499. The Maze III

Problem Description

In this problem, we are given a m x n maze represented by a 2D grid with 0 indicating empty spaces and 1 indicating walls. A ball can roll in the maze in the cardinal directions (up, down, left, or right), but it only stops when it hits a wall. The ball stops moving further in the same direction once it encounters a wall. Additionally, there is a hole in the maze; if the ball goes over the hole, it will fall into it, and that is the target location the ball needs to reach.

The ball's start position and the hole's position are both given. The goal is to find a sequence of directions (as a string) that will lead the ball into the hole using the shortest path possible. If the shortest path is not unique, the lexicographically smallest sequence of directions should be returned. A sequence is considered lexicographically smaller if it has the smallest alphabetical order possible. If there is no possible way for the ball to fall into the hole, the function should return the string "impossible".

The valid directions the ball can roll are denoted by: 'u' for up, 'd' for down, 'l' for left, and 'r' for right. The distance is measured by the number of empty spaces the ball rolls over from its start to the hole, not counting the starting square but including the final square over the hole.

The maze is framed by walls, so the ball cannot roll beyond the limits of the grid.

Intuition

The intuition behind the given solution is founded on Breadth-first Search (BFS), a common algorithm for exploring paths in a graph or grid-like structures like a maze.

The BFS works by exploring all potential moves from a given position before moving on to the next set of positions. In this context, the valid moves for the ball are to keep rolling in one of the four cardinal directions until it hits a wall.

We use BFS to explore each possible path the ball can take and keep track of the distance travelled and the path taken for each visited position in the maze:

1. Keep Distance and Path Information: Each position in the maze maintains two pieces of information. One is the minimum number of steps taken to reach that position from the initial ball position. The other is the lexicographically smallest sequence of moves that the ball takes to reach that position.

2. Exploring the Maze: From the current position, we consider all four possible directions the ball can roll. For each direction, we roll the ball until it hits a wall or falls into the hole.

3. Tracking Shortest Paths: When the ball stops, we check if the distance to this new stopped position is less than any previously recorded distance. If it is less, or if the distances are equal but the new path is lexicographically smaller, we update the distance and path for that position.

4. Queue for BFS: We add each new position that we can roll to (except the hole) to a queue to explore in the future. This ensures that all possible moves from each position are considered before moving further away.

5. Final Check: Once the entire maze has been explored, we check the information for the hole's position. If a path was found, it will contain the lexicographically smallest sequence of directions to get there. If no path was found, the original None assignment will remain, and we will return "impossible" instead.

The provided solution neatly consolidates this approach using a queue data structure to perform the BFS, a 2D array for storing the distance, and another 2D array for storing the path strings for each position the ball stops at.

Learn more about Depth-First Search, Breadth-First Search, Graph, Shortest Path and Heap (Priority Queue) patterns.

Solution Approach

The implementation uses a BFS algorithm to traverse the maze and search for the shortest path to the hole. Here's the detailed breakdown of the steps and the data structures used:

1. Initialization: We define the maze's dimensions m and n. The starting position r, c (row, column) of the ball is extracted from the ball list. Similarly, the hole's position rh, ch is extracted from the hole list. We initialize a queue q using deque from the collections module, which will hold the positions to explore. We also define two 2D lists: dist to keep track of the distances and path to keep track of the paths.

2. Distance and Path Setup: The dist list is filled with inf (infinity) to indicate an unknown distance, while path is filled with None, except for the starting position, which is set to 0 in dist and the empty string '' in path.

3. BFS Loop: A while loop runs as long as there are positions in the queue q. At each iteration, we dequeue a position and examine all four possible movements from that position. The movements are encoded as tuples (a, b, d), where a and b are the directional increments, and d is the corresponding direction character ('u', 'd', 'l', 'r').

4. Valid Movement Checking: Inside this loop, we check if moving in a particular direction will stop the ball at a different position. We simulate the ball's movement by incrementally adding the direction increments a and b to the current position x, y until the ball hits a wall or the hole.

5. Updating Distances and Paths: When the ball cannot move any further in a direction, we check if this new position is either closer than any previously recorded position or if it's the same distance but with a lexicographically smaller path. If either is true, we update dist and path.

6. Enqueueing Next Positions: If the new stopped position is not the hole, it is added to the queue for further exploration.

7. Returning the Result: Once the BFS completes, the path at the hole's position path[rh][ch] represents the shortest path to the hole. If no path was found, path[rh][ch] will be None, and we will return 'impossible'.

The algorithm efficiently finds the shortest path by only updating positions when a better path is found. It ensures that the path followed is both the shortest in terms of distance and lexicographically minimal due to the updates in dist and path.

The data structures used — the queue for BFS and 2D arrays for distances and paths — are well-suited for the task. These choices allow the algorithm to queue future positions to visit and maintain necessary state information for each position in the maze.

This breadth-first search ensures that we examine all moves from a starting point before moving outward, thus always finding the shortest path to any reachable position.

Example Walkthrough

Let's consider a small maze, where 0 indicates an empty space, and 1 indicates a wall. We are given the ball's start position and the hole's target position. We will demonstrate the steps of the BFS algorithm to find the shortest path.

For the example, our maze (grid) will look like this:

11 1 1 1
21 0 0 1
31 0 1 1
41 0 0 0
51 1 1 1

Let the ball start position be at (1, 1) and the hole's target position be at (3, 2). Note that we are using (row, column) indexing.

Initial Setup

1Start position (S): (1, 1)
2Hole position (H): (3, 2)
3
4Maze:
5    1 1 1 1
6    1 S 0 1
7    1 0 1 1
8    1 0 H 0
9    1 1 1 1

We initialize the dist with inf (indicating unknown distance) and path with None for each cell except the start (1, 1), which has a distance of 0 and a path of ''. We enqueue the start position into our queue q.

Step-by-Step BFS

1. We dequeue (1, 1) and start exploring all four directions.

2. From (1, 1), we can roll down (d) to (3, 1), since it is the next wall or hole position vertically. We update dist[3][1] to 2 and path[3][1] to 'd', and enqueue (3, 1).

3. Enqueue order: (3, 1).

4. Dequeue (3, 1) and explore. It can go right (r) to the hole (3, 2). We update dist[3][2] to 3 and path[3][2] to 'dr'. There are no more enqueues because we've reached the hole.

5. No more positions to explore in the queue; the BFS ends.

Final Check

We consult the final dist and path values at the hole position (3, 2) and find a distance of 3 and a path of 'dr'.

Hence, the shortest path for the ball to fall into the hole is dr (down, then right), with only 3 moves disregarding the starting square.

This example walk-through demonstrates the BFS approach, where we systematically explore all directions from the starting point, updating distances and paths, until we find the shortest lexicographically smallest path to the hole.

Solution Implementation

Time and Space Complexity

Time Complexity

The given code implements a breadth-first search (BFS) algorithm on a matrix to find the shortest path for a ball to roll from its starting position to the hole. Each move consists of the ball rolling in a straight line until it hits a wall or falls into the hole.

The time complexity of the BFS algorithm is generally O(V + E), where V is the number of vertices and E is the number of edges in the graph. However, for a two-dimensional matrix, we should consider the following:

• Each cell in the matrix can be visited multiple times because the ball can reach the same cell from different directions. There are m * n cells, where m is the number of rows and n is the number of columns.
• For each cell (i, j), the ball can roll in four directions until it hits a wall or a boundary of the maze. Therefore, in the worst-case scenario, it might traverse the entire length or width of the maze.
• On average, the ball rolls m/2 or n/2 steps before hitting a wall or the boundary in either direction.

Taking these into consideration, the worst-case time complexity can be approximated as:

O((m * n) * (m + n))

This is because, for each of the m * n possible starting points, the ball may have to traverse an entire row or column (m + n steps in the worst case).

Space Complexity

The space complexity of the BFS algorithm depends on the size of the queue used, as well as the additional space for the dist and path arrays:

• The queue can grow up to m * n in size, as it contains positions representing the cells visited by the ball.
• The dist and path arrays are both sized m * n.

Hence, the overall space complexity is:

O(m * n + m * n) = O(2 * m * n) = O(m * n)

Overall, both dist and path arrays contribute to the space complexity equally, but constants are dropped in Big O notation, resulting in a total space complexity of O(m * n).

Learn more about how to find time and space complexity quickly using problem constraints.