# 490. The Maze

## Descriptio

There is a ball in a maze with empty spaces and walls. The ball can go through empty spaces 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 ball's start position, the destination and the maze, determine whether the ball could stop at the destination.

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

Example 1

Input 1: a maze represented by a 2D array
0 0 1 0 0
0 0 0 0 0
0 0 0 1 0
1 1 0 1 1
0 0 0 0 0

Input 2: start coordinate (rowStart, colStart) = (0, 4)
Input 3: destination coordinate (rowDest, colDest) = (4, 4)

Output: true
Explanation: One possible way is : left -> down -> left -> down -> right -> down -> right.

image

Example 2

Input 1: a maze represented by a 2D array
0 0 1 0 0
0 0 0 0 0
0 0 0 1 0
1 1 0 1 1
0 0 0 0 0

Input 2: start coordinate (rowStart, colStart) = (0, 4)
Input 3: destination coordinate (rowDest, colDest) = (3, 2)

Output: false
Explanation: There is no way for the ball to stop at the destination.

image

Note:

1. There is only one ball and one destination in the maze.
2. Both the ball and the destination exist on an empty space, and they will not be at the same position initially.
3. The given maze does not contain border (like the red rectangle in the example pictures), but you could assume the border of the maze are all walls.
4. The maze contains at least 2 empty spaces, and both the width and height of the maze won't exceed 100.

## Solution

### DFS

``````class Solution {
public static final int[][] DIRECTIONS = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}};

public boolean hasPath(int[][] maze, int[] start, int[] destination) {
return dfs(maze, start, destination, new HashSet<>());
}

private boolean dfs(int[][] maze, int[] start, int[] dest, Set<Integer> visited) {
if (start[0] == dest[0] && start[1] == dest[1]) {   // meet destination
return true;
}

int index = start[0] * maze[0].length + start[1];   // convert 2D to 1D
if (!visited.add(index)) {  // decides if start[] has already been set as start and failed
return false;
}

for (int[] d : DIRECTIONS) {    // try each direction
int x = start[0];
int y = start[1];

while (isEmpty(maze, x + d[0], y + d[1])) {
x += d[0];
y += d[1];
}

if (dfs(maze, new int[] {x, y}, dest, visited)) {
return true;
}
}
// important: no need to backtrack!!!
return false;
}

public boolean isEmpty(int[][] maze, int i, int j) {
return isValid(maze, i, j) && maze[i][j] != 1;
}

public boolean isValid(int[][] maze, int i, int j) {
return i >= 0 && i < maze.length && j >= 0 && j < maze[0].length;
}
}
``````

``````class Solution {
public static final int[][] DIRECTIONS = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}};

public boolean hasPath(int[][] maze, int[] start, int[] destination) {
return dfs(maze, start, destination);
}

private boolean dfs(int[][] maze, int[] start, int[] dest) {
if (start[0] == dest[0] && start[1] == dest[1]) {   // meet destination
return true;
}

if (maze[start[0]][start[1]] == -1) {   // decide if already visited as start and failed
return false;
}

maze[start[0]][start[1]] = -1;  // mark visited as start

for (int[] d : DIRECTIONS) {    // try each direction
int x = start[0];
int y = start[1];

while (isEmpty(maze, x + d[0], y + d[1])) {
x += d[0];
y += d[1];
}

if (dfs(maze, new int[] {x, y}, dest)) {
return true;
}
}
// important: no need to backtrack!!!
return false;
}

public boolean isEmpty(int[][] maze, int i, int j) {
return isValid(maze, i, j) && maze[i][j] != 1;
}

public boolean isValid(int[][] maze, int i, int j) {
return i >= 0 && i < maze.length && j >= 0 && j < maze[0].length;
}
}
``````

### BFS

``````class Solution {
public static final int[][] DIRECTIONS = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}};

public boolean hasPath(int[][] maze, int[] start, int[] destination) {
int m = maze.length;
int n = maze[0].length;
Queue<int[]> queue = new LinkedList<>();
boolean[][] visited = new boolean[m][n];

queue.offer(start);
visited[start[0]][start[1]] = true;

while (!queue.isEmpty()) {
int[] pos = queue.poll();
if (pos[0] == destination[0] && pos[1] == destination[1]) {
return true;
}

for (int[] direction : DIRECTIONS) {
int x = pos[0];
int y = pos[1];

while (isEmpty(maze, x + direction[0], y + direction[1])) {
x += direction[0];
y += direction[1];
}

if (!visited[x][y]) {
queue.offer(new int[] {x, y});
visited[x][y] = true;
}
}
}

return false;
}

public boolean isEmpty(int[][] maze, int i, int j) {
return isValid(maze, i, j) && maze[i][j] != 1;
}

public boolean isValid(int[][] maze, int i, int j) {
return i >= 0 && i < maze.length && j >= 0 && j < maze[0].length;
}
}
``````

``````class Solution {
public static final int[][] DIRECTIONS = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}};

public boolean hasPath(int[][] maze, int[] start, int[] destination) {
int m = maze.length;
int n = maze[0].length;
Queue<int[]> queue = new LinkedList<>();
queue.offer(start);
maze[start[0]][start[1]] = -1;

while (!queue.isEmpty()) {
int[] curr = queue.poll();
if (isEqual(curr, destination)) {
return true;
}

for (int[] d : DIRECTIONS) {
int x = curr[0];
int y = curr[1];

while (canMoveTo(maze, x + d[0], y + d[1])) {
x += d[0];
y += d[1];
}

if (maze[x][y] != 0) {
continue;
}

queue.offer(new int[] {x, y});
maze[x][y] = -1;
}
}

return false;
}

private boolean isEqual(int[] a, int[] b) {
for (int i = 0; i < a.length; ++i) {
if (a[i] != b[i]) {
return false;
}
}

return true;
}

private boolean canMoveTo(int[][] maze, int i, int j) {
return i >= 0 && i < maze.length && j >= 0 && j < maze[0].length
&& maze[i][j] != 1;
}
}
``````

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