<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:
    def catMouseGame(self, graph: List[List[int]]) -> int:
        n = len(graph)
        MOUSE, CAT = 1, 2
        color = [[[0] * 3 for i in range(n)] for j in range(n)]
        q = deque()
        for i in range(1, n): 
            for t in range(1, 3): 
                color[0][i][t] = 1
                q.append((0, i, t))
                color[i][i][t] = 2
                q.append((i, i, t))

        while q:
            curStatus = q.popleft()
            mouse, cat, turn = curStatus
            for preStatus in self.allprevstatus(graph, curStatus): 
                premouse, precat, preturn = preStatus
                if color[premouse][precat][preturn] != 0: 
                    continue
                if color[mouse][cat][turn] == 3 - turn: 
                    color[premouse][precat][preturn] = preturn
                    q.append(preStatus)
                elif self.allneighbourswin(color, graph, preStatus): 
                    color[premouse][precat][preturn] = 3 - preturn
                    q.append(preStatus)
        
        return color[1][2][1]

    def allprevstatus(self, graph, status): 
        ans = []
        mouse, cat, turn = status
        if turn == 1:
            for precat in graph[cat]: 
                if precat == 0:
                    continue
                ans.append((mouse, precat, 2))
        else:
            for premouse in graph[mouse]: 
                ans.append((premouse, cat, 1))
        return ans

    def allneighbourswin(self, color, graph, status): 
        mouse, cat, turn = status
        if turn == 1: 
            for nextmouse in graph[mouse]: 
                if color[nextmouse][cat][2] != 2: 
                    return False
        elif turn == 2: 
            for nextcat in graph[cat]: 
                if nextcat == 0:
                    continue
                if color[mouse][nextcat][1] != 1: 
                    return False
        return True
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