<p>A game is played by a cat and a mouse named Cat and Mouse.</p>

<p>The environment is represented by a <code>grid</code> of size <code>rows x cols</code>, where each element is a wall, floor, player (Cat, Mouse), or food.</p>

<ul>
	<li>Players are represented by the characters <code>&#39;C&#39;</code>(Cat)<code>,&#39;M&#39;</code>(Mouse).</li>
	<li>Floors are represented by the character <code>&#39;.&#39;</code> and can be walked on.</li>
	<li>Walls are represented by the character <code>&#39;#&#39;</code> and cannot be walked on.</li>
	<li>Food is represented by the character <code>&#39;F&#39;</code> and can be walked on.</li>
	<li>There is only one of each character <code>&#39;C&#39;</code>, <code>&#39;M&#39;</code>, and <code>&#39;F&#39;</code> in <code>grid</code>.</li>
</ul>

<p>Mouse and Cat play according to the following rules:</p>

<ul>
	<li>Mouse <strong>moves first</strong>, then they take turns to move.</li>
	<li>During each turn, Cat and Mouse can jump in one of the four directions (left, right, up, down). They cannot jump over the wall nor outside of the <code>grid</code>.</li>
	<li><code>catJump, mouseJump</code> are the maximum lengths Cat and Mouse can jump at a time, respectively. Cat and Mouse can jump less than the maximum length.</li>
	<li>Staying in the same position is allowed.</li>
	<li>Mouse can jump over Cat.</li>
</ul>

<p>The game can end in 4 ways:</p>

<ul>
	<li>If Cat occupies the same position as Mouse, Cat wins.</li>
	<li>If Cat reaches the food first, Cat wins.</li>
	<li>If Mouse reaches the food first, Mouse wins.</li>
	<li>If Mouse cannot get to the food within 1000 turns, Cat wins.</li>
</ul>

<p>Given a <code>rows x cols</code> matrix <code>grid</code> and two integers <code>catJump</code> and <code>mouseJump</code>, return <code>true</code><em> if Mouse can win the game if both Cat and Mouse play optimally, otherwise return </em><code>false</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/12/sample_111_1955.png" style="width: 580px; height: 239px;" />
<pre>
<strong>Input:</strong> grid = [&quot;####F&quot;,&quot;#C...&quot;,&quot;M....&quot;], catJump = 1, mouseJump = 2
<strong>Output:</strong> true
<strong>Explanation:</strong> Cat cannot catch Mouse on its turn nor can it get the food before Mouse.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/12/sample_2_1955.png" style="width: 580px; height: 175px;" />
<pre>
<strong>Input:</strong> grid = [&quot;M.C...F&quot;], catJump = 1, mouseJump = 4
<strong>Output:</strong> true
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> grid = [&quot;M.C...F&quot;], catJump = 1, mouseJump = 3
<strong>Output:</strong> false
</pre>

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

<ul>
	<li><code>rows == grid.length</code></li>
	<li><code>cols = grid[i].length</code></li>
	<li><code>1 &lt;= rows, cols &lt;= 8</code></li>
	<li><code>grid[i][j]</code> consist only of characters <code>&#39;C&#39;</code>, <code>&#39;M&#39;</code>, <code>&#39;F&#39;</code>, <code>&#39;.&#39;</code>, and <code>&#39;#&#39;</code>.</li>
	<li>There is only one of each character <code>&#39;C&#39;</code>, <code>&#39;M&#39;</code>, and <code>&#39;F&#39;</code> in <code>grid</code>.</li>
	<li><code>1 &lt;= catJump, mouseJump &lt;= 8</code></li>
</ul>

class Solution {
public:
    bool canMouseWin(vector<string>& grid, int catJump, int mouseJump) {
        vector<int> direction = {0,1,0,-1,0};
        int m = grid.size(), n = grid[0].size();
        int mouse_pos=0, cat_pos = 0, num_floor=0;
        auto hash_pos = [n](int x, int y){
            return x*n+y;
        };

        for(int i=0;i<m;i++)
        {
            for(int j=0;j<n;j++)
            {
                if(grid[i][j]!='#')
                {
                    num_floor++;
                }
                if(grid[i][j]=='C')
                {
                    cat_pos = hash_pos(i,j);
                }
                else if(grid[i][j]=='M')
                {
                    mouse_pos = hash_pos(i,j);
                }
            }
        }

        unordered_map<int,int> dp;

        function<bool(int,int,int)> dfs = [&](int cat, int mouse, int turn)->bool{
            int state = (cat<<16) | (mouse<<8) | turn;
            if(dp.find(state)!=dp.end())
            {
                return dp[state];
            }
            if(turn==2*num_floor)
            {
                return dp[state]=false;
            }
            if(turn%2==0)
            {
                int i = mouse/n, j = mouse%n; 
                for(int k=0;k<4;k++)
                {
                    for(int jump=0;jump<=mouseJump;jump++)
                    {
                        int x = i+direction[k]*jump;
                        int y = j+direction[k+1]*jump;
                        if(x<0 || x>=m || y<0 || y>=n || grid[x][y]=='#')
                        {
                            break;
                        }
                        if(grid[x][y]=='F')
                        {
                            return dp[state] = true;
                        }
                        if(dfs(cat, hash_pos(x,y), turn+1))
                        {
                            return dp[state] = true;
                        }
                    }
                }
                dp[state] = false;
            }
            else
            {
                int i = cat/n, j = cat%n; 
                for(int k=0;k<4;k++)
                {
                    for(int jump=0;jump<=catJump;jump++)
                    {
                        int x = i+direction[k]*jump;
                        int y = j+direction[k+1]*jump;
                        if(x<0 || x>=m || y<0 || y>=n || grid[x][y]=='#')
                        {
                            break;
                        }
                        if(grid[x][y]=='F' || hash_pos(x,y)==mouse)
                        {
                            return dp[state] = false;
                        }
                        if(!dfs( hash_pos(x,y), mouse, turn+1))
                        {
                            return dp[state] = false;
                        }
                    }
                }
                dp[state] = true;
            }
            return dp[state];
        };

        return dfs(cat_pos, mouse_pos, 0);
    }
};
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