<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 {
    vector<vector<int>> gr;
    int dp[50][50][3];
    
    int DP(int mouse, int cat, int turn) {
        if (dp[mouse][cat][turn] != -1) return dp[mouse][cat][turn];
        if (mouse == cat) return dp[mouse][cat][turn] = 2; 
        if (mouse == 0) return dp[mouse][cat][turn] = 1;   
        
        dp[mouse][cat][turn] = 0; 
        
        bool isDraw = false;
        if (turn == 1) { 
            for (auto &nxt : gr[mouse]) {
                if (nxt == cat) continue; 
                int res_nxt = DP(nxt, cat, 2);
                if (res_nxt == 1) {
                    return dp[mouse][cat][turn] = 1; 
                } else if (res_nxt == 0) {
                    isDraw = true;
                }
            }
            if (isDraw) return dp[mouse][cat][turn] = 0; 
            return dp[mouse][cat][turn] = 2; 
        } else { 
            for (auto &nxt : gr[cat]) {
                if (nxt == 0) continue; 
                int res_nxt = DP(mouse, nxt, 1);
                if (res_nxt == 2) {
                    return dp[mouse][cat][turn] = 2; 
                } else if (res_nxt == 0) {
                    isDraw = true;
                }
            }
            if (isDraw) return dp[mouse][cat][turn] = 0; 
            return dp[mouse][cat][turn] = 1;
        }
    }
    
public:
    int catMouseGame(vector<vector<int>>& graph) {
        this->gr = graph;
        int n = graph.size();
        memset(dp, -1, sizeof(dp)); 
        
        vector<tuple<int, int, int>> reV;
        DP(1, 2, 1); 
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                for (int k = 1; k <= 2; ++k) { 
                    if (dp[i][j][k] == 0) {
                        reV.push_back({i, j, k});
                    }
                }
            }
        }
        
        int lastRound = -1;
        while (lastRound != reV.size() && !reV.empty()) {
            lastRound = reV.size();
            vector<tuple<int, int, int>> newReV;
            
            for (const auto& state : reV) {
                int mouse, cat, turn;
                tie(mouse, cat, turn) = state;
                dp[mouse][cat][turn] = -1; 
                DP(mouse, cat, turn); 
            }
            
            for (int i = 0; i < n; ++i) {
                for (int j = 0; j < n; ++j) {
                    for (int k = 1; k <= 2; ++k) {
                        if (dp[i][j][k] == 0) {
                            newReV.push_back({i, j, k});
                        }
                    }
                }
            }
            reV = newReV;
        }
        
        return dp[1][2][1] == -1 ? 0 : dp[1][2][1];
    }
};

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