Cat and mouse

Time: O(N^3); Space: O(N^2); hard

A game on an undirected graph is played by two players, Mouse and Cat, who alternate turns.

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

Mouse starts at node 1 and goes first, Cat starts at node 2 and goes second, and there is a Hole at node 0.

During each player’s turn, they must travel along one edge of the graph that meets where they are. For example, if the Mouse is at node 1, it must travel to any node in graph[1].

Additionally, it is not allowed for the Cat to travel to the Hole (node 0.)

Then, the game can end in 3 ways:

  1. If ever the Cat occupies the same node as the Mouse, the Cat wins.

  2. If ever the Mouse reaches the Hole, the Mouse wins.

  3. If ever a position is repeated (ie. the players are in the same position as a previous turn, and it is the same player’s turn to move), the game is a draw.

Given a graph, and assuming both players play optimally, return 1 if the game is won by Mouse, 2 if the game is won by Cat, and 0 if the game is a draw.

Example 1:

4---3---1
|   |
2---5
 \ /
  0

Input: graph = [[2,5],[3],[0,4,5],[1,4,5],[2,3],[0,2,3]]

Output: 0

Constraints:

  • 3 <= len(graph) <= 50

  • It is guaranteed that graph[1] is non-empty.

  • It is guaranteed that graph[2] contains a non-zero element.

[1]:

class Solution1(object):
    """
    Time: O(N^3)
    Space: O(N^2)
    """
    def catMouseGame(self, graph):
        """
        :type graph: List[List[int]]
        :rtype: int
        """
        HOLE, MOUSE_START, CAT_START = range(3)
        DRAW, MOUSE, CAT = range(3)

        def move(graph, lookup, i, other_i, is_mouse_turn):
            key = (i, other_i, is_mouse_turn)
            if key in lookup:
                return lookup[key]

            lookup[key] = DRAW
            if is_mouse_turn:
                skip, target, win, lose = other_i, HOLE, MOUSE, CAT
            else:
                skip, target, win, lose = HOLE, other_i, CAT, MOUSE

            for nei in graph[i]:
                if nei == target:
                    result = win
                    break
            else:
                result = lose
                for nei in graph[i]:
                    if nei == skip:
                        continue
                    tmp = move(graph, lookup, other_i, nei, not is_mouse_turn)
                    if tmp == win:
                        result = win
                        break
                    if tmp == DRAW:
                        result = DRAW
            lookup[key] = result
            return result

        return move(graph, {}, MOUSE_START, CAT_START, True)

[3]:

s = Solution1()

graph = [[2,5],[3],[0,4,5],[1,4,5],[2,3],[0,2,3]]
assert s.catMouseGame(graph) == 0