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English Version

题目描述

有一堆石头,用整数数组 stones 表示。其中 stones[i] 表示第 i 块石头的重量。

每一回合,从中选出任意两块石头,然后将它们一起粉碎。假设石头的重量分别为 x 和 y,且 x <= y。那么粉碎的可能结果如下:

  • 如果 x == y,那么两块石头都会被完全粉碎;
  • 如果 x != y,那么重量为 x 的石头将会完全粉碎,而重量为 y 的石头新重量为 y-x

最后,最多只会剩下一块 石头。返回此石头 最小的可能重量 。如果没有石头剩下,就返回 0

 

示例 1:

输入:stones = [2,7,4,1,8,1]
输出:1
解释:
组合 2 和 4,得到 2,所以数组转化为 [2,7,1,8,1],
组合 7 和 8,得到 1,所以数组转化为 [2,1,1,1],
组合 2 和 1,得到 1,所以数组转化为 [1,1,1],
组合 1 和 1,得到 0,所以数组转化为 [1],这就是最优值。

示例 2:

输入:stones = [31,26,33,21,40]
输出:5

 

提示:

  • 1 <= stones.length <= 30
  • 1 <= stones[i] <= 100

解法

方法一:动态规划

两个石头的重量越接近,粉碎后的新重量就越小。同样的,两堆石头的重量越接近,它们粉碎后的新重量也越小。

所以本题可以转换为,计算容量为 sum / 2 的背包最多能装多少重量的石头。

定义 dp[i][j] 表示从前 i 个石头中选出若干个,使得所选石头重量之和为不超过 j 的最大重量。

Python3

动态规划——0-1 背包朴素做法:

class Solution:
    def lastStoneWeightII(self, stones: List[int]) -> int:
        s = sum(stones)
        m, n = len(stones), s >> 1
        dp = [[0] * (n + 1) for _ in range(m + 1)]
        for i in range(1, m + 1):
            for j in range(n + 1):
                dp[i][j] = dp[i - 1][j]
                if stones[i - 1] <= j:
                    dp[i][j] = max(
                        dp[i][j], dp[i - 1][j - stones[i - 1]] + stones[i - 1]
                    )
        return s - 2 * dp[-1][-1]

动态规划——0-1 背包空间优化:

class Solution:
    def lastStoneWeightII(self, stones: List[int]) -> int:
        s = sum(stones)
        m, n = len(stones), s >> 1
        dp = [0] * (n + 1)
        for v in stones:
            for j in range(n, v - 1, -1):
                dp[j] = max(dp[j], dp[j - v] + v)
        return s - dp[-1] * 2

Java

class Solution {
    public int lastStoneWeightII(int[] stones) {
        int s = 0;
        for (int v : stones) {
            s += v;
        }
        int m = stones.length;
        int n = s >> 1;
        int[][] dp = new int[m + 1][n + 1];
        for (int i = 1; i <= m; ++i) {
            for (int j = 0; j <= n; ++j) {
                dp[i][j] = dp[i - 1][j];
                if (stones[i - 1] <= j) {
                    dp[i][j] = Math.max(dp[i][j], dp[i - 1][j - stones[i - 1]] + stones[i - 1]);
                }
            }
        }
        return s - dp[m][n] * 2;
    }
}
class Solution {
    public int lastStoneWeightII(int[] stones) {
        int s = 0;
        for (int v : stones) {
            s += v;
        }
        int m = stones.length;
        int n = s >> 1;
        int[] dp = new int[n + 1];
        for (int v : stones) {
            for (int j = n; j >= v; --j) {
                dp[j] = Math.max(dp[j], dp[j - v] + v);
            }
        }
        return s - dp[n] * 2;
    }
}

C++

class Solution {
public:
    int lastStoneWeightII(vector<int>& stones) {
        int s = accumulate(stones.begin(), stones.end(), 0);
        int m = stones.size(), n = s >> 1;
        vector<vector<int>> dp(m + 1, vector<int>(n + 1));
        for (int i = 1; i <= m; ++i) {
            for (int j = 0; j <= n; ++j) {
                dp[i][j] = dp[i - 1][j];
                if (stones[i - 1] <= j) dp[i][j] = max(dp[i][j], dp[i - 1][j - stones[i - 1]] + stones[i - 1]);
            }
        }
        return s - dp[m][n] * 2;
    }
};
class Solution {
public:
    int lastStoneWeightII(vector<int>& stones) {
        int s = accumulate(stones.begin(), stones.end(), 0);
        int n = s >> 1;
        vector<int> dp(n + 1);
        for (int& v : stones)
            for (int j = n; j >= v; --j)
                dp[j] = max(dp[j], dp[j - v] + v);
        return s - dp[n] * 2;
    }
};

Go

func lastStoneWeightII(stones []int) int {
	s := 0
	for _, v := range stones {
		s += v
	}
	m, n := len(stones), s>>1
	dp := make([][]int, m+1)
	for i := range dp {
		dp[i] = make([]int, n+1)
	}
	for i := 1; i <= m; i++ {
		for j := 0; j <= n; j++ {
			dp[i][j] = dp[i-1][j]
			if stones[i-1] <= j {
				dp[i][j] = max(dp[i][j], dp[i-1][j-stones[i-1]]+stones[i-1])
			}
		}
	}
	return s - dp[m][n]*2
}

func max(a, b int) int {
	if a > b {
		return a
	}
	return b
}
func lastStoneWeightII(stones []int) int {
	s := 0
	for _, v := range stones {
		s += v
	}
	n := s >> 1
	dp := make([]int, n+1)
	for _, v := range stones {
		for j := n; j >= v; j-- {
			dp[j] = max(dp[j], dp[j-v]+v)
		}
	}
	return s - dp[n]*2
}

func max(a, b int) int {
	if a > b {
		return a
	}
	return b
}

JavaScript

/**
 * @param {number[]} stones
 * @return {number}
 */
var lastStoneWeightII = function (stones) {
    let s = 0;
    for (let v of stones) {
        s += v;
    }
    const n = s >> 1;
    let dp = new Array(n + 1).fill(0);
    for (let v of stones) {
        for (let j = n; j >= v; --j) {
            dp[j] = Math.max(dp[j], dp[j - v] + v);
        }
    }
    return s - dp[n] * 2;
};

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