The Maze III

Hard

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GraphsDynamic Programming

There is a ball in a maze with empty cells and walls. The ball can go through empty cells by rolling up, down, left or right, but it won't stop rolling until hitting a wall. When the ball stops, it could choose the next direction.

Given the maze, the ball's start position and the hole position, where the ball should drop into, implement the function to find the shortest distance for the ball to stop at the hole. If the ball cannot reach the hole, output -1.

The maze is represented by a binary 2D array. 1 means the wall and 0 means the empty cell. You may assume that the borders of the maze are all walls. The start and hole coordinates are represented by row and column indexes.

Example 1:

Input 1: a maze represented by a 2D array

0 0 0 0 0
1 1 0 0 1
0 0 0 0 0
0 1 0 0 1
0 1 0 0 0

Input 2: start coordinate (rowStart, colStart) = (4, 3)
Input 3: hole coordinate (rowHole, colHole) = (0, 1)

Output: 12

Explanation:

One shortest way is : left -> up -> left -> up -> right -> right. The total distance is 1 + 1 + 3 + 1 + 2 + 4 = 12.

Example 2:

Input 1: a maze represented by a 2D array

0 0 0 0 0
1 1 0 0 1
0 0 0 0 0
0 1 0 0 1
0 1 0 0 0

Input 2: start coordinate (rowStart, colStart) = (4, 3)
Input 3: hole coordinate (rowHole, colHole) = (3, 0)

Output: -1

Explanation:
There is no way for the ball to stop at the hole.

Note:

There is only one hole in the maze.
Both the ball and the hole exist on an empty cell, and they will not be at the same position initially.
The given maze does not contain border (like the boundary walls in the figure).
You could assume the maze is non-empty, and its width and height will not exceed 100.

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the dimensions of the maze (rows x columns), and what are the maximum possible values for each dimension?
What values represent walls, empty spaces, the start, and the hole? Can I assume these will always be consistent (e.g., 0 for empty, 1 for wall)?
If there is no path from the start to the hole, what should I return (e.g., null, an empty string, or a specific error code)?
Is the maze guaranteed to be rectangular (i.e., all rows have the same length)? What happens if the start or hole are outside the bounds of the maze?
Can the ball start at the same location as the hole? If so, what is the expected output?

Brute Force Solution

Approach

The brute force strategy to solve the maze tries every possible path the ball could take. We systematically explore each direction at every step, continuing until we find the shortest path to the hole or exhaust all possibilities. This means exploring potentially many invalid or very long paths before potentially finding the shortest.

Here's how the algorithm would work step-by-step:

Start with the ball's initial position.
Try moving the ball in all four possible directions: up, down, left, and right.
For each direction, move the ball until it hits a wall or goes out of bounds.
Once the ball stops, check if it has reached the hole. If yes, we have found a possible path.
If the ball hasn't reached the hole, and the path hasn't been explored before, repeat the process (try all four directions again) from the ball's new position.
Keep track of the distance traveled for each path.
If we reach a point where all directions have already been explored or lead to a dead end (no way to reach the hole), backtrack to a previous position and try a different direction.
Continue exploring all possible paths until the hole is found or all paths have been exhausted.
If the hole is found, return the shortest distance among all valid paths found.

Code Implementation

def the_maze_iii_brute_force(maze, ball, hole):
    row_count = len(maze)
    column_count = len(maze[0])
    shortest_distance = float('inf')

    def find_shortest_path(current_position, current_distance, visited):
        nonlocal shortest_distance
        row, column = current_position

        # Check if the ball has reached the hole.
        if (row, column) == hole:
            shortest_distance = min(shortest_distance, current_distance)
            return

        # Explore all four directions.
        for direction_row, direction_column in [(0, 1), (0, -1), (1, 0), (-1, 0)]:
            new_row = row
            new_column = column
            distance = 0

            # Move the ball until it hits a wall.
            while (
                0 <= new_row + direction_row < row_count
                and 0 <= new_column + direction_column < column_count
                and maze[new_row + direction_row][new_column + direction_column] == 0
            ):
                new_row += direction_row
                new_column += direction_column
                distance += 1

                # Check if we reached the hole while rolling
                if (new_row, new_column) == hole:
                    break

            if distance > 0:
                new_position = (new_row, new_column)

                # Avoid cycles by checking visited positions
                if new_position not in visited:

                    # Mark the new position as visited.
                    new_visited = visited.copy()
                    new_visited.add(new_position)

                    # Recursively call the function
                    find_shortest_path(new_position, current_distance + distance, new_visited)

    find_shortest_path(tuple(ball), 0, {tuple(ball)})

    if shortest_distance == float('inf'):
        return -1
    else:
        return shortest_distance

Big(O) Analysis

Time Complexity

O(4^(m*n)) – In the worst-case scenario, the ball might explore almost every cell in the maze (m rows and n columns) multiple times before finding the hole or determining that no path exists. From each cell, there are up to four possible directions to move. This could lead to exploring paths akin to a 4-ary tree where the depth is roughly the number of cells in the maze (m*n). Consequently, the time complexity grows exponentially with the maze's dimensions. Therefore, the brute force approach results in O(4^(m*n)) time complexity, representing the exploration of all possible paths.

Space Complexity

O(N^2) – The algorithm uses a queue or stack (implicitly or explicitly) to store the paths to explore, which in the worst case, could contain all possible cells in the maze, leading to a space complexity of O(N^2), where N is the dimension of the maze. A visited set (or matrix) to keep track of explored cells is also used, further contributing O(N^2). Backtracking and storing the distance for each path explored can lead to a depth-first search and storage of O(N^2) elements in the call stack as well. Thus, the auxiliary space is dominated by these structures leading to O(N^2).

Optimal Solution

Approach

The optimal approach to solving this maze problem involves finding the shortest path from a starting point to a target location, considering that a ball can roll until it hits a wall. It uses a smart search method to efficiently explore possible paths, keeping track of the shortest distance found so far.

Here's how the algorithm would work step-by-step:

Imagine the ball rolling through the maze until it hits a wall. We want to find the shortest distance it needs to roll to reach the hole.
Start by virtually rolling the ball from its starting position in all four directions: up, down, left, and right.
When the ball stops at a wall, check if that spot has already been visited. If it has, and the previous visit used a shorter path, then we can stop and avoid going further with that direction.
If the spot hasn't been visited, or if we've found a shorter way to get there, record the new, shorter distance it took to reach that spot.
If the ball lands directly in the hole, we have a potential solution, so we need to track the total distance from the start to the hole.
Now, repeat the process from each of the new spots where the ball stopped, always choosing the shortest distance spots first, until all reachable spots have been explored.
Compare all the distances recorded when the ball landed in the hole. Pick the shortest one. This is the shortest path to the hole.
If the ball can't reach the hole, it means there is no path from the start, so return 'impossible'.

Code Implementation

import heapq

def find_shortest_path(
    maze,
    start_position,
    hole_position
):
    rows = len(maze)
    cols = len(maze[0])

    distance_matrix = {}
    distance_matrix[start_position] = 0

    priority_queue = [(0, start_position)]

    while priority_queue:
        current_distance, current_position = heapq.heappop(priority_queue)

        if current_distance > distance_matrix.get(current_position, float('inf')):
            continue

        for direction_row, direction_col in [(0, 1), (0, -1), (1, 0), (-1, 0)]:
            row_position, col_position = current_position
            distance = current_distance

            while (
                0 <= row_position + direction_row < rows and
                0 <= col_position + direction_col < cols and
                maze[row_position + direction_row][col_position + direction_col] == 0
            ):
                row_position += direction_row
                col_position += direction_col
                distance += 1

                if (row_position, col_position) == hole_position:
                    break

            if (row_position, col_position) != current_position:
                # We moved, so update the path if shorter
                if distance < distance_matrix.get((row_position, col_position), float('inf')):
                    distance_matrix[(row_position, col_position)] = distance
                    heapq.heappush(priority_queue, (distance, (row_position, col_position)))

    if hole_position in distance_matrix:
        return distance_matrix[hole_position]
    else:
        return -1

def the_maze_iii(maze, start_row, start_col, hole_row, hole_col):
    shortest_path_length = find_shortest_path(
        maze,
        (start_row, start_col),
        (hole_row, hole_col)
    )

    if shortest_path_length == -1:
        return "impossible"
    else:
        return str(shortest_path_length)

Big(O) Analysis

Time Complexity

O(m * n * max(m, n)) – The algorithm uses a priority queue (or similar data structure) to explore possible paths. In the worst case, we might visit each cell of the maze (m rows and n columns) multiple times. For each cell, we explore up to four directions, rolling the ball until it hits a wall. The rolling process in each direction can take at most max(m, n) steps. Therefore, the overall time complexity is roughly proportional to the number of cells (m * n) multiplied by the maximum rolling distance (max(m, n)), leading to O(m * n * max(m, n)).

Space Complexity

O(M * N) – The algorithm utilizes a distance matrix to keep track of the shortest distance from the start to each cell in the maze, where M and N are the dimensions of the maze. It also implicitly uses a queue (though not explicitly mentioned, the 'choosing the shortest distance spots first' implies a priority queue which can be implemented with a queue) which, in the worst case, could contain all the cells in the maze. Therefore, the space complexity is determined by the size of the distance matrix which is proportional to the maze size (M * N).

Edge Cases

Case	How to Handle
Maze is null or empty	Return an empty string or null to indicate no path exists.
Ball and hole start at the same location	The shortest path is an empty string, indicating no movement needed.
The ball cannot reach the hole because it's blocked	Return "impossible" or a similar string representing no solution.
Maze dimensions are very large, potentially causing memory issues or long run times	Prioritize using efficient data structures like priority queues and optimize distance calculations to handle large mazes within time limits.
Multiple shortest paths exist; return lexicographically smallest one	Maintain path string during search and when distances are equal, update the path only if the new path is lexicographically smaller.
Integer overflow during distance calculation	Use a larger integer type (e.g., long) or appropriate bounds checking to prevent overflow.
The maze contains cycles or loops that the ball could endlessly traverse	The shortest path algorithm should implicitly handle cycles by tracking visited nodes and prioritizing shorter paths.
The hole is located at the maze boundary	The algorithm should handle boundary conditions correctly, ensuring calculations stay within bounds.

Maze is null or empty

How to Handle:

Return an empty string or null to indicate no path exists.

Ball and hole start at the same location

How to Handle:

The shortest path is an empty string, indicating no movement needed.

The ball cannot reach the hole because it's blocked

How to Handle:

Return "impossible" or a similar string representing no solution.

Maze dimensions are very large, potentially causing memory issues or long run times

How to Handle:

Prioritize using efficient data structures like priority queues and optimize distance calculations to handle large mazes within time limits.

Multiple shortest paths exist; return lexicographically smallest one

How to Handle:

Maintain path string during search and when distances are equal, update the path only if the new path is lexicographically smaller.

Integer overflow during distance calculation

How to Handle:

Use a larger integer type (e.g., long) or appropriate bounds checking to prevent overflow.

The maze contains cycles or loops that the ball could endlessly traverse

How to Handle:

The shortest path algorithm should implicitly handle cycles by tracking visited nodes and prioritizing shorter paths.

The hole is located at the maze boundary

How to Handle:

The algorithm should handle boundary conditions correctly, ensuring calculations stay within bounds.