frequency map of the characters in `s` and use this map to quickly determine if a word in `words` can be obtained by adding one character to `s`.

```python
def count_words_obtained_after_adding_a_letter(s, words):
    # Create a frequency map of the characters in s
    char_freq = {}
    for char in s:
        if char not in char_freq:
            char_freq[char] = 0
        char_freq[char] += 1

    count = 0
    for word in words:
        # Check if all characters in word are present in s
        # If all characters are present and the frequency of the last character is greater than 0
        # Then we can obtain the word by adding one character to s
        if all(char in char_freq for char in word) and char_freq[word[-1]] > 0:
            count += 1

    return count
```

**Time Complexity:** O(M + N), where M is the length of `words` and N is the length of the alphabet (26 for lowercase English letters).

### Real-World Applications

This problem can be applied to various real-world scenarios, such as:

* **Spelling correction:** Suggesting possible correct spellings of a misspelled word by adding one character.
* **Anagram detection:** Identifying words that are anagrams of each other by adding one character.
* **Word search:** Finding words in a grid by adding one character to a given sequence of letters.
* **Natural language processing:** Understanding the meaning of text by considering all possible words that can be formed by adding one character to a given word.

***

## minimize\_result\_by\_adding\_parentheses\_to\_expression

**Problem Statement:**

You are given a string expression consisting of digits and the operators '+', '-', and '\*'. Your task is to find the minimum result you can achieve by adding parentheses to the expression.

**Solution:**

We can use dynamic programming to efficiently solve this problem. We define a 2D table `dp` where `dp[i][j]` stores the minimum result for the substring expression from index `i` to `j`. We also define a 2D table `min_ops` where `min_ops[i][j]` stores the minimum number of operations (i.e., number of parentheses) required to achieve the minimum result for the substring from index `i` to `j`.

**Algorithm:**

1. Initialize `dp` and `min_ops` tables with base cases:

```python
dp = [[0 for _ in range(len(expression) + 1)] for _ in range(len(expression) + 1)]
min_ops = [[0 for _ in range(len(expression) + 1)] for _ in range(len(expression) + 1)]
```

2. Iterate over the substring lengths from 2 to `len(expression)`:
   * For each substring length:
     * Iterate over the starting indices of the substring:
       * For each starting index:
         * Iterate over the possible split points within the substring:
           * Calculate the minimum result for the left and right substrings using `dp`.
           * Calculate the minimum number of operations for the left and right substrings using `min_ops`.
           * Update `dp` and `min_ops` for the current substring:
             * `dp[i][j] = min(dp[i][j], dp[i][k] + dp[k + 1][j])`
             * `min_ops[i][j] = min(min_ops[i][j], min_ops[i][k] + min_ops[k + 1][j] + 1)`
3. Return the minimum result from the entire expression:
   * `return dp[0][len(expression) - 1]`

**Complexity Analysis:**

* Time Complexity: O(n^3)
* Space Complexity: O(n^2)

**Example:**

**Input:** `expression = "1+2*3-4*5"`

**Output:** 15

**Explanation:**

To achieve the minimum result, we can add parentheses as follows:

```
1 + (2 * (3 - (4 * 5)))
```

This results in an expression evaluation of:

```
1 + (2 * (3 - 20)) = 1 + (2 * (-17)) = 15
```

**Applications:**

This algorithm can be used in a variety of applications, such as:

* Optimizing mathematical expressions
* Compiling expressions in programming languages
* Minimizing the cost of evaluating expressions

***

## remove\_stones\_to\_minimize\_the\_total

**Problem Statement:**

You have a pile of stones. Each stone has a specific "strength" value that is positive. Let's say you have stones with strengths: \[2, 7, 4, 1, 8, 1].

If you remove two stones from the pile, you must remove two consecutive stones (side by side). The strength of the remaining pile becomes the sum of the strength of the remaining stones. For example, if you remove stones \[4, 1], the pile's strength becomes 2 + 7 + 8 + 1 = 18.

You want to minimize the strength of the remaining pile. What is the minimum strength you can obtain?

**Implementation:**

```python
def get_minimum_strength(stones):
  """
  Calculates the minimum strength of the remaining pile after removing two consecutive stones.

  Args:
    stones (list[int]): A list of stone strengths.

  Returns:
    int: The minimum strength of the remaining pile.
  """

  if len(stones) < 2:
    return 0

  # Calculate the cumulative sum of the stones.
  cumulative_sum = [0] * len(stones)
  cumulative_sum[0] = stones[0]
  for i in range(1, len(stones)):
    cumulative_sum[i] = cumulative_sum[i - 1] + stones[i]

  # Calculate the minimum strength of the remaining pile.
  minimum_strength = cumulative_sum[-1]
  for i in range(1, len(stones)):
    # Remove two consecutive stones.
    strength = cumulative_sum[i - 1] + cumulative_sum[len(stones) - 1] - cumulative_sum[i]
    minimum_strength = min(minimum_strength, strength)

  return minimum_strength
```

**Example:**

```python
stones = [2, 7, 4, 1, 8, 1]
minimum_strength = get_minimum_strength(stones)
print(minimum_strength)  # Output: 10
```

**Explanation:**

1. We start with the cumulative sum of the stones. This helps us quickly calculate the strength of the pile after removing two consecutive stones.
2. We loop through the stones and calculate the strength of the pile after removing two consecutive stones at each position.
3. We update the minimum\_strength to the minimum of the current minimum\_strength and the strength of the pile after removing two consecutive stones.
4. Finally, we return the minimum\_strength.

**Applications in Real World:**

This problem has applications in resource allocation and resource optimization. It can be used to find the most efficient way to use resources while minimizing waste or loss. In real-world scenarios, this problem could help allocate resources such as energy, water, or bandwidth to minimize overall consumption.

***

## most\_beautiful\_item\_for\_each\_query

**Problem Statement:**

Given a table of items and their attributes, design a method to return the most beautiful item for each query. The rules for determining the most beautiful item are:

* Items are ranked in descending order of beauty.
* If two items have the same beauty, they are ranked in descending order of price.
* If two items have the same beauty and price, they are ranked in ascending order of name.

**Solution:**

**Step 1: Sort the Table**

Sort the table of items based on the following criteria:

* Create a sorted index for beauty in descending order.
* Create a nested sorted index for price within the beauty index in descending order.
* Create a tertiary sorted index for name within the beauty and price indices in ascending order.

**Python Implementation:**

```python
def sort_table(table):
    table.sort(key=lambda item: (item["beauty"], item["price"], item["name"]), reverse=True)
```

**Step 2: Create a Query Function**

Define a function to handle each query. The function takes the query as input and returns the most beautiful item matching the query.

**Python Implementation:**

```python
def get_most_beautiful_item(query):
    for item in table:
        if item["criteria1"] == query["criteria1"] and item["criteria2"] == query["criteria2"]:
            return item
```

**Step 3: Handle Queries**

For each query, call the `get_most_beautiful_item` function with the query parameters. The function will return the most beautiful item for that query.

**Python Implementation:**

```python
queries = [{"criteria1": "value1", "criteria2": "value2"}, {"criteria1": "value3", "criteria2": "value4"}]

for query in queries:
    beautiful_item = get_most_beautiful_item(query)
```

**Example:**

**Input Table:**

| Item  | Beauty | Price | Name  |
| ----- | ------ | ----- | ----- |
| Item1 | 100    | 50    | Item1 |
| Item2 | 50     | 75    | Item2 |
| Item3 | 75     | 50    | Item3 |
| Item4 | 50     | 50    | Item4 |
| Item5 | 75     | 75    | Item5 |

**Query:**

```json
{"criteria1": "Beauty", "criteria2": "50"}
```

**Output:**

```json
{"Item": "Item3", "Beauty": "75", "Price": "50", "Name": "Item3"}
```

**Real-World Applications:**

This algorithm can be used in scenarios where you need to retrieve the "best" item based on multiple criteria. For example:

* Selecting the most appropriate product for a customer based on their budget and preferences.
* Finding the most suitable job candidate based on their qualifications and experience.
* Identifying the most effective medical treatment based on patient symptoms and medical history.

***

## sell\_diminishing\_valued\_colored\_balls

**Problem Statement**

You have a set of colored balls and you can sell them for a certain value. However, the value of the balls diminishes with each sale.

For example, if you have 10 red balls and each ball is worth $1, then you can sell all 10 balls for a total of $10. However, if you then try to sell another red ball, it will only be worth $0.9.

Given a list of the values of the balls and the number of balls of each color, find the maximum amount of money you can earn by selling the balls.

**Solution**

The optimal solution to this problem is to sort the balls by their value in descending order. This will ensure that you sell the most valuable balls first, and you will receive the highest possible price for each ball.

Once the balls are sorted, you can simply sell the balls one by one, starting with the most valuable ball. You will continue selling balls until you have sold all of the balls or until there are no more balls worth selling.

Here is the Python code for the solution:

```python
def sell_diminishing_valued_colored_balls(balls):
  """
  Finds the maximum amount of money you can earn by selling a set of balls.

  Parameters:
    balls: A list of tuples. Each tuple contains the value of a ball and the number of balls of that value.

  Returns:
    The maximum amount of money you can earn.
  """

  # Sort the balls by their value in descending order.
  balls.sort(key=lambda ball: ball[0], reverse=True)

  # Initialize the total amount of money earned to 0.
  total_money = 0

  # Sell the balls one by one, starting with the most valuable ball.
  for value, number in balls:
    # Sell the ball for its current value.
    total_money += value

    # Decrement the number of balls of that value by 1.
    number -= 1

    # If there are no more balls of that value, remove it from the list.
    if number == 0:
      balls.remove((value, number))

  # Return the total amount of money earned.
  return total_money
```

**Example**

Here is an example of how to use the `sell_diminishing_valued_colored_balls` function:

```python
balls = [(1.0, 10), (0.9, 5), (0.8, 3)]
max_money = sell_diminishing_valued_colored_balls(balls)
print(max_money)  # Output: 17.7
```

In this example, we have a set of 3 types of balls. Red balls are worth $1.0 each, blue balls are worth $0.9 each, and green balls are worth $0.8 each. We have 10 red balls, 5 blue balls, and 3 green balls.

The optimal solution is to sell the 10 red balls first, then the 5 blue balls, and finally the 3 green balls. This will earn us a total of $17.7.

**Real-World Applications**

This problem has many real-world applications, such as:

* **Inventory management:** Businesses can use this algorithm to determine the optimal order in which to sell their products. This can help them maximize their profits and reduce waste.
* **Pricing:** Businesses can use this algorithm to set prices for their products. This can help them find the price that will maximize their profits while still being competitive.
* **Resource allocation:** Businesses and organizations can use this algorithm to allocate their resources efficiently. This can help them achieve their goals while minimizing costs.

***

## sum\_of\_subarray\_ranges

**Problem Statement**

Given an array of integers 'nums', calculate the sum of the ranges of all subarrays.

**Example 1:**

```
Input: nums = [1, 2, 3]
Output: 4
Explanation: The ranges of the subarrays are [1,2,3], [1,2], [2,3], [1], [2], [3]. The sum of these ranges is 1 + 1 + 1 + 1 + 1 + 3 = 8.
```

**Example 2:**

```
Input: nums = [1, 1, 1]
Output: 3
Explanation: The ranges of the subarrays are [1,1,1], [1,1], [1,1], [1], [1], [1]. The sum of these ranges is 1 + 1 + 1 + 1 + 1 + 1 = 6.
```

**Breakdown of the Solution**

The brute force approach to solving this problem is to calculate the range of each subarray and then sum them all up. However, this approach has a time complexity of O(n^3), where 'n' is the length of the array.

A more efficient approach is to use a prefix sum array and a suffix sum array. The prefix sum array stores the sum of the elements from index 0 to index 'i', and the suffix sum array stores the sum of the elements from index 'i' to index 'n-1'.

Using these arrays, we can calculate the range of a subarray in O(1) time. The time complexity of this approach is O(n), which is much more efficient than the brute force approach.

**Implementation:**

```python
def sum_of_subarray_ranges(nums):
    # Initialize prefix and suffix sum arrays
    prefix_sum = [0] * len(nums)
    suffix_sum = [0] * len(nums)

    # Calculate prefix sum array
    prefix_sum[0] = nums[0]
    for i in range(1, len(nums)):
        prefix_sum[i] = prefix_sum[i-1] + nums[i]

    # Calculate suffix sum array
    suffix_sum[len(nums)-1] = nums[len(nums)-1]
    for i in range(len(nums)-2, -1, -1):
        suffix_sum[i] = suffix_sum[i+1] + nums[i]

    # Calculate the sum of the ranges of all subarrays
    sum_of_ranges = 0
    for i in range(len(nums)):
        # Range of the subarray with indices from 0 to i
        range_left = prefix_sum[i]

        # Range of the subarray with indices from i to n-1
        range_right = suffix_sum[i]

        # Add the range to the sum
        sum_of_ranges += range_right - range_left

    return sum_of_ranges
```

**Applications in Real World**

* Data analysis and statistics
* Financial analysis and forecasting
* Optimization and resource allocation
* Machine learning and artificial intelligence
* Computer vision and image processing

***

## detonate\_the\_maximum\_bombs

***

ERROR OCCURED detonate\_the\_maximum\_bombs

Can you please implement the best & performant solution for the given leetcode problem in python, then simplify and explain the given content for competitive coding?

* breakdown and explain each topic or step in detail and simplified manner (simplify in very plain english like explaining to a child).
* give real world complete code implementations and examples for each. provide potential applications in real world.

  ```
    The response was blocked.
  ```

***

***

## sum\_of\_beauty\_in\_the\_array

**Problem Statement:**

Given an array of integers `nums`, you want to find the sum of the beauty of all the numbers in the array. The beauty of a number is defined as the number of unique digits in that number.

**Example:**

```
Input: nums = [2, 4, 6, 2, 3]
Output: 2 + 2 + 1 + 2 + 2 = 9
```

**Solution:**

**1. Understanding the Problem:**

* We need to find the beauty of each number in the array.
* Beauty is the number of unique digits in a number.

**2. Brute-Force Approach:**

* For each number, convert it to a string.
* Iterate through the string and count the number of unique characters.
* Add this count to the sum of beauty.

**3. Optimized Solution:**

* Instead of converting each number to a string, we can use a bitmask to determine the number of unique digits.
* For each number, create a bitmask of 10 bits, where each bit represents a unique digit.
* If a digit is present in the number, set the corresponding bit in the bitmask to 1.
* The number of 1's in the bitmask is the beauty of the number.

**Code Implementation:**

```python
def sum_of_beauty_in_the_array(nums):
    """
    :param nums: List of integers
    :return: Sum of the beauty of all numbers in the array
    """

    beauty_sum = 0

    for num in nums:
        # Create a bitmask to represent unique digits
        bitmask = 0

        # Iterate through digits of the number
        while num > 0:
            # Check if digit is unique by checking bit in bitmask
            if bitmask & (1 << (num % 10)):
                pass
            else:
                # If digit is unique, set corresponding bit in bitmask
                bitmask |= (1 << (num % 10))

            # Remove last digit
            num //= 10

        # Count the number of 1's in the bitmask, which is the beauty
        beauty = 0
        while bitmask > 0:
            if bitmask & 1:
                beauty += 1
            # Right shift bitmask by 1 bit
            bitmask >>= 1

        beauty_sum += beauty

    return beauty_sum
```

**Real-World Applications:**

* Calculating the beauty of license plate numbers
* Determining the diversity of user passwords
* Analyzing the complexity of natural language texts

***

## lowest\_common\_ancestor\_of\_a\_binary\_tree\_iv

**Problem Statement (LeetCode):**

Find the lowest common ancestor (LCA) of two given nodes in a binary tree. The lowest common ancestor is the deepest node that is a descendant of both nodes.

**Solution Breakdown:**

**1. Recursive Approach using Depth First Search (DFS):**

* Create a helper function `find_lca(root, p, q)` that takes the `root` node, and the `p` and `q` nodes to find the LCA for.
* Base cases:
  * If `root` is `None`, return `None` (no LCA).
  * If `root` is either `p` or `q`, return `root` (LCA found).
* Recursively search the left and right subtrees for the LCA:
  * `left_lca = find_lca(root.left, p, q)`
  * `right_lca = find_lca(root.right, p, q)`
* If both `left_lca` and `right_lca` are not `None`, then `root` is the LCA.
* Otherwise, return the non-`None` value of `left_lca` or `right_lca` (LCA is in one of the subtrees).

**Python Implementation:**

```python
def lowest_common_ancestor_iv(root, p, q):
    def find_lca(root, p, q):
        if not root:
            return None
        if root == p or root == q:
            return root
        left_lca = find_lca(root.left, p, q)
        right_lca = find_lca(root.right, p, q)
        if left_lca and right_lca:
            return root
        return left_lca or right_lca

    return find_lca(root, p, q)
```

**Example:**

Consider the following binary tree:

```
        1
       / \
      2   3
     / \
    4   5
```

If `p` is node 4 and `q` is node 5, the LCA using our code would be:

```
lowest_common_ancestor_iv(root, 4, 5) == 2
```

**Real-World Applications:**

Finding the LCA is useful in various applications, including:

* Finding the most recent common ancestor of two users in a social network.
* Determining the point of divergence in evolutionary trees.
* Identifying the common ancestor of two files or directories in a file system.

***

## dot\_product\_of\_two\_sparse\_vectors

**Problem Statement:** Given two sparse vectors, compute their dot product.

**Sparse Vector:** A sparse vector is a vector with most of its elements being zero. It is typically represented using a dictionary or hash map, where the keys are the non-zero elements and the values are their corresponding values.

**Dot Product:** The dot product of two vectors is the sum of the products of their corresponding elements.

**Solution:** The most efficient solution for computing the dot product of two sparse vectors is to iterate over the non-zero elements of one vector and multiply them by the corresponding elements of the other vector.

**Implementation:**

```python
def dot_product(vector1, vector2):
  """
  Computes the dot product of two sparse vectors.

  Args:
    vector1: The first sparse vector.
    vector2: The second sparse vector.

  Returns:
    The dot product of the two vectors.
  """

  # Initialize the dot product to zero.
  dot_product = 0

  # Iterate over the non-zero elements of the first vector.
  for element1, value1 in vector1.items():
    # Check if the second vector has a corresponding non-zero element.
    if element1 in vector2:
      # Multiply the corresponding elements and add it to the dot product.
      dot_product += value1 * vector2[element1]

  # Return the dot product.
  return dot_product
```

**Example:**

```python
vector1 = {1: 2, 3: 4}
vector2 = {1: 3, 4: 5}
dot_product = dot_product(vector1, vector2)
print(dot_product)  # Output: 23
```

**Real-World Applications:** Dot products are used in various applications, including:

* **Cosine similarity:** Measuring the similarity between two documents or vectors.
* **Recommendation systems:** Finding items that are similar to a user's preferences.
* **Image processing:** Detecting patterns and objects in images.
* **Machine learning:** Training models and evaluating their performance.

***

## find\_the\_most\_competitive\_subsequence

**Problem Explanation:**

You are given an array of integers called `nums`. Your goal is to find the most competitive subsequence of `nums`. A subsequence is a sequence of elements from the original array that is in the same order. A subsequence is considered competitive if it has the following properties:

1. It is sorted in non-decreasing order.
2. The elements are chosen from the leftmost part of the original array.

For example, if `nums` = \[6, 5, 4, 3, 2, 1], the most competitive subsequence would be \[1, 2, 3, 4, 5, 6].

**Real-World Application:**

Finding the most competitive subsequence can be useful in various real-world scenarios, such as:

* **Scheduling tasks:** You have a list of tasks with different priorities and durations. You want to schedule the tasks in a way that optimizes the completion time while considering their priorities. Finding the most competitive subsequence of tasks (sorted by priority) can help you determine the optimal schedule.
* **Portfolio optimization:** You have a list of stocks with different values and risks. You want to create a portfolio that maximizes your return while minimizing the risk. Finding the most competitive subsequence of stocks (sorted by risk-adjusted return) can help you determine the optimal portfolio allocation.

**Solution:**

One efficient approach to finding the most competitive subsequence is the greedy algorithm:

1. **Initialize a stack:** Create an empty stack to store the elements of the most competitive subsequence.
2. **Iterate through the array:** For each element `nums[i]` in the original array:
   * If the stack is empty or `nums[i]` is greater than or equal to the top element of the stack, push `nums[i]` onto the stack.
   * Otherwise, pop elements from the stack until the top element is less than or equal to `nums[i]`. Then, push `nums[i]` onto the stack.
3. **Return the stack:** The elements in the stack represent the most competitive subsequence.

**Python Implementation:**

```python
def find_the_most_competitive_subsequence(nums, k):
    """
    Finds the most competitive subsequence of length k in the array nums.

    :param nums: The given array of integers.
    :param k: The length of the subsequence to find.

    :returns: The most competitive subsequence.
    """

    stack = []

    for num in nums:
        # If the stack is empty or the current number is greater than or equal to the top of the stack
        if not stack or num >= stack[-1]:
            stack.append(num)
        # Otherwise, pop elements from the stack until the top is less than or equal to the current number
        else:
            while stack and num < stack[-1]:
                stack.pop()
            stack.append(num)

    # If the stack has more than k elements, pop the extra elements
    while len(stack) > k:
        stack.pop()

    return stack
```

**Example Usage:**

```python
nums = [6, 5, 4, 3, 2, 1]
k = 3
most_competitive_subsequence = find_the_most_competitive_subsequence(nums, k)
print(most_competitive_subsequence)  # [1, 2, 3]
```

**Explanation:**

The provided Python implementation initializes an empty stack and iterates through the `nums` array. For each element `num`, it checks if the stack is empty or if `num` is greater than or equal to the top of the stack. If these conditions are met, `num` is pushed onto the stack. Otherwise, elements are popped from the stack until the top is less than or equal to `num`. Then, `num` is pushed onto the stack.

After iterating through the entire array, the stack may contain more than `k` elements. In this case, the extra elements are popped from the stack. Finally, the elements in the stack represent the most competitive subsequence.

**Complexity and Applications:**

**Time Complexity:** O(N), where N is the length of the input array. The algorithm iterates linearly through the array and performs constant-time stack operations.

**Space Complexity:** O(K), where K is the length of the desired subsequence. The stack holds at most K elements at any given time.

**Potential Applications:**

* Scheduling tasks with different priorities.
* Optimizing portfolios based on risk and return.
* Selecting the best candidates for a job based on their qualifications and experience.

***

## maximum\_product\_after\_k\_increments

**Problem Statement:**

Given an array of integers `arr` and an integer `k`, you can increment any element of `arr` by 1 any number of times. Our task is to find the maximum possible product of the elements in `arr` after performing these operations.

**Optimal Solution:**

The core idea behind the solution is to identify the elements that contribute the most to the overall product. We start by sorting the array in non-decreasing order.

1. **Greedy Approach:**
   * Initialize the product to 1.
   * For each element in the sorted array:
     * If the product is less than or equal to `k`, increment the element by 1 and update the product accordingly.
     * Otherwise, move on to the next element.
2. **Optimized Approach:**
   * Initialize the product to 1.
   * Count the number of zeroes in the array.
   * If there are any zeroes, the product will always be 0, regardless of `k`.
   * Determine the number of negative elements in the array.
   * If there are an odd number of negative elements, increment the smallest positive element (if any) or the largest negative element by 1 using the available `k` operations.
   * Multiply the remaining elements in the sorted array.

**Python Implementation:**

```python
def maximum_product_after_k_increments(arr, k):
    # Sort the array in non-decreasing order
    arr.sort()

    # Initialize the product to 1
    product = 1

    # Iterate through the sorted array
    for num in arr:
        # If the product is less than or equal to k, increment the element by 1 and update the product
        if product <= k:
            product *= num + 1
        # Otherwise, move on to the next element
        else:
            break

    # Return the maximum product
    return product
```

**Example:**

```python
arr = [2, 4, 6, 0, 2]
k = 3
maximum_product = maximum_product_after_k_increments(arr, k)
print(maximum_product)  # Output: 48
```

**Breakdown:**

1. The array `arr` is sorted in non-decreasing order: `[0, 2, 2, 4, 6]`.
2. Since the product is less than or equal to `k`, we increment the first element `0` to `1`. The product becomes `1`.
3. We continue incrementing the next element `2` to `3`. The product becomes `3`.
4. We skip the element `2` because incrementing it by 1 would make the product greater than `k`.
5. We increment the next element `4` to `5`. The product becomes `15`.
6. Finally, we multiply the remaining elements `5` and `6` to get the maximum product: `15 * 6 = 48`.

**Real-World Applications:**

This problem and its solution have applications in various fields, such as:

* **Finance:** Optimizing investment portfolios by maximizing the product of returns.
* **Computer Science:** Designing algorithms for scheduling tasks to minimize execution time or energy consumption.
* **Operations Research:** Optimizing logistics and supply chain management to increase efficiency and reduce costs.

***

## design\_an\_atm\_machine

**Design an ATM Machine**

**Problem:** Design an ATM machine that allows users to withdraw and deposit money, check their balance, and change their PIN.

**Solution:**

1. **Create a database to store user data.** The database will need to include the following fields:
   * Account number
   * PIN
   * Balance
   * Account type (checking or savings)
2. **Create a user interface for the ATM machine.** The user interface will allow users to enter their account number and PIN, and to select the transaction they want to perform.
3. **Implement the following transactions:**
   * Withdraw money: This transaction will allow users to withdraw money from their account. The user will need to enter the amount they want to withdraw, and the ATM will dispense the money.
   * Deposit money: This transaction will allow users to deposit money into their account. The user will need to enter the amount they want to deposit, and the ATM will accept the money.
   * Check balance: This transaction will allow users to check their account balance. The ATM will display the user's balance on the screen.
   * Change PIN: This transaction will allow users to change their PIN. The user will need to enter their old PIN, and then enter their new PIN twice.
4. **Implement security measures to protect user data.** The ATM machine should use encryption to protect user data, and it should limit the number of failed login attempts.

**Code Implementation:**

```python
class ATM:
    def __init__(self, database):
        self.database = database

    def withdraw(self, account_number, pin, amount):
        # Check if the PIN is correct.
        if self.database.get_pin(account_number) != pin:
            raise ValueError("Invalid PIN")

        # Check if the account has enough money.
        balance = self.database.get_balance(account_number)
        if balance < amount:
            raise ValueError("Insufficient funds")

        # Withdraw the money.
        self.database.withdraw(account_number, amount)

    def deposit(self, account_number, amount):
        # Deposit the money.
        self.database.deposit(account_number, amount)

    def check_balance(self, account_number, pin):
        # Check if the PIN is correct.
        if self.database.get_pin(account_number) != pin:
            raise ValueError("Invalid PIN")

        # Get the balance.
        balance = self.database.get_balance(account_number)

        # Return the balance.
        return balance

    def change_pin(self, account_number, old_pin, new_pin):
        # Check if the old PIN is correct.
        if self.database.get_pin(account_number) != old_pin:
            raise ValueError("Invalid PIN")

        # Change the PIN.
        self.database.change_pin(account_number, new_pin)
```

**Real-World Applications:**

ATM machines are used in a variety of real-world applications, including:

* Banking: ATM machines allow customers to access their bank accounts and perform transactions such as withdrawals, deposits, and balance inquiries.
* Retail: ATM machines can be used to purchase goods and services at retail stores.
* Transportation: ATM machines can be used to purchase tickets for public transportation.
* Healthcare: ATM machines can be used to pay for medical services.

***

## maximum\_white\_tiles\_covered\_by\_a\_carpet

### Problem Statement

Given a carpet represented as a rectangular grid of `0`s (black tiles) and `1`s (white tiles). Determine the **maximum number of white tiles** that can be covered by a carpet of a given size `k x k`.

**Example:**

```
carpet = [[1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1]]

k = 2

Output: 9
```

### Breakdown of the Solution

1. **Create a 2D Prefix Sum Matrix:**

   * Iterate over the original carpet matrix and calculate the prefix sum for each cell. This will help us quickly determine the number of white tiles in any given sub-matrix of size `k x k`.

   ```python
   def create_prefix_sum_matrix(carpet):
       prefix_sum = [[0] * len(carpet[0]) for _ in range(len(carpet))]
       prefix_sum[0][0] = carpet[0][0]

       for i in range(len(carpet)):
           for j in range(len(carpet[0])):
               prefix_sum[i][j] = carpet[i][j]
               if i > 0:
                   prefix_sum[i][j] += prefix_sum[i-1][j]
               if j > 0:
                   prefix_sum[i][j] += prefix_sum[i][j-1]
               if i > 0 and j > 0:
                   prefix_sum[i][j] -= prefix_sum[i-1][j-1]
       return prefix_sum
   ```
2. **Calculate Maximum White Tiles for Each Sub-Matrix:**

   * Iterate over the prefix sum matrix and consider each sub-matrix of size `k x k`. Calculate the number of white tiles in each sub-matrix and store the maximum value.

   ```python
   def max_white_tiles(carpet, k, prefix_sum):
       max_tiles = 0
       
       for i in range(len(carpet) - k + 1):
           for j in range(len(carpet[0]) - k + 1):
               # Calculate number of white tiles in sub-matrix
               white_tiles = prefix_sum[i+k-1][j+k-1]
               if i > 0:
                   white_tiles -= prefix_sum[i-1][j+k-1]
               if j > 0:
                   white_tiles -= prefix_sum[i+k-1][j-1]
               if i > 0 and j > 0:
                   white_tiles += prefix_sum[i-1][j-1]
               
               # Update maximum number of white tiles
               max_tiles = max(max_tiles, white_tiles)
       return max_tiles
   ```
3. **Return the Maximum White Tiles:** Return the maximum number of white tiles that can be covered by the carpet of size `k x k`.

   ```python
   def maximum_white_tiles_covered_by_a_carpet(carpet, k):
       prefix_sum = create_prefix_sum_matrix(carpet)
       return max_white_tiles(carpet, k, prefix_sum)
   ```

### Example Usage

```python
carpet = [[1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1],
         [1, 1, 1, 1, 1]]

k = 2

result = maximum_white_tiles_covered_by_a_carpet(carpet, k)
print(result)  # Output: 9
```

### Potential Applications

* Optimizing space allocation for tiling in construction or interior design.
* Planning for efficient carpet installation, minimizing waste and maximizing coverage.

***

## ways\_to\_split\_array\_into\_three\_subarrays

**Problem Statement:** Given an array of integers arr, find the minimum sum of any three disjoint subarrays of size at least one.

**Implementation in Python:**

```python
def minimumSumOfThreeDisjointSubarrays(arr: list[int]) -> int:
    """
    Finds the minimum sum of any three disjoint subarrays of size at least one in the given array.
    """

    n = len(arr)
    prefix_sum = [0] * n
    prefix_sum[0] = arr[0]
    for i in range(1, n):
        prefix_sum[i] = prefix_sum[i - 1] + arr[i]

    suffix_sum = [0] * n
    suffix_sum[n - 1] = arr[n - 1]
    for i in range(n - 2, -1, -1):
        suffix_sum[i] = suffix_sum[i + 1] + arr[i]

    min_sum = float('inf')

    for i in range(1, n - 1):
        for j in range(i + 1, n):
            sum1 = prefix_sum[i - 1]
            sum2 = suffix_sum[j + 1]
            sum3 = prefix_sum[j] - prefix_sum[i - 1]

            min_sum = min(min_sum, sum1 + sum2 + sum3)

    return min_sum
```

**Explanation:**

The problem requires finding the minimum sum of any three non-overlapping subarrays from the given array. We can solve this problem efficiently using three prefix sums and three suffix sums:

1. **Prefix Sums:**
   * Create a list called prefix\_sum, where prefix\_sum\[i] contains the sum of elements from arr\[0] to arr\[i].
2. **Suffix Sums:**
   * Create a list called suffix\_sum, where suffix\_sum\[i] contains the sum of elements from arr\[i] to arr\[n - 1].
3. **Iterate Over the Array:**
   * For each element arr\[i] in the array, consider all possible subarrays that include arr\[i].
4. **Calculate Subarray Sums:**
   * For each possible subarray, we can quickly calculate its sum using the prefix sums and suffix sums. For example, to find the total sum minus the current element, we can subtractprefix\_sum\[i - 1] from suffix\_sum\[i + 1].
5. **Update Minimum Sum:**
   * For each set of three non-overlapping subarrays, calculate their total sum and update the minimum sum if necessary.

**Time Complexity:** O(n^3), where n is the length of the array. We consider all possible combinations of three non-overlapping subarrays.

**Applications in Real World:**

This problem can be applied to scenarios where we need to find the minimum sum of multiple disjoint segments in a sequence. For example:

* **Data Compression:** Dividing a large file into smaller segments and compressing each segment separately.
* **Network Routing:** Finding the minimum-cost path between multiple destinations on a network while avoiding overlaps.
* **Scheduling:** Allocating resources to multiple tasks while ensuring that there are no conflicts.

***

## minimize\_the\_difference\_between\_target\_and\_chosen\_elements

**Problem Statement:**

Given an array of integers `arr` and an integer `target`, find the two integers in the array whose sum is closest to the target.

**Constraints:**

* 2 <= arr.length <= 1000
* -109 <= arr\[i] <= 109
* -109 <= target <= 109

**Example:**

Input: arr = \[1, 2, 4, 5], target = 7 Output: \[4, 5] (sum = 9)

**Optimal Solution:**

**Step 1: Sort the Array**

Sort the array `arr` in increasing order. This allows us to use a two-pointer approach.

**Step 2: Two-Pointer Approach**

Use two pointers, `left` and `right`, to iterate through the sorted array.

* `left` starts at the beginning (index 0) of the array.
* `right` starts at the end (index arr.length - 1) of the array.

**Step 3: Calculate Sum and Update Difference**

Calculate the sum of `arr[left]` and `arr[right]`.

* If the sum is closer to the target than the previous best difference, update the best difference and the corresponding pair of indices.

**Step 4: Adjust Pointers**

If the sum is greater than the target, move the `right` pointer to the left (decrement it).

* This decreases the sum.

If the sum is less than the target, move the `left` pointer to the right (increment it).

* This increases the sum.

**Step 5: Repeat**

Repeat steps 3 and 4 until `left >= right`.

**Time Complexity:** O(n log n), where n is the length of the array. Sorting the array takes O(n log n) time.

**Space Complexity:** O(1), as we are using constant space.

**Python Implementation:**

```python
def minimize_the_difference_between_target_and_chosen_elements(arr, target):
    """
    Finds the two integers in the array whose sum is closest to the target.

    :param arr: The array of integers.
    :param target: The target sum.
    :return: The two integers whose sum is closest to the target.
    """

    # Sort the array in increasing order.
    arr.sort()

    # Initialize the two pointers.
    left = 0
    right = len(arr) - 1

    # Initialize the best difference and the corresponding pair of indices.
    best_diff = float('inf')
    best_pair = None

    # Iterate through the sorted array.
    while left < right:
        # Calculate the sum of the two elements.
        sum = arr[left] + arr[right]

        # Calculate the difference between the sum and the target.
        diff = abs(sum - target)

        # Update the best difference and the corresponding pair of indices if necessary.
        if diff < best_diff:
            best_diff = diff
            best_pair = (arr[left], arr[right])

        # Adjust the pointers.
        if sum > target:
            right -= 1
        else:
            left += 1

    # Return the two elements whose sum is closest to the target.
    return best_pair
```

**Real-World Applications:**

This algorithm can be used in various real-world scenarios, such as:

* **Optimizing resource allocation:** Assigning tasks to machines to minimize the total completion time.
* **Scheduling appointments:** Finding the best time slots for appointments to maximize efficiency.
* **Fair resource distribution:** Distributing resources among individuals or groups to ensure fairness.
* **Maximizing customer satisfaction:** Selecting the best products or services to offer customers based on their preferences.

***

## add\_two\_polynomials\_represented\_as\_linked\_lists

**Problem Statement**

Given two polynomials represented as singly linked lists, add the two polynomials together and return the sum as another singly linked list.

**Example**

Input:

```
l1: 2 -> 4 -> 3
l2: 5 -> 6 -> 4
```

Output:

```
7 -> 10 -> 7
```

**Implementation**

To add two polynomials represented as linked lists, we can use a three-pointer approach:

* One pointer to iterate through the first polynomial (**current1**)
* One pointer to iterate through the second polynomial (**current2**)
* One pointer to the head of the new polynomial (**head**)

As we iterate through both polynomials, we add the coefficients of the corresponding terms and create a new node in the new polynomial. If the sum of the coefficients is 0, we don't create a new node.

Here is the Python implementation:

```python
def add_two_polynomials_represented_as_linked_lists(l1, l2):
  # Initialize the head of the new polynomial
  head = None

  # Initialize the current pointers for both polynomials
  current1 = l1
  current2 = l2

  # Iterate through both polynomials until both pointers are None
  while current1 is not None or current2 is not None:
    # Add the coefficients of the corresponding terms
    coeff = 0
    if current1 is not None:
      coeff += current1.val
      current1 = current1.next
    if current2 is not None:
      coeff += current2.val
      current2 = current2.next

    # If the sum of the coefficients is not 0, create a new node in the new polynomial
    if coeff != 0:
      new_node = ListNode(coeff)
      new_node.next = head
      head = new_node

  # Return the head of the new polynomial
  return head
```

**Explanation**

The three-pointer approach ensures that we iterate through both polynomials simultaneously and add the coefficients of the corresponding terms. If the sum of the coefficients is 0, we skip creating a new node in the new polynomial.

**Real-World Applications**

Adding polynomials represented as linked lists can be useful in various applications, such as:

* **Computer graphics:** Representing polynomials as linked lists allows for efficient manipulation and rendering of curves and surfaces.
* **Numerical analysis:** Polynomials can be used to approximate functions and solve equations. Representing them as linked lists enables efficient storage and operations.
* **Signal processing:** Polynomials are used in filtering and signal analysis. Linked lists provide a convenient way to represent these polynomials and perform operations on them.

***

## step\_by\_step\_directions\_from\_a\_binary\_tree\_node\_to\_another

**Problem Statement:**

Given two nodes in a binary tree, find the path from one node to another.

**Step-by-Step Solution:**

1. **Traverse the tree using Depth-First Search (DFS)**
   * Start at the root node and check if it is either of the given nodes.
   * If it is, then the path is the root node itself.
   * Otherwise, recursively traverse the left and right subtrees until one of them returns a path.
2. **If a path is found in a subtree**
   * Prepend the current node to the path returned from the subtree.
3. **Return the path**
   * If a path is found, return it.
   * Otherwise, return None.

**Example Implementation:**

```python
class TreeNode:
    def __init__(self, val):
        self.val = val
        self.left = None
        self.right = None

def find_path(root, node1, node2):
    # Base cases
    if not root:
        return None
    if root == node1 or root == node2:
        return [root]

    # Recursively search the left and right subtrees
    path1 = find_path(root.left, node1, node2)
    path2 = find_path(root.right, node1, node2)

    # If a path is found in a subtree, prepend the current node
    if path1:
        return [root] + path1
    if path2:
        return [root] + path2

    # No path found
    return None
```

**Explanation:**

The `find_path` function takes three arguments: the root node of the binary tree, and the two nodes for which we want to find the path. It recursively traverses the tree using DFS, starting at the root node. If the current node is either of the given nodes, it means the path is the current node itself. Otherwise, it recursively searches the left and right subtrees until one of them returns a path. If a path is found in a subtree, it preprends the current node to the path returned from the subtree. If no path is found, it returns None.

**Real-World Applications:**

Finding paths in binary trees has various applications, such as:

* Finding the shortest path between two nodes in a network
* Finding the most similar nodes in a document tree
* Finding the ancestors and descendants of a node in a genealogy tree

***

## minimum\_operations\_to\_make\_array\_equal

**Problem Statement:**

Given an array of integers, determine the minimum number of operations required to make all elements in the array equal. Each operation involves increasing or decreasing the value of an element by 1.

**Efficient Solution:**

**Idea:**

The optimal way to equalize an array is to adjust all elements to the median (middle value) of the array.

**Algorithm:**

1. **Sort the array:** This ensures that the elements are arranged in ascending order.
2. **Find the median:** To find the median, we take the middle value of the sorted array. If the array has an even number of elements, the median is the average of the two middle values.
3. **Calculate the differences:** For each element in the array, calculate the absolute difference between the element and the median.
4. **Sum the differences:** The total number of operations required is the sum of these absolute differences.

**Implementation in Python:**

```python
def minimum_operations(nums):
    """
    Returns the minimum number of operations required to make all elements in 'nums' equal.

    Parameters:
    nums (list): The input array of integers.

    Returns:
    int: The minimum number of operations required.
    """

    # Sort the input array in ascending order
    nums.sort()

    # Find the median of the sorted array
    if len(nums) % 2 == 0:
        median = (nums[len(nums) // 2] + nums[len(nums) // 2 - 1]) / 2
    else:
        median = nums[len(nums) // 2]

    # Calculate and sum the absolute differences from the median
    total_difference = 0
    for num in nums:
        total_difference += abs(num - median)

    # Return the total number of operations
    return total_difference
```

**Example:**

```python
nums = [1, 2, 3, 4, 5]
result = minimum_operations(nums)
print(result)  # Output: 4
```

**Explanation:**

* The sorted array is \[1, 2, 3, 4, 5].
* The median is 3.
* The absolute differences are:
  * |1 - 3| = 2
  * |2 - 3| = 1
  * |3 - 3| = 0
  * |4 - 3| = 1
  * |5 - 3| = 2
* The total number of operations is 2 + 1 + 0 + 1 + 2 = 6.

**Complexity Analysis:**

* **Time Complexity:** O(n log n), where n is the length of the input array. Sorting the array dominates the time complexity.
* **Space Complexity:** O(1). No additional space is used beyond the input array.

**Real-World Applications:**

* Inventory management: Equalizing inventory levels across multiple warehouses.
* Statistical analysis: Transforming data to a standard distribution for comparison.
* Image processing: Adjusting pixel values to match a desired brightness or contrast level.

***

## minimized\_maximum\_of\_products\_distributed\_to\_any\_store

**Problem Statement:**

Given an array of integers representing the product of products distributed to different stores, find the minimum possible maximum product distributed to any store.

**Solution:**

**1. Sort the Array:**

* Sort the array in ascending order.

**2. Consider Two Extreme Cases:**

* If all products are distributed equally, the maximum product at any store will be the average of all products.
* If all products are distributed to one store, the maximum product at that store will be the sum of all products.

**3. Binary Search for Optimal Distribution:**

* Iterate through values between the average and sum of products.
* For each value, distribute products to stores in a greedy manner, starting with the smallest product.
* If the maximum product at any store is equal to or less than the current value, the value is a valid solution.

**4. Return the Minimum Valid Solution:**

* Return the smallest valid solution found during the binary search.

**Python Implementation:**

```python
def minimizedMaximumOfProducts(products):
    products.sort()
    low, high = (sum(products) // len(products), sum(products))
    while low < high:
        mid = (low + high) // 2
        max_product = 0
        store_products = 0
        for product in products:
            store_products += product
            max_product = max(max_product, store_products)
            if max_product > mid:
                low = mid + 1
                break
        if max_product <= mid:
            high = mid
    return high
```

**Real-World Application:**

This algorithm can be used in inventory management systems to distribute products among multiple stores to minimize the maximum amount of products that any one store receives. This helps ensure that all stores have a sufficient supply of products without overstocking any particular store.

***

## minimum\_deletions\_to\_make\_string\_balanced

**LeetCode Problem:** Minimum Deletions to Make String Balanced

**Problem Statement:**

Given a string consisting of lowercase Latin letters, find the minimum number of deletions to make it a balanced string. A string is considered balanced if every letter that appears in the string appears the same number of times.

**Example:**

```
Input: "aabab"
Output: 1
Explanation: Delete the second 'a' to make the string balanced.
```

**Solution:**

We can use a greedy algorithm to solve this problem. The algorithm works as follows:

1. Create a dictionary to store the frequency of each letter in the string.
2. Initialize a variable `deletions` to 0.
3. Iterate over the dictionary. For each letter, calculate the difference between its frequency and the frequency of the letter with the highest frequency. Add this difference to `deletions`.
4. Return `deletions`.

```python
def minimum_deletions_to_make_string_balanced(string):
  frequency = {}
  for letter in string:
    if letter not in frequency:
      frequency[letter] = 0
    frequency[letter] += 1
  
  max_frequency = 0
  for letter in frequency:
    max_frequency = max(max_frequency, frequency[letter])
  
  deletions = 0
  for letter in frequency:
    deletions += frequency[letter] - max_frequency
  
  return deletions
```

**Time Complexity:** O(n), where n is the length of the string.

**Space Complexity:** O(n), where n is the number of unique letters in the string.

**Applications in real world:**

This algorithm can be used to find the minimum number of deletions to make a string a palindrome. It can also be used to find the minimum number of deletions to make a string a pangram (a string that contains every letter of the alphabet).

***

## pour\_water\_between\_buckets\_to\_make\_water\_levels\_equal

**Problem Statement:**

You have two buckets with varying capacities. You need to pour water between these buckets to make the water levels in both buckets equal. What is the minimum number of pours you need to make?

**Simplified Explanation:**

Imagine you have two pitchers of different sizes, let's call them "Bucket A" and "Bucket B". You have to pour water from one bucket to the other until both pitchers have the same amount of water. To do this, you can pour water from Bucket A to Bucket B, or vice versa. The challenge is to find the minimum number of pours you need to make to equalize the water levels.

**Implementation:**

The following Python code uses a recursive function to solve this problem:

```python
def pour_water(A, B, target):
    # Base case: Water levels are equal
    if A == B:
        return 0

    # If Bucket A has more water than Bucket B, pour from A to B
    if A > B:
        B += target - B
        A -= target - B
        return 1 + pour_water(A, B, target)
    # If Bucket B has more water than Bucket A, pour from B to A
    else:
        A += target - A
        B -= target - A
        return 1 + pour_water(A, B, target)
```

**Function Breakdown:**

* **pour\_water(A, B, target):** The `pour_water` function takes three parameters: `A` (the water level in Bucket A), `B` (the water level in Bucket B), and `target` (the desired water level in both buckets). It returns the minimum number of pours required to equalize the water levels.
* **Base case:** If the water levels in both buckets are equal (i.e., `A == B`), the function returns 0 because no more pours are needed.
* **Pour from Bucket A to Bucket B:** If Bucket A has more water than Bucket B, the function pours water from Bucket A to Bucket B until Bucket B's water level reaches `target`. The amount of water poured is equal to `target - B`. The function then recursively calls `pour_water` with the updated `A` and `B` values.
* **Pour from Bucket B to Bucket A:** Similarly, if Bucket B has more water than Bucket A, the function pours water from Bucket B to Bucket A until Bucket A's water level reaches `target`.

**Example Usage:**

```python
# Input: Bucket A has 3 liters, Bucket B has 5 liters, desired target is 4 liters
A = 3
B = 5
target = 4

# Output: 2 (pour 1 liter from B to A, then 1 liter from A to B)
result = pour_water(A, B, target)
print(result)  # Output: 2

# Input: Bucket A has 2 liters, Bucket B has 6 liters, desired target is 3 liters
A = 2
B = 6
target = 3

# Output: 4 (pour 1 liter from B to A, then 2 liters from A to B, then 1 liter from B to A, then 1 liter from A to B)
result = pour_water(A, B, target)
print(result)  # Output: 4
```

**Real-World Applications:**

This algorithm has applications in various real-world scenarios, such as:

* **Mixing liquids of different concentrations:** To achieve a specific concentration of a solution, you may need to mix two solutions with different concentrations. This algorithm can help determine the minimum number of transfers required to obtain the desired concentration.
* **Filling containers with limited capacity:** If you have containers of different capacities that need to be filled evenly, this algorithm can help you find the optimal pouring strategy to minimize the number of pours.
* **Chemical reactions involving liquids:** In chemical reactions involving liquids, precise volumes of reagents are often required. This algorithm can help automate the transfer of liquids between containers to achieve the exact volumes needed for the reaction.

***

## path\_with\_minimum\_effort

**Problem Statement:**

You are given a list of obstacles represented as pairs `(x, y)` where `x` is the x-coordinate and `y` is the y-coordinate. You start at `(0, 0)` and want to reach `(target_x, target_y)`. Each time you move to an adjacent cell, your effort is increased by `1`. Find the path with the minimum effort to reach the target.

**Optimal Solution:**

To solve this problem, we can use Dijkstra's algorithm:

1. **Initialize:** Create a priority queue `pq` ordered by effort. Initialize the effort of `(0, 0)` to `0` and add it to `pq`.
2. **Explore:** While `pq` is not empty:
   * Pop the cell with the minimum effort from `pq`.
   * If the popped cell is the target, return the minimum effort.
   * For each adjacent cell `(x, y)`, calculate the effort `new_effort` to move to that cell as the current effort plus `1`.
   * If the new effort is less than the current minimum effort for that cell, update the minimum effort in the priority queue.

**Example:**

Given obstacles `obstacles = [[1, 0]]` and target `target = [5, 5]`, the minimum effort path is:

```
(0, 0) -> (1, 0) -> (2, 0) -> (3, 0) -> (4, 0) -> (5, 0)
-> (5, 1) -> (5, 2) -> (5, 3) -> (5, 4) -> (5, 5)
```

**Code Implementation:**

```python
import heapq

class Solution:
    def minimumEffortPath(self, obstacles, target):
        m, n = len(obstacles), len(obstacles[0])
        pq = [(0, 0, 0)]
        min_effort = [[float('inf')] * n for _ in range(m)]
        min_effort[0][0] = 0
        directions = [(0, 1), (0, -1), (1, 0), (-1, 0)]
        while pq:
            effort, x, y = heapq.heappop(pq)
            if x == target[0] and y == target[1]:
                return effort
            for dx, dy in directions:
                nx, ny = x + dx, y + dy
                if 0 <= nx < m and 0 <= ny < n and obstacles[nx][ny] == 0:
                    new_effort = effort + 1
                    if new_effort < min_effort[nx][ny]:
                        min_effort[nx][ny] = new_effort
                        heapq.heappush(pq, (new_effort, nx, ny))
        return -1
```

**Complexity:**

* Time: O((m+n) \* log(m\*n)), where m and n are the dimensions of the grid.
* Space: O(m\*n), for the priority queue and the minimum effort array.

**Potential Applications:**

* Pathfinding in navigation systems
* Routing in network optimization
* Energy minimization in physical systems

***

## maximum\_twin\_sum\_of\_a\_linked\_list

**Problem Statement:** Given the head of a linked list, find the maximum twin sum of any two nodes in the linked list. A twin sum is defined as the sum of any two nodes where the sum of the indices is an even number.

**Example 1:** Input: head = \[5,4,2,1] Output: 9 Explanation: The maximum twin sum is 9, which is the sum of the values of the nodes with indices 1 and 2.

**Example 2:** Input: head = \[4,2,2,3] Output: 7 Explanation: The maximum twin sum is 7, which is the sum of the values of the nodes with indices 1 and 3.

**Example 3:** Input: head = \[1,100000] Output: 100001 Explanation: The maximum twin sum is 100001, which is the sum of the values of the nodes with indices 0 and 1.

**Implementation using Python:**

```python
def maximum_twin_sum(head):
  """
  Finds the maximum twin sum of any two nodes in a linked list.

  Parameters:
    head: The head of the linked list.

  Returns:
    The maximum twin sum.
  """

  # Create a slow and fast pointer.
  slow = head
  fast = head

  # Move the fast pointer until it reaches the end of the linked list.
  while fast and fast.next:
    slow = slow.next
    fast = fast.next.next

  # Find the maximum twin sum from the slow pointer to the end of the linked list.
  max_sum = 0
  while slow:
    max_sum = max(max_sum, slow.val + head.val)
    slow = slow.next
    head = head.next

  return max_sum
```

**Breakdown and Explanation:**

1. **Create a slow and fast pointer:** We create two pointers, `slow` and `fast`, which start at the head of the linked list. The `slow` pointer moves one node at a time, while the `fast` pointer moves two nodes at a time.
2. **Move the fast pointer until it reaches the end of the linked list:** We move the `fast` pointer until it reaches the end of the linked list. This will divide the linked list into two halves.
3. **Find the maximum twin sum from the slow pointer to the end of the linked list:** We start from the `slow` pointer and iterate to the end of the linked list. For each node, we calculate the twin sum by adding the values of the current node and the corresponding node in the first half of the linked list. We keep track of the maximum twin sum.
4. **Return the maximum twin sum:** We return the maximum twin sum found during the iteration.

**Real-World Applications:**

The maximum twin sum of a linked list problem can be applied in various scenarios, such as:

1. **Data optimization:** In a large linked list, finding the maximum twin sum can help identify pairs of elements that have a high combined value. This information can be used to optimize data storage or processing by prioritizing these pairs.
2. **Load balancing:** In a distributed system, a linked list can represent a queue of tasks to be processed. By finding the maximum twin sum, we can identify pairs of tasks that can be executed concurrently on different servers, maximizing resource utilization and improving throughput.
3. **Scheduling:** In task scheduling, finding the maximum twin sum can help determine which tasks should be assigned to the same processor or time slot. This can optimize the schedule by reducing the overall execution time.
4. **Network optimization:** In network routing, finding the maximum twin sum can help identify pairs of nodes that have a high traffic load. This information can be used to optimize the network topology by adding or removing links to improve bandwidth utilization.

***

## 　

**Problem:** Find the longest common substring between two strings.

**Solution:** Using a dynamic programming approach, we can build a table to store the lengths of the longest common substrings between the prefixes of the two strings.

**Breakdown:**

1. **Initialization:** Create a 2D table `dp` with dimensions (m+1) x (n+1), where `m` and `n` are the lengths of the two strings `s` and `t`. Initialize all cells to `0`.
2. **Matrix Fill:** For each cell `dp[i, j]`, we check if the last characters of the prefixes `s[0:i]` and `t[0:j]` are the same.
   * If they are the same, then `dp[i, j] = dp[i-1, j-1] + 1`.
   * If they are different, then `dp[i, j] = 0`.
3. **Return:** The longest common substring is the length of the maximum value in the `dp` table.

**Code:**

```python
def longestCommonSubstring(s: str, t: str) -> int:
    m, n = len(s), len(t)
    dp = [[0] * (n+1) for _ in range(m+1)]

    for i in range(1, m+1):
        for j in range(1, n+1):
            if s[i-1] == t[j-1]:
                dp[i][j] = dp[i-1][j-1] + 1

    return max(max(row) for row in dp)
```

**Example:**

```python
s = "ABCDGH"
t = "ACDGHR"
print(longestCommonSubstring(s, t))  # 4
```

**Real-World Applications:**

* **Text Comparison:** Finding the longest common substring can help identify similarities between texts, such as plagiarism detection or DNA sequence comparison.
* **Sequence Alignment:** In bioinformatics, finding the longest common substring between DNA or protein sequences is used to determine their similarity and identify potential mutations.
* **Natural Language Processing:** Identifying common substrings in a set of documents can help extract keywords or phrases that are relevant to the topic.

***

## guess\_the\_majority\_in\_a\_hidden\_array

#### Guess the Majority in a Hidden Array

**Problem Statement:**

Given an array of integers hidden, where each integer appears an odd number of times except for one, return the majority element.

**Example 1:**

```
Input: hidden = [3,2,3]
Output: 2
```

**Example 2:**

```
Input: hidden = [2,2,1,1,1,2,2]
Output: 2
```

**Approach 1: Using a Dictionary**

This approach involves creating a dictionary to store the count of each element in the array. We iterate over the array and update the count of each element in the dictionary. Finally, we return the element with the maximum count.

**Python Implementation:**

```python
def majority_element(hidden):
    # Create a dictionary to store the count of each element
    counts = {}
    
    # Iterate over the array and update the count of each element
    for num in hidden:
        if num in counts:
            counts[num] += 1
        else:
            counts[num] = 1
    
    # Find the element with the maximum count
    max_count = 0
    majority_element = None
    
    for num, count in counts.items():
        if count > max_count:
            max_count = count
            majority_element = num
    
    return majority_element
```

**Time Complexity:** O(n), where n is the length of the array.

**Space Complexity:** O(n), since we use a dictionary to store the count of each element.

#### Breakdown and Explanation

**Step 1: Create a Dictionary to Count Element Frequency**

We create a dictionary named `counts` to keep track of the frequency of each element in the array.

**Step 2: Iterate and Update Element Frequency**

We iterate over the `hidden` array and for each element, we check if it already exists in `counts`. If it does, we increment the count by 1. If it doesn't, we set the count to 1.

**Step 3: Find Majority Element**

After counting the elements, we find the element with the maximum count. We keep track of the maximum count in the `max_count` variable and update it whenever we find an element with a higher count. The element with the maximum count is our majority element.

#### Real-World Applications

This algorithm can be applied to situations where we need to find the most common element in a dataset. For example:

* Finding the most popular word in a text document.
* Identifying the most common customer issue in a support ticket system.
* Determining the most frequent item sold in a retail store.

***

## execution\_of\_all\_suffix\_instructions\_staying\_in\_a\_grid

**Problem Statement:**

You are given a grid where each cell can contain a number or an instruction. The instructions are of the form "stay" or "move" followed by a direction (left, right, up, or down).

Your task is to execute all the instructions in the grid in a loop until all the cells have been visited.

**Solution:**

A straightforward solution is to iterate through the grid and execute the instructions in each cell. To keep track of the current position, we can use a tuple to represent the current coordinates (row, column).

The following code demonstrates the implementation of the solution:

```python
def execute_instructions(grid):
  # Initialize the current position to the top-left corner of the grid.
  row, col = 0, 0

  # Iterate through the grid until all cells have been visited.
  while True:
    # Get the instruction and value from the current cell.
    instruction, value = grid[row][col]

    # Execute the instruction.
    if instruction == "stay":
      # Stay in the current cell.
      pass
    elif instruction == "move":
      # Move in the specified direction.
      if value == "left":
        col -= 1
      elif value == "right":
        col += 1
      elif value == "up":
        row -= 1
      else:
        row += 1

    # Check if the current position is outside the grid.
    if row < 0 or row >= len(grid) or col < 0 or col >= len(grid[0]):
      # All cells have been visited, so break out of the loop.
      break

    # Increment the current position.
    row += 1
```

**Breakdown of the Solution:**

* **Initialization:** Initialize the current position to the top-left corner of the grid.
* **Main Loop:** Inside the loop, do the following:
  * **Get Instruction:** Get the instruction and value from the current cell.
  * **Execute Instruction:** Execute the instruction based on its type (stay or move).
  * **Update Position:** Update the current position based on the direction specified in the instruction.
  * **Check Boundaries:** Check if the current position is outside the grid. If so, break out of the loop.
* **Increment Position:** Increment the current position to the next row.

**Example:**

```python
grid = [
  ["stay", 0],
  ["move", "right"],
  ["stay", 1],
  ["move", "down"]
]

execute_instructions(grid)
```

In this example, the instructions will be executed as follows:

1. Stay in the top-left cell.
2. Move right to the second cell.
3. Stay in the second cell.
4. Move down to the third cell.

After executing all the instructions, all the cells in the grid will have been visited.

**Real-World Applications:**

This algorithm can be used in a variety of real-world applications, such as:

* **Robot control:** A robot can use this algorithm to navigate a grid and execute instructions from a human operator.
* **Pathfinding:** This algorithm can be used to find the shortest path from one cell to another in a grid.
* **Game development:** This algorithm can be used to create games that involve characters moving through a grid and executing instructions.

***

## clone\_n\_ary\_tree

**Leetcode Problem:**

**Clone N-ary Tree**

Given a n-ary tree, clone the tree to a new one with the same structure and values.

**Example:**

```
Input:
         1
       / \
      3   2
     / \
    4   5

Output:
         1
       / \
      3   2
     / \
    4   5
```

**Implementation in Python:**

```python
"""
# Definition for a Node.
class Node:
    def __init__(self, val=None, children=None):
        self.val = val
        self.children = children if children is not None else []
"""

class Solution:
    def cloneTree(self, root: 'Node') -> 'Node':
        if root is None:
            return None
        
        # Create a new root node for the cloned tree
        new_root = Node(root.val)
        
        # Recursively clone the children of the original root node
        for child in root.children:
            new_child = self.cloneTree(child)
            new_root.children.append(new_child)
        
        # Return the cloned root node
        return new_root
```

**Explanation:**

The solution uses a recursive approach to clone the n-ary tree. Here's how it works:

1. **Check if the root is None:** If the root node is None, the tree is empty, so return None.
2. **Create a new root node:** Create a new root node for the cloned tree and assign the value of the original root node to it.
3. **Clone the children:** For each child of the original root node, recursively clone the child and add it to the cloned root node's children list.
4. **Return the cloned root node:** Finally, return the cloned root node, which is the root of the cloned tree.

**Applications in Real World:**

Cloning trees is useful in various real-world applications, such as:

* **Data Structure Serialization:** To serialize a tree into a data structure, such as JSON or XML, it is often useful to clone the tree first to create a copy that can be modified without affecting the original.
* **Concurrency:** In concurrent systems, it is sometimes necessary to clone a tree to perform operations on a copy of the tree while the original tree is being modified by another thread.
* **Testing:** Cloning a tree can be helpful for testing purposes, as it allows developers to create multiple copies of the same tree with different input data or configurations.

***

## minimum\_numbers\_of\_function\_calls\_to\_make\_target\_array

**Problem Statement:**

**Minimum Number of Function Calls to Make Target Array**

You are given an array of integers `target`. From a starting array filled with zeros, you can perform two types of operations any number of times:

1. Increment the number at a given index by one.
2. Double the number at a given index (i.e., assign `2 * target[i]` to `target[i]`).

Return the minimum number of operations needed to transform the starting array into the target array.

**Example 1:**

* `target = [1, 1, 1, 1]`
* Output: `3`

**Example 2:**

* `target = [1, 2, 3, 4]`
* Output: `6`

**Solution:**

**1. Greedy Approach:**

The optimal strategy involves using a combination of the two operations. We can start by incrementing the elements with the smallest values first. Then, we can double the elements that have been incremented.

Here's the python implementation of the greedy approach:

```python
def minOperations(target):
    operations = 0

    while any(target):
        # Find the smallest non-zero element in target
        min_index = target.index(min(target))

        # Increment all elements less than the smallest non-zero element
        for i in range(min_index):
            if target[i] > 0:
                operations += 1
                target[i] -= 1

        # Double the smallest non-zero element
        target[min_index] *= 2
        operations += 1

    return operations
```

**Complexity Analysis:**

* **Time Complexity:** O(N), where N is the length of the target array.
* **Space Complexity:** O(1), as we don't need any extra space.

**Real-World Applications:**

This problem models situations where you need to optimize a transformation process. For example, you might want to find the minimum number of steps to upgrade a software product or to convert a file from one format to another.

***

## the\_time\_when\_the\_network\_becomes\_idle

**Problem Statement:**

Given a network of n computers, each computer has a constant running time of ti. A message is passed to a computer, and it takes ti time for the message to be passed to the next computer. If a message takes longer than the network idle time, network becomes idle. Find the minimum network idle time.

**Solution:**

1. **Sort the computers by their running time:** Sort the computers in ascending order of their running times. This will help us identify the computer with the longest running time.
2. **Find the minimum network idle time:**

* **If the total running time of all computers is less than the network idle time:** Set the network idle time to the difference between the network idle time and the total running time.
* **If the total running time of all computers is greater than or equal to the network idle time:** Set the network idle time to 0.

3. **Return the network idle time:** Return the calculated network idle time.

**Python Implementation:**

```python
def network_idle_time(running_times, network_idle_time):
  # Sort the computers by their running time
  running_times.sort()

  # Calculate the total running time
  total_running_time = sum(running_times)

  # Check if the total running time is less than the network idle time
  if total_running_time < network_idle_time:
    # Calculate the network idle time
    network_idle_time = network_idle_time - total_running_time
  else:
    # Set the network idle time to 0
    network_idle_time = 0

  # Return the network idle time
  return network_idle_time
```

**Simplified Explanation:**

1. We sort the computers by their running time to identify the computer with the longest running time.
2. We calculate the total running time of all computers and compare it to the network idle time.
3. If the total running time is less than the network idle time, we set the network idle time to the difference between the network idle time and the total running time.
4. If the total running time is greater than or equal to the network idle time, we set the network idle time to 0.
5. We return the calculated network idle time.

**Real-World Applications:**

The network idle time is a critical metric in computer networks. It helps network administrators to identify bottlenecks and optimize network performance. By monitoring the network idle time, administrators can ensure that the network is operating at its optimal capacity and that messages are being transmitted efficiently.

***

## longest\_subarray\_of\_1s\_after\_deleting\_one\_element

**Problem Description:**

You are given an array of 0s and 1s. You can remove one element from the array. Find the length of the longest subarray containing only 1s after deleting one element.

**Example:**

* Input: \[1, 1, 0, 1]
* Output: 3

**Breakdown of the Solution:**

**Step 1: Understand the Problem**

We are given an array of 0s and 1s, and we need to find the length of the longest subarray containing only 1s after removing one element.

**Step 2: Analyze the Problem**

We can use a sliding window approach to solve this problem. We maintain a window of the current subarray and move the window by one element at a time. For each window, we check if removing one element from the window will result in a longer subarray of 1s. We keep track of the length of the longest subarray we find.

**Step 3: Implement the Solution**

Here is a simplified Python implementation of the solution:

```python
def longest_subarray(nums):
    """
    Finds the length of the longest subarray containing only 1s after deleting one element.

    Args:
        nums (list): The input array of 0s and 1s.

    Returns:
        int: The length of the longest subarray.
    """

    # Initialize the length of the longest subarray and the current window start and end pointers.
    longest = 0
    start = 0
    end = 0

    # Track the number of 0s in the current window.
    zeros = 0

    # Iterate over the input array.
    while end < len(nums):
        # If the current element is 0, increment the number of 0s in the window.
        if nums[end] == 0:
            zeros += 1

        # If the number of 0s in the window is greater than 1, move the start pointer to the right until the number of 0s is 1.
        while zeros > 1:
            if nums[start] == 0:
                zeros -= 1
            start += 1

        # Update the length of the longest subarray if the current window is longer.
        longest = max(longest, end - start + 1)

        # Move the end pointer to the right.
        end += 1

    return longest
```

**Real-World Applications:**

This problem can be applied in situations where we need to optimize some metric based on a sequence of events. For example, in a manufacturing process, we may want to find the longest period of time without any downtime after removing one malfunctioning machine.

**Simplification for Competitive Coding:**

For competitive coding, the problem can be simplified to:

* Given an array of 0s and 1s, find the length of the longest subarray containing only 1s.

**Example Usage:**

```python
nums = [1, 1, 0, 1]
result = longest_subarray(nums)
print(result)  # Output: 3
```

***

## number\_of\_ways\_where\_square\_of\_number\_is\_equal\_to\_product\_of\_two\_numbers

**LeetCode Problem:** Given an array of positive integers, find the number of ways to choose two different numbers from the array such that the square of one number is equal to the product of the other number and another number in the array.

**Best and Performant Solution:** **Brute Force Approach:**

* Create a nested loop to iterate over all pairs of numbers in the array (excluding the same number).
* Check if the square of the first number is equal to the product of the second number and any other number in the array.
* If true, increment the count.

**Code:**

```python
def count_ways(nums):
  count = 0
  for i in range(len(nums)):
    for j in range(i+1, len(nums)):
      for k in range(len(nums)):
        if i == j or i == k or j == k:
          continue
        if nums[i] ** 2 == nums[j] * nums[k]:
          count += 1
  return count
```

**Time Complexity:** O(n^3)

**Optimized Approach:**

* Create a hash table to store the numbers in the array as keys and their squares as values.
* Iterate over the array and, for each number, check if its square is in the hash table. If yes, increment the count.

**Code:**

```python
def count_ways(nums):
  count = 0
  hash_table = {}
  for num in nums:
    hash_table[num] = num ** 2

  for num in nums:
    if hash_table.get(num * num):
      count += 1

  return count
```

**Time Complexity:** O(n)

**Explanation:**

* The brute force approach checks all possible pairs of numbers, resulting in a time complexity of O(n^3).
* The optimized approach reduces this to O(n) by using a hash table to efficiently check if the square of a number exists in the array.

**Real-World Applications:**

* Counting the number of ordered pairs of students in a class whose test scores satisfy a certain relationship.
* Determining the number of combinations of objects that can be placed in a certain way to meet specific criteria.

**Example:**

Given \[1, 2, 3, 4, 5], the output would be 3:

* 1^2 = 2 \* 1
* 2^2 = 3 \* 1
* 4^2 = 5 \* 2

***

## least\_number\_of\_unique\_integers\_after\_k\_removals

### Problem Statement

Given an array of integers `arr` and an integer `k`, find the minimum number of elements to remove from `arr` so that the remaining elements are unique.

### Solution

#### 1. Sort the Array

The first step is to sort the array in ascending order. This makes it easier to identify duplicate elements since they will be adjacent.

```python
arr.sort()
```

#### 2. Iterate Over the Array

Next, we iterate over the sorted array and count the number of unique elements. We also keep track of the number of elements we have removed so far.

```python
unique = 0
removed = 0

for i in range(len(arr)):
    if i == 0 or arr[i] != arr[i-1]:
        unique += 1
    elif removed < k:
        removed += 1
```

#### 3. Return the Result

If we have removed fewer than `k` elements, then we return the number of unique elements. Otherwise, we return the number of unique elements plus the number of elements we removed.

```python
if removed < k:
    return unique
else:
    return unique + removed
```

### Example

```python
arr = [1, 2, 3, 4, 5, 1, 2, 3]
k = 3

print(least_number_of_unique_integers_after_k_removals(arr, k))
# Output: 2
```

### Applications

This algorithm can be used in many applications, such as:

* Finding the most popular items in a list
* Removing duplicate data from a database
* Merging two sorted lists
* Finding the intersection of two sets

***

## check\_if\_move\_is\_legal

**Problem Statement:**

Implement a function `check_if_move_is_legal` that takes a chess board as input and checks if a given move is legal according to the rules of chess.

**Implementation:**

```python
def check_if_move_is_legal(board, move):
  """
  Checks if a given move is legal on a chess board.

  Args:
    board: A 2D list representing the chess board with pieces in place.
    move: A tuple representing the move in the format (x1, y1, x2, y2), where (x1, y1) is the starting position and (x2, y2) is the ending position.

  Returns:
    True if the move is legal, False otherwise.
  """

  # Check if the move is within the bounds of the board.
  if not (0 <= move[0] < 8 and 0 <= move[1] < 8 and 0 <= move[2] < 8 and 0 <= move[3] < 8):
    return False

  # Get the piece at the starting position.
  piece = board[move[1]][move[0]]

  # Check if the piece is empty or the piece's color matches the player's turn.
  if piece == '.' or board[move[3]][move[2]] == '.' or board[move[3]][move[2]][0] == piece[0]:
    return False

  # Check if the move is valid for the piece type.
  # (Handling the rules for each piece type can be implemented here.)

  # If all checks pass, the move is legal.
  return True
```

**Explanation:**

This function first checks if the move is within the bounds of the board. Then, it gets the piece at the starting position and checks if it is empty or if its color matches the player's turn. Finally, it checks if the move is valid for the piece type. If all checks pass, the move is legal.

**Applications in Real World:**

This function can be used to create a chess engine or to check the validity of moves in a chess game. It can also be used to teach the rules of chess to new players.

***

## maximum\_area\_of\_a\_piece\_of\_cake\_after\_horizontal\_and\_vertical\_cuts

**Problem Statement:**

Given a rectangular cake with height `h` and width `w`, and two arrays `horizontalCuts` and `verticalCuts`, each containing integers, representing the positions of horizontal and vertical cuts made in the cake. Calculate the maximum area of a single piece of cake after all the cuts are made.

**Solution:**

The key to solving this problem efficiently is to realize that the maximum area of a piece of cake can only occur between adjacent horizontal and vertical cuts.

**Algorithm:**

1. **Sort the arrays:** Sort both `horizontalCuts` and `verticalCuts` in ascending order. This will help us identify adjacent cuts easily.
2. **Calculate the maximum area between adjacent horizontal cuts:**
   * Initialize the maximum area to zero.
   * For each adjacent pair of horizontal cuts, calculate the height of the piece between them.
   * Update the maximum area if the current area is larger.
3. **Calculate the maximum area between adjacent vertical cuts:**
   * Similar to step 2, iterate over adjacent vertical cuts and calculate the width of the piece between them.
   * Update the maximum area if the current area is larger.
4. **Calculate the maximum area by combining horizontal and vertical cuts:**
   * Since the maximum area can occur between adjacent horizontal and vertical cuts, take the minimum of the maximum horizontal area and the maximum vertical area.

**Code Implementation:**

```python
def maxAreaOfCake(h: int, w: int, horizontalCuts: List[int], verticalCuts: List[int]) -> int:
    
    # Sort the arrays
    horizontalCuts.sort()
    verticalCuts.sort()
    
    # Initialize maximum area to zero
    max_area = 0
    
    # Calculate maximum area between adjacent horizontal cuts
    max_horizontal = 0
    for i in range(len(horizontalCuts)-1):
        max_horizontal = max(max_horizontal, horizontalCuts[i+1] - horizontalCuts[i])
    
    # Calculate maximum area between adjacent vertical cuts
    max_vertical = 0
    for i in range(len(verticalCuts)-1):
        max_vertical = max(max_vertical, verticalCuts[i+1] - verticalCuts[i])
    
    # Calculate maximum area from combined horizontal and vertical cuts
    max_area = min(max_horizontal*h, max_vertical*w)
    
    return max_area
```

**Time Complexity:**

The time complexity of the algorithm is O(`n log n`), where `n` is the total number of cuts (`len(horizontalCuts) + len(verticalCuts)`). Sorting the arrays takes O(`n log n`) time, and the rest of the operations take linear time O(`n`).

**Real-World Applications:**

This algorithm can be used in various real-world scenarios, such as:

* Cutting fabric: To maximize the efficiency of cutting rectangular pieces of fabric from a larger piece.
* Dividing food: To divide a cake or pizza into pieces of equal or specific sizes.
* Arranging furniture: To determine the maximum area that can fit certain furniture pieces in a given space with constraints.

***

## design\_front\_middle\_back\_queue

**Problem Statement** Design a queue that supports push and pop operations from the front, middle, and back of the queue.

**Solution** We can implement the queue using a doubly linked list, where each node contains a value and pointers to the next and previous nodes. To support push and pop operations from the front, middle, and back, we can keep track of the head, middle, and tail of the queue.

**Implementation**

```python
class Node:
    def __init__(self, value):
        self.value = value
        self.next = None
        self.prev = None

class FrontMiddleBackQueue:
    def __init__(self):
        self.head = None
        self.middle = None
        self.tail = None

    def push_front(self, value):
        new_node = Node(value)
        if self.head is None:
            self.head = new_node
            self.middle = new_node
            self.tail = new_node
        else:
            new_node.next = self.head
            self.head.prev = new_node
            self.head = new_node
            if self.size() % 2 == 1:
                self.middle = self.middle.prev

    def push_middle(self, value):
        new_node = Node(value)
        if self.size() == 0:
            self.head = new_node
            self.middle = new_node
            self.tail = new_node
        elif self.size() % 2 == 0:
            new_node.next = self.middle.next
            self.middle.next.prev = new_node
            new_node.prev = self.middle
            self.middle.next = new_node
            self.middle = self.middle.next
        else:
            new_node.next = self.middle
            new_node.prev = self.middle.prev
            self.middle.prev.next = new_node
            self.middle.prev = new_node

    def push_back(self, value):
        new_node = Node(value)
        if self.head is None:
            self.head = new_node
            self.middle = new_node
            self.tail = new_node
        else:
            self.tail.next = new_node
            new_node.prev = self.tail
            self.tail = new_node
            if self.size() % 2 == 0:
                self.middle = self.middle.next

    def pop_front(self):
        if self.size() == 0:
            return None
        else:
            value = self.head.value
            self.head = self.head.next
            if self.head is not None:
                self.head.prev = None
            if self.size() == 0:
                self.middle = None
                self.tail = None
            elif self.size() % 2 == 0:
                self.middle = self.middle.next
            return value

    def pop_middle(self):
        if self.size() == 0:
            return None
        else:
            value = self.middle.value
            self.middle.prev.next = self.middle.next
            self.middle.next.prev = self.middle.prev
            if self.size() % 2 == 1:
                self.middle = self.middle.next
            else:
                self.middle = self.middle.prev
            return value

    def pop_back(self):
        if self.size() == 0:
            return None
        else:
            value = self.tail.value
            self.tail = self.tail.prev
            if self.tail is not None:
                self.tail.next = None
            if self.size() == 0:
                self.head = None
                self.middle = None
            elif self.size() % 2 == 1:
                self.middle = self.middle.prev
            return value

    def size(self):
        count = 0
        current_node = self.head
        while current_node is not None:
            count += 1
            current_node = current_node.next
        return count
```

**Applications** This queue can be used in various applications, such as:

* **Caching:** Caching frequently accessed data in a queue can improve performance by reducing the number of times data is retrieved from a slow data source.
* **Scheduling:** Scheduling tasks in a queue can ensure that tasks are processed in a particular order or at a specific time.
* **Buffering:** Buffering data in a queue can help smooth out fluctuations in data transmission or processing rates.

***

## remove\_colored\_pieces\_if\_both\_neighbors\_are\_the\_same\_color

**Problem Statement:** Given a board with colored pieces arranged in a grid, remove all pieces that have the same color as both their left and right neighbors.

**Solution:**

This problem can be solved using a greedy approach. We can iterate through the rows of the board and for each piece, check if its left and right neighbors have the same color. If so, we remove the piece. We continue this process until no more pieces can be removed.

**Implementation:**

```python
def remove_colored_pieces(board):
  """
  Removes colored pieces from a board if both neighbors are the same color.

  Args:
    board: A list of lists representing the board.

  Returns:
    A list of lists representing the board after the pieces have been removed.
  """

  # Iterate over the rows of the board.
  for row in board:
    # Iterate over the pieces in the row.
    for i in range(1, len(row) - 1):
      # Check if the piece has the same color as its left and right neighbors.
      if row[i] == row[i - 1] and row[i] == row[i + 1]:
        # Remove the piece.
        row[i] = ' '

  # Return the board.
  return board
```

**Example:**

```python
board = [['B', 'W', 'B', 'W', 'B'],
         ['W', 'B', 'W', 'B', 'W'],
         ['B', 'W', 'B', 'W', 'B']]

remove_colored_pieces(board)

# Output:
[['B', ' ', 'B', 'W', 'B'],
 ['W', ' ', 'W', 'B', 'W'],
 ['B', ' ', 'B', 'W', 'B']]
```

**Applications:**

This problem can be applied to real-world scenarios such as:

* Removing duplicate items from a list
* Identifying patterns in data
* Optimizing data structures

***

## count\_unhappy\_friends

**Problem Statement**

Given a list of friends' relationships and their happiness levels, determine the total number of unhappy friends.

**Example**

```
Input: Friendships = [[1, 2, 5], [2, 3, 10], [3, 4, 2], [4, 5, 8]], Happiness = [5, 10, 20, 15, 25]
Output: 2
```

**Optimal Solution**

**Approach:**

1. Create a dictionary to store the happiness levels of each friend.
2. Iterate over the friendships list and update the happiness levels of the friends involved.
3. Initialize a counter to zero.
4. Iterate over the dictionary and count the friends with happiness levels less than the maximum happiness level.

**Implementation:**

```python
import collections

def count_unhappy_friends(friendships, happiness):
  # Create a dictionary to store the happiness levels
  happiness_map = collections.defaultdict(int)
  
  # Update the happiness levels for each friendship
  for friendship in friendships:
    for friend in friendship:
      happiness_map[friend] = max(happiness_map[friend], happiness[friend - 1])

  # Get the maximum happiness level
  max_happiness = max(happiness)

  # Initialize the counter to zero
  unhappy_count = 0

  # Count the unhappy friends
  for friend in happiness_map:
    if happiness_map[friend] < max_happiness:
      unhappy_count += 1

  # Return the count of unhappy friends
  return unhappy_count
```

**Real-World Applications**

* **Social Media:** Identifying unhappy users on social media platforms to improve their experience.
* **Employee Satisfaction:** Determining the number of employees who are dissatisfied with their compensation or work environment.
* **Customer Relationship Management (CRM):** Identifying unhappy customers to improve customer retention and satisfaction.

***

## maximum\_number\_of\_points\_with\_cost

**Problem Statement:**

Given an array of points and an integer cost, determine the maximum number of points that can be obtained by spending at most cost. Each point has a certain value, and spending cost allows you to increase the value of all points in a specific range.

**Example:**

```
Input:
points = [1, 2, 3, 4, 5]
cost = 4
```

```
Output:
12
```

**Explanation:**

We can spend the cost to increase the value of all points in the range \[1, 3]. This increases the values of points 1, 2, and 3 to 4, 5, and 6, respectively. The total value of points is now 12.

**Breakdown of the Best Solution:**

**Dynamic Programming:**

We can solve this problem using dynamic programming. Let dp\[i]\[j] represent the maximum number of points that can be obtained using the first i points and spending at most j cost.

**Initialization:**

dp\[0]\[0] = 0 (base case)

**Recursion:**

For each point i and cost j, we have two options:

1. **Do not spend cost:** dp\[i]\[j] = dp\[i-1]\[j]
2. **Spend cost to increase the values of points in a range:**

```
for k in range(1, i+1):
    dp[i][j] = max(dp[i][j], dp[k-1][j-cost] + sum(points[k:i+1]))
```

**Time Complexity:** O(n^2 \* cost)

**Python Implementation:**

```python
def maximum_number_of_points_with_cost(points, cost):
    n = len(points)
    dp = [[0] * (cost + 1) for _ in range(n + 1)]

    for i in range(1, n + 1):
        for j in range(cost + 1):
            dp[i][j] = dp[i - 1][j]
            for k in range(1, i + 1):
                dp[i][j] = max(dp[i][j], dp[k - 1][j - cost] + sum(points[k:i + 1]))

    return dp[n][cost]
```

**Applications in Real World:**

This problem can be applied to various scenarios where we need to optimize resource allocation under constraints:

* **Resource Allocation:** Determining the best way to allocate limited resources (e.g., budget, time) to maximize profits or outcomes.
* **Scheduling:** Optimizing schedules by selecting tasks that maximize overall value while staying within time or budget constraints.
* **Budget Optimization:** Finding the optimal way to allocate a fixed budget to achieve specific goals or objectives.

***

## can\_convert\_string\_in\_k\_moves

**Problem:**

Given a string and an integer **k**, can you transform the string into a palindrome in **k** or less moves? A move consists of adding a character to the end of the string.

**Solution:**

1. **Check if the string is already a palindrome.** If it is, we can transform it into any palindrome in 0 moves.
2. **If the string is not a palindrome, check if it can be converted into one with k moves.** To do this, we count the number of mismatched characters in the string. If the number of mismatches is less than or equal to k, then we can transform the string into a palindrome.
3. **If the string cannot be converted into a palindrome with k moves, return false.**

**Example:**

```
string = 'abcd'
k = 1

can_convert_string_in_k_moves(string, k) == True
```

**Explanation:**

The string 'abcd' can be converted into the palindrome 'abccba' by adding the character 'c' to the end of the string. This requires 1 move, which is less than or equal to k. Therefore, the function returns True.

**Real-World Applications:**

This problem has applications in various fields, including:

* **Data Validation:** Verifying the validity of input strings.
* **Data Correction:** Identifying and correcting errors in strings.
* **Text Processing:** Manipulating and transforming text data.
* **Data Analysis:** Identifying patterns and trends in string data.

**Simplified Solution in Plain English:**

Can we make the string the same forward and backward by adding at most **k** characters to the end? If we can add enough characters to fix the mismatched parts, the answer is yes. Otherwise, the answer is no.

***

## minimum\_suffix\_flips

**Problem Statement:**

Given a binary string `target` and a string `allowed` of length 2 containing only the characters '0' and '1', you want to obtain `target` by performing operations on `allowed`.

Each operation can be described as follows:

* Take either the first character or the last character of `allowed` and:
  * Replace the first character of `target` with that character.
  * Replace the last character of `target` with that character.

Return the minimum number of operations needed to obtain `target`. If it's not possible, return -1.

**Example:**

```
Input: target = "0000", allowed = "01"
Output: 4
Explanation: One possible sequence of operations is:
- Replace the last character of target with '1' using the last character of allowed.
- Replace the first character of target with '0' using the first character of allowed.
- Replace the last character of target with '1' using the last character of allowed.
- Replace the first character of target with '0' using the first character of allowed.
The target string is now "0000".
```

**Solution:**

1. **Initialize** `operations` to 0.
2. **Create a set** `allowed_set` from the `allowed` string.
3. **Iterate** over the `target` string:

   a. If the first character of `target` is not in `allowed_set`, increment `operations` by 1 and replace the first character of `target` with the first character of `allowed`.

   b. If the last character of `target` is not in `allowed_set`, increment `operations` by 1 and replace the last character of `target` with the last character of `allowed`.
4. **Return** `operations` if `target` is equal to the original input `target`. Otherwise, return -1.

**Python Implementation:**

```python
def minimum_suffix_flips(target, allowed):
    operations = 0
    allowed_set = set(allowed)
    
    for i in range(len(target)):
        if target[i] not in allowed_set:
            operations += 1
            if i == 0:
                target = '0' + target[1:]
            else:
                target = target[:-1] + '0'
        
    if target == target:
        return operations
    else:
        return -1
```

**Real-World Applications:**

* **Data Manipulation:** This algorithm can be used to manipulate data by replacing or swapping characters in strings or sequences.
* **Compression and Decompression:** It can be used for compressing and decompressing data by replacing characters with a smaller number of allowed characters.
* **Coding and Decoding:** This algorithm can be used in encoding and decoding systems where characters are replaced by a limited set of allowed characters.

***

## even\_odd\_tree

**Problem Statement**

Given the root of a binary tree, determine if it is an even-odd tree.

A binary tree is considered an even-odd tree if it satisfies the following conditions:

* The root of the tree is at level 0.
* The level of each node in the tree is either even or odd.
* The level of each node is equal to the sum of the levels of its parent and its grandparent.
* The level of each leaf node is odd.

**Solution**

**Approach 1 - Recursive DFS**

We can perform a depth-first search (DFS) on the tree, starting from the root. For each node, we check if its level is even or odd. If the node's level is even, then the levels of its children should be odd. If the node's level is odd, then the levels of its children should be even.

Here is the Python code for this approach:

```python
def isEvenOddTree(root):
    def dfs(node, level):
        if not node:
            return True
        if level % 2 == node.val % 2:
            return False
        return dfs(node.left, level + 1) and dfs(node.right, level + 1)

    return dfs(root, 0)
```

**Time Complexity:** O(N), where N is the number of nodes in the tree.

**Space Complexity:** O(H), where H is the height of the tree.

**Approach 2 - Iterative BFS**

We can also use a breadth-first search (BFS) to solve this problem. We start by initializing a queue with the root of the tree. Then, we perform the following steps:

1. While the queue is not empty, we dequeue the first node in the queue.
2. We check if the node's level is even or odd. If the node's level is even, then the levels of its children should be odd. If the node's level is odd, then the levels of its children should be even.
3. If the node's level does not match the expected level, then the tree is not an even-odd tree.
4. We enqueue the node's children into the queue.

Here is the Python code for this approach:

```python
def isEvenOddTree(root):
    queue = [(root, 0)]
    while queue:
        node, level = queue.pop(0)
        if level % 2 == node.val % 2:
            return False
        if node.left:
            queue.append((node.left, level + 1))
        if node.right:
            queue.append((node.right, level + 1))
    return True
```

**Time Complexity:** O(N), where N is the number of nodes in the tree.

**Space Complexity:** O(W), where W is the maximum width of the tree.

**Real-World Applications**

Even-odd trees can be used to model a variety of real-world problems. For example, they can be used to model the levels of a hierarchy, such as the organizational structure of a company or the levels of a government. They can also be used to model the structure of a computer network or the levels of a software program.

***

## reorder\_routes\_to\_make\_all\_paths\_lead\_to\_the\_city\_zero

**Problem Statement:**

You have a set of routes, where each route is a pair of cities. You want to reorder the routes so that all paths from any city eventually lead to city 0.

**Example:**

```
Input:
[[0,1],[1,2],[2,0]]

Output:
[[1,2],[2,0],[0,1]]
```

**Solution:**

The simplest way to solve this problem is to use the Union Find data structure.

* **Union Find:** A Union Find is a data structure that can find and merge disjoint sets. In this problem, we will use a Union Find to track which cities are connected.
* **Algorithm:**
  1. Initialize a Union Find with each city as its own set.
  2. For each route `[u, v]`, merge the sets containing `u` and `v`.
  3. If after merging all routes, all cities are in the same set, then return the routes in any order.
  4. Otherwise, there is no valid reordering of the routes.

**Implementation:**

```python
from typing import List, Tuple

class UnionFind:
    def __init__(self):
        self.parent = {}
        self.rank = {}

    def find(self, x):
        if x not in self.parent:
            self.parent[x] = x
            self.rank[x] = 0
        if x != self.parent[x]:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]

    def union(self, x, y):
        px = self.find(x)
        py = self.find(y)
        if px == py:
            return
        if self.rank[px] > self.rank[py]:
            self.parent[py] = px
        elif self.rank[px] < self.rank[py]:
            self.parent[px] = py
        else:
            self.parent[py] = px
            self.rank[px] += 1

def reorder_routes_to_make_all_paths_lead_to_the_city_zero(
    routes: List[Tuple[int, int]]
) -> bool:
    uf = UnionFind()
    for u, v in routes:
        uf.union(u, v)
    if uf.find(0) != uf.find(max(uf.parent)):
        return False
    return True
```

**Real World Applications:**

Union Finds have many applications in real-world problems:

* **Network Connectivity:** Determining which computers in a network are connected.
* **Social Networking:** Finding connected components in social networks.
* **Image Processing:** Identifying connected components in images.
* **Circuit Analysis:** Finding loops in circuits.

***

## destroying\_asteroids

**LeetCode Problem:** <https://leetcode.com/problems/asteroid-collision/>

**Problem Statement:** There are several asteroids in a row, and each asteroid has a positive integer mass. You are moving toward the right, and every time you encounter an asteroid, you smash it by the following rules:

* If the mass of your spaceship is greater than or equal to the mass of the asteroid, the asteroid is destroyed, and your spaceship's mass increases by the mass of the asteroid.
* If the mass of your spaceship is less than the mass of the asteroid, your spaceship is destroyed instead.

Given the masses of the asteroids, determine the final mass of your spaceship after destroying all the asteroids.

**Example:**

```
Input: asteroids = [5, 10, 5]
Output: 20
Explanation:
- You smash the first asteroid with mass 5 and your spaceship's mass becomes 10 + 5 = 15.
- You smash the second asteroid with mass 10 and your spaceship's mass becomes 15 + 10 = 25.
- You smash the third asteroid with mass 5 and your spaceship's mass becomes 25 + 5 = 30.
- Finally, your spaceship's mass is 30.
```

**Optimal Solution:** The optimal solution to this problem uses a stack to keep track of asteroids that have not been destroyed yet. Here's how it works:

1. Iterate through the asteroids from left to right.
2. Push the first asteroid onto the stack.
3. For each subsequent asteroid, compare its mass to the mass of the asteroid on top of the stack:
   * If the current asteroid's mass is greater than or equal to the top of the stack, pop the top of the stack and add the current asteroid's mass to your spaceship's mass.
   * If the current asteroid's mass is less than the top of the stack, push the current asteroid onto the stack.
4. After iterating through all the asteroids, your spaceship's mass will be the sum of the masses of all the asteroids that were not destroyed.

**Python Implementation:**

```python
def asteroid_collision(asteroids):
    stack = []

    for asteroid in asteroids:
        while stack and stack[-1] > 0 and asteroid < -stack[-1]:
            stack.pop()
        if not stack or stack[-1] < 0 or asteroid > 0:
            stack.append(asteroid)

    return sum(stack)
```

**Time Complexity:** O(n), where n is the number of asteroids.

**Space Complexity:** O(n), for the stack.

**Applications in Real World:**

This problem can be applied to various real-world scenarios, such as:

* **Space Exploration:** Determining the trajectory of asteroids and their potential impact on spacecraft.
* **Robotics:** Planning the path of robots in environments with obstacles.
* **Game Development:** Simulating collisions between objects in a game engine.

***

## check\_if\_a\_parentheses\_string\_can\_be\_valid

**Problem Statement:**

Given a string containing only parentheses '(' and ')', determine if it can be made valid by removing some or no characters. A string is valid if every left parenthesis '(' has a corresponding right parenthesis ')'.

**Solution:**

**High-Level Approach:**

We can iterate through the string and count the number of left and right parentheses. If the number of left parentheses is less than or equal to the number of right parentheses, we can remove some '(' and ')' characters to make the string valid.

**Detailed Algorithm:**

1. Initialize two variables: `left` to 0 and `right` to 0.
2. Iterate through the string:
   * If the current character is '(' increment `left`.
   * If the current character is ')' and `left` is greater than 0, decrement `left` and increment `right`.
   * If the current character is ')' and `left` is 0, the string cannot be made valid. Return False.
3. After iterating through the string, check if `left` is equal to `right`. If they are equal, return True; otherwise, return False.

**Example:**

```python
def check_if_a_parentheses_string_can_be_valid(s):
  """
  Checks if a parentheses string can be made valid by removing some or no characters.

  Args:
    s (str): The string to check.

  Returns:
    bool: True if the string can be made valid, False otherwise.
  """

  left = 0
  right = 0

  for char in s:
    if char == '(':
      left += 1
    elif char == ')' and left > 0:
      left -= 1
      right += 1
    elif char == ')':
      return False

  return left == right
```

**Applications:**

This algorithm can be used in various applications, such as:

* **Syntax checking:** Verifying the validity of parentheses in programming code.
* **Data parsing:** Ensuring that XML or JSON data has properly balanced parentheses.
* **Text processing:** Cleaning up text that may contain mismatched parentheses.
