LeetCode 877. Stone Game

Alex and Lee play a game with piles of stones.  There are an even number of piles arranged in a row, and each pile has a positive integer number of stones piles[i].

The objective of the game is to end with the most stones.  The total number of stones is odd, so there are no ties.

Alex and Lee take turns, with Alex starting first.  Each turn, a player takes the entire pile of stones from either the beginning or the end of the row.  This continues until there are no more piles left, at which point the person with the most stones wins.

Assuming Alex and Lee play optimally, return True if and only if Alex wins the game.

 

Example 1:

Input: [5,3,4,5]
Output: true
Explanation: 
Alex starts first, and can only take the first 5 or the last 5.
Say he takes the first 5, so that the row becomes [3, 4, 5].
If Lee takes 3, then the board is [4, 5], and Alex takes 5 to win with 10 points.
If Lee takes the last 5, then the board is [3, 4], and Alex takes 4 to win with 9 points.
This demonstrated that taking the first 5 was a winning move for Alex, so we return true.

 

Note:

1. 2 <= piles.length <= 500
2. piles.length is even.
3. 1 <= piles[i] <= 500
4. sum(piles) is odd.

"""
dp[i][j] 表示只有[i,j]给你选择的时候，可以取的最大值

dp[i][i] = piles[i]

解题思路：画转移树（n的个数尽可能小，保证叶子节点覆盖所有情况就行）

"""
class Solution(object):
    def stoneGame(self, piles):
        """
        :type piles: List[int]
        :rtype: bool
        """
        num = len(piles)
        
        dp = [[0 for col in xrange(num)] for row in xrange(num)]
        visited = [[0 for col in xrange(num)] for row in xrange(num)]
        
        return self.memorySearch(0 ,num - 1,dp,visited,piles) > sum(piles) * 1.0 / 2
        
    def memorySearch(self,start,end,dp,visited,piles):
        if start == end:
            return piles[i]
        
        if start < 0 or end >= len(piles) or start > end:
            return 0
        
        if visited[start][end] == 1:
            return dp[start][end]
        
        dp[start][end] = max(
            #左边
            min(
                #左
                self.memorySearch(start + 2 ,end,dp,visited,piles),
                #右
                self.memorySearch(start + 1 ,end - 1,dp,visited,piles)
               ) + piles[start],
        
            #右
            min(
                #右
                self.memorySearch(start ,end - 2,dp,visited,piles),
                #左
                self.memorySearch(start + 1 ,end - 1,dp,visited,piles)
               ) + piles[end]
        )
        
        visited[start][end] = 1
        return dp[start][end]

可以从数学的角度解O(1)时间和空间

https://leetcode.com/problems/stone-game/solution/