<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>

class Solution {
public:
    int traverse(vector<vector<int>>& grid, int cur_r, int cur_c, int empty_count) {
        if (grid[cur_r][cur_c] == 2) {
            return empty_count == -1?1:0;
        }
        grid[cur_r][cur_c] = -1;

        vector<pair<int, int>> directions;
        if (cur_r - 1 >= 0 && grid[cur_r - 1][cur_c] != -1) {
            directions.emplace_back(cur_r - 1, cur_c);
        }
        if (cur_r + 1 < grid.size() && grid[cur_r + 1][cur_c] != -1) {
            directions.emplace_back(cur_r + 1, cur_c);
        }
        if (cur_c - 1 >= 0 && grid[cur_r][cur_c - 1] != -1) {
            directions.emplace_back(cur_r, cur_c - 1);
        }
        if (cur_c + 1 < grid[0].size() && grid[cur_r][cur_c + 1] != -1) {
            directions.emplace_back(cur_r, cur_c + 1);
        }

        if (directions.empty()) {
            grid[cur_r][cur_c] = 0;
            return 0;
        }

        int path_count = 0;
        for (int i = 0; i < directions.size(); i++) {
            path_count += traverse(grid, directions[i].first, directions[i].second, empty_count - 1);
        }
        grid[cur_r][cur_c] = 0;
        return path_count;
    }

    int uniquePathsIII(vector<vector<int>>& grid) {
        int r = 0;
        int c = 0;
        int empty_count = 0;
        for (int i = 0;i < grid.size(); i++) {
            for (int j = 0;j < grid[i].size(); j++) {
                if (grid[i][j] == 1) {
                    r = i;
                    c = j;
                }
                if (grid[i][j] == 0) {
                    empty_count++;
                }
            }
        }
        return traverse(grid, r, c, empty_count);
    }
};
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