<p>You are given an <code>m x n</code> integer array <code>grid</code> where <code>grid[i][j]</code> could be:</p>

<ul>
	<li><code>1</code> representing the starting square. There is exactly one starting square.</li>
	<li><code>2</code> representing the ending square. There is exactly one ending square.</li>
	<li><code>0</code> representing empty squares we can walk over.</li>
	<li><code>-1</code> representing obstacles that we cannot walk over.</li>
</ul>

<p>Return <em>the number of 4-directional walks from the starting square to the ending square, that walk over every non-obstacle square exactly once</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/02/lc-unique1.jpg" style="width: 324px; height: 245px;" />
<pre>
<strong>Input:</strong> grid = [[1,0,0,0],[0,0,0,0],[0,0,2,-1]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> We have the following two paths: 
1. (0,0),(0,1),(0,2),(0,3),(1,3),(1,2),(1,1),(1,0),(2,0),(2,1),(2,2)
2. (0,0),(1,0),(2,0),(2,1),(1,1),(0,1),(0,2),(0,3),(1,3),(1,2),(2,2)
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/02/lc-unique2.jpg" style="width: 324px; height: 245px;" />
<pre>
<strong>Input:</strong> grid = [[1,0,0,0],[0,0,0,0],[0,0,0,2]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> We have the following four paths: 
1. (0,0),(0,1),(0,2),(0,3),(1,3),(1,2),(1,1),(1,0),(2,0),(2,1),(2,2),(2,3)
2. (0,0),(0,1),(1,1),(1,0),(2,0),(2,1),(2,2),(1,2),(0,2),(0,3),(1,3),(2,3)
3. (0,0),(1,0),(2,0),(2,1),(2,2),(1,2),(1,1),(0,1),(0,2),(0,3),(1,3),(2,3)
4. (0,0),(1,0),(2,0),(2,1),(1,1),(0,1),(0,2),(0,3),(1,3),(1,2),(2,2),(2,3)
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/08/02/lc-unique3-.jpg" style="width: 164px; height: 165px;" />
<pre>
<strong>Input:</strong> grid = [[0,1],[2,0]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There is no path that walks over every empty square exactly once.
Note that the starting and ending square can be anywhere in the grid.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 20</code></li>
	<li><code>1 &lt;= m * n &lt;= 20</code></li>
	<li><code>-1 &lt;= grid[i][j] &lt;= 2</code></li>
	<li>There is exactly one starting cell and one ending cell.</li>
</ul>

using grid = std::vector<std::vector<int>>;
using dir = std::vector<std::pair<int, int>>;
using visited = std::vector<std::vector<bool>>;

class Solution {
private:
    enum Legend 
      { Obstacle = -1,
        Empty = 0,
        Source = 1, 
        Sink = 2
      };
    int M;
    int N;
    dir dirs = {{0, -1}, {-1, 0}, {0, 1}, {1, 0}};
    bool checkBoundary(int r, int c) 
      { return r >= 0 && r < M && c >= 0 && c < N; }
    int dfs(const grid& maze, int r, int c, int cells, visited mask)
    {
        if(maze[r][c] == Sink && cells == -1)
            return 1;
        if(maze[r][c] == Obstacle || 
           maze[r][c] == Sink && cells >= 0)
          return 0;

        int paths = 0;
        for(auto d : dirs)
        {
            int y = r + d.first;
            int x = c + d.second;
            if(checkBoundary(y, x) && 
               !mask[y][x])
            {
                auto x_y_mask = mask;
                x_y_mask[y][x] = true;
                paths += dfs(maze, y, x, cells - 1, x_y_mask);
            }
        }
        return paths;
    }
public:
    int uniquePathsIII(vector<vector<int>>& maze) 
    {
        M = maze.size();
        N = maze[0].size();
        std::pair<int, int> src;
        int cells = 0;
        for(int r = 0; r < M; r++)
          for(int c = 0; c < N; c++)
          {
            if(maze[r][c] == Source) src = {r, c};
            if(maze[r][c] == Empty) cells++;
          }
        visited mask(M, std::vector<bool>(N, false));
        mask[src.first][src.second] = true;
        return dfs(maze, src.first, src.second, cells, mask); 
    }
};
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