<p>A game on an <strong>undirected</strong> graph is played by two players, Mouse and Cat, who alternate turns.</p>

<p>The graph is given as follows: <code>graph[a]</code> is a list of all nodes <code>b</code> such that <code>ab</code> is an edge of the graph.</p>

<p>The mouse starts at node <code>1</code> and goes first, the cat starts at node <code>2</code> and goes second, and there is a hole at node <code>0</code>.</p>

<p>During each player&#39;s turn, they <strong>must</strong> travel along one&nbsp;edge of the graph that meets where they are.&nbsp; For example, if the Mouse is at node 1, it <strong>must</strong> travel to any node in <code>graph[1]</code>.</p>

<p>Additionally, it is not allowed for the Cat to travel to the Hole (node <code>0</code>).</p>

<p>Then, the game can end in three&nbsp;ways:</p>

<ul>
	<li>If ever the Cat occupies the same node as the Mouse, the Cat wins.</li>
	<li>If ever the Mouse reaches the Hole, the Mouse wins.</li>
	<li>If ever a position is repeated (i.e., the players are in the same position as a previous turn, and&nbsp;it is the same player&#39;s turn to move), the game is a draw.</li>
</ul>

<p>Given a <code>graph</code>, and assuming both players play optimally, return</p>

<ul>
	<li><code>1</code>&nbsp;if the mouse wins the game,</li>
	<li><code>2</code>&nbsp;if the cat wins the game, or</li>
	<li><code>0</code>&nbsp;if the game is a draw.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/17/cat1.jpg" style="width: 300px; height: 300px;" />
<pre>
<strong>Input:</strong> graph = [[2,5],[3],[0,4,5],[1,4,5],[2,3],[0,2,3]]
<strong>Output:</strong> 0
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/17/cat2.jpg" style="width: 200px; height: 200px;" />
<pre>
<strong>Input:</strong> graph = [[1,3],[0],[3],[0,2]]
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>3 &lt;= graph.length &lt;= 50</code></li>
	<li><code>1&nbsp;&lt;= graph[i].length &lt; graph.length</code></li>
	<li><code>0 &lt;= graph[i][j] &lt; graph.length</code></li>
	<li><code>graph[i][j] != i</code></li>
	<li><code>graph[i]</code> is unique.</li>
	<li>The mouse and the cat can always move.&nbsp;</li>
</ul>

class Solution {
public:
    vector<vector<vector<int>>> dp;
    
    int dfs(vector<vector<int>>& graph, int cat, int mouse, int move) {
        // Base cases
             // Mouse wins
        if (dp[cat][mouse][move] != -1) return dp[cat][mouse][move];
        if (move > 5 * graph.size()) {
            dp[cat][mouse][move]=0;
            return 0;  // Draw if moves exceed a limit
        }
        if (cat == mouse) {
            dp[cat][mouse][move]=2;
            return 2;         // Cat wins
        }
        if (mouse == 0) {
            dp[cat][mouse][move]=1;
            return 1;   
        }   

        

        if (move % 2 == 0) {  // Mouse's turn
            bool isDraw = false;
            for (int child : graph[mouse]) {
                int result = dfs(graph, cat, child, move + 1);
                if (result == 1)  {
                    dp[cat][mouse][move] = 1;  // Mouse wins
                    return 1;
                }
                if (result == 0) isDraw = true;  // Possible draw
            }
            if (isDraw) {
                dp[cat][mouse][move] = 0;  // Draw if found
                return 0;
            } 
            dp[cat][mouse][move] = 2;
            return 2;  // Otherwise, Cat wins
        } else {  // Cat's turn
            bool isDraw = false;
            for (int child : graph[cat]) {
                if (child == 0) continue;  // Cat cannot move to the hole
                int result = dfs(graph, child, mouse, move + 1);
                if (result == 2) return dp[cat][mouse][move] = 2;  // Cat wins
                if (result == 0) isDraw = true;  // Possible draw
            }
            if (isDraw) return dp[cat][mouse][move] = 0;  // Draw if found
            return dp[cat][mouse][move] = 1;  // Otherwise, Mouse wins
        }
    }

    int catMouseGame(vector<vector<int>>& graph) {
        int n = graph.size();
        dp = vector<vector<vector<int>>>(n, vector<vector<int>>(n, vector<int>(6 * n+1, -1)));
        return dfs(graph, 2, 1, 0);
    }
};

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