<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)
        dp = [ [{"c":0, "m":0} for j in range(n)] for i in range(n) ]
        outdegree = [ [{"c":len(graph[j])-(0 in graph[j]), "m":len(graph[i])} for j in range(n)] for i in range(n) ]
        nodes = []
        for j in range(1,n):
            dp[0][j]["m"] = dp[0][j]["c"] = 1
            nodes.extend( [(0,j,"m",1), (0,j,"c",1)] )
            dp[j][j]["m"] =  dp[j][j]["c"] = 2
            nodes.extend( [(j,j,"m",2), (j,j,"c",2)])
        
        while nodes:
            nodes_new = []
            for i, j, move, result in nodes:
                if i==1 and j==2 and move=="m": return result
                premove = "c" if move=="m" else "m"
                if premove=="c":
                    for j_ in graph[j]:
                        if j_==0: continue
                        if dp[i][j_]["c"]: continue
                        if result==2:
                            dp[i][j_]["c"] = 2; nodes_new.append((i,j_,"c",2))
                        else:
                            outdegree[i][j_]["c"]-=1
                            if outdegree[i][j_]["c"]==0:
                                dp[i][j_]["c"] = 1; nodes_new.append((i,j_,"c",1))
                else: # premove=="m"
                    for i_ in graph[i]:
                        if dp[i_][j]["m"]: continue
                        if result==1:
                            dp[i_][j]["m"] = 1; nodes_new.append((i_,j,"m",1))
                        else:
                            outdegree[i_][j]["m"]-=1
                            if outdegree[i_][j]["m"]==0:
                                dp[i_][j]["m"] = 2; nodes_new.append((i_,j,"m",2))
            nodes = nodes_new
        return dp[1][2]["m"]
18000