<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(object):
    def catMouseGame(self, graph):
        """
        :type graph: List[List[int]]
        :rtype: int
        """
        N = len(graph)
        MOUSE, CAT = 1, 2
        # mouse, cat, turn
        color = [[[0] * 3 for i in range(N)] for j in range(N)]
        q = collections.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()
            cat, mouse, turn = curStatus
            for preStatus in self.findAllPrevStatus(graph, curStatus):
                preCat, preMouse, preTurn = preStatus
                if color[preCat][preMouse][preTurn] != 0:
                    continue
                if color[cat][mouse][turn] == 3 - turn:
                    color[preCat][preMouse][preTurn] = preTurn
                    q.append(preStatus)
                elif self.allNeighboursWin(color, graph, preStatus):
                    color[preCat][preMouse][preTurn] = 3 - preTurn
                    q.append(preStatus)
        return color[1][2][1]
    
    def findAllPrevStatus(self, graph, curStatus):
        ret = []
        mouse, cat, turn = curStatus
        if turn == 1:
            for preCat in graph[cat]:
                if preCat == 0:
                    continue
                ret.append((mouse, preCat, 2))
        else:
            for preMouse in graph[mouse]:
                ret.append((preMouse, cat, 1))
        return ret
    
    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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