<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 uniquePathsIII(vector<vector<int>>& grid) {
        long long ans = 0;
        string s = "";
        int n = grid.size();
        int m = grid[0].size();
        for ( int i = 1 ; i <= n * m ; i++ ) {
            s += '0';
        }
        int x , y;
        int block = 0;
        for ( int i = 0 ; i < n ; i++ ) {
            for ( int j = 0 ; j < m ; j++ ) {
                if ( grid[i][j] == 1 ) {
                    x = i;
                    y = j;
                }
                if ( grid[i][j] == -1 ) {
                    block++;
                }
            }
        }
        int pos = ( x * m ) + y;
        s[pos] = '1';
        queue <pair<string , int>> st;
        st.push( { s , pos } );
        while ( st.size() != 0 ) {
            pair<string , int> way = st.front();
            st.pop();
            x = way.second / m;
            y = way.second % m;
            int num = 0;
            for ( auto it : way.first ) {
                if ( it == '1' ) {
                    num++;
                }
            }
            if( grid[x][y] == 2 ) {
                if ( num == n * m - block ) {
                    ans++;
                }
                continue;
            }
            if ( y > 0 && grid[x][y-1] != -1 && way.first[ x * m + y - 1 ] == '0' ) {
                string new1 = way.first;
                new1[x * m + y - 1] = '1';
                st.push( { new1 , x * m + y - 1 } );
            }
            if ( x > 0 && grid[x-1][y] != -1 && way.first[ ( x - 1 ) * m + y ] == '0') {
                string new1 = way.first;
                new1[( x - 1 ) * m + y ] = '1';
                st.push( { new1 , ( x - 1 ) * m + y } );
            }
            if ( x < n - 1 && grid[x+1][y] != -1 && way.first[( x + 1 ) * m + y ] == '0' ) {
                string new1 = way.first;
                new1[( x + 1 ) * m + y ] = '1';
                st.push( { new1 ,( x + 1 ) * m + y  } );
            }
            if ( y < m - 1 && grid[x][y+1] != -1 && way.first[x * m + y + 1] == '0' ) {
                string new1 = way.first;
                new1[x * m + y + 1] = '1';
                st.push( { new1 , x * m + y + 1 } );
            }
        }
        return ans;
    }
};
16138