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689. 三个无重叠子数组的最大和

English Version

题目描述

给你一个整数数组 nums 和一个整数 k ，找出三个长度为 k 、互不重叠、且全部数字和（3 * k 项）最大的子数组，并返回这三个子数组。

以下标的数组形式返回结果，数组中的每一项分别指示每个子数组的起始位置（下标从 0 开始）。如果有多个结果，返回字典序最小的一个。

 

示例 1：

输入：nums = [1,2,1,2,6,7,5,1], k = 2
输出：[0,3,5]
解释：子数组 [1, 2], [2, 6], [7, 5] 对应的起始下标为 [0, 3, 5]。
也可以取 [2, 1], 但是结果 [1, 3, 5] 在字典序上更大。

示例 2：

输入：nums = [1,2,1,2,1,2,1,2,1], k = 2
输出：[0,2,4]

 

提示：

  • 1 <= nums.length <= 2 * 104
  • 1 <= nums[i] < 216
  • 1 <= k <= floor(nums.length / 3)

解法

方法一：滑动窗口

滑动窗口，枚举第三个子数组的位置，同时维护前两个无重叠子数组的最大和及其位置。

时间复杂度 $O(n)$，其中 $n$ 是数组 $nums$ 的长度。空间复杂度 $O(1)$。

class Solution:
    def maxSumOfThreeSubarrays(self, nums: List[int], k: int) -> List[int]:
        s = s1 = s2 = s3 = 0
        mx1 = mx12 = 0
        idx1, idx12 = 0, ()
        ans = []
        for i in range(k * 2, len(nums)):
            s1 += nums[i - k * 2]
            s2 += nums[i - k]
            s3 += nums[i]
            if i >= k * 3 - 1:
                if s1 > mx1:
                    mx1 = s1
                    idx1 = i - k * 3 + 1
                if mx1 + s2 > mx12:
                    mx12 = mx1 + s2
                    idx12 = (idx1, i - k * 2 + 1)
                if mx12 + s3 > s:
                    s = mx12 + s3
                    ans = [*idx12, i - k + 1]
                s1 -= nums[i - k * 3 + 1]
                s2 -= nums[i - k * 2 + 1]
                s3 -= nums[i - k + 1]
        return ans
class Solution {
    public int[] maxSumOfThreeSubarrays(int[] nums, int k) {
        int[] ans = new int[3];
        int s = 0, s1 = 0, s2 = 0, s3 = 0;
        int mx1 = 0, mx12 = 0;
        int idx1 = 0, idx121 = 0, idx122 = 0;
        for (int i = k * 2; i < nums.length; ++i) {
            s1 += nums[i - k * 2];
            s2 += nums[i - k];
            s3 += nums[i];
            if (i >= k * 3 - 1) {
                if (s1 > mx1) {
                    mx1 = s1;
                    idx1 = i - k * 3 + 1;
                }
                if (mx1 + s2 > mx12) {
                    mx12 = mx1 + s2;
                    idx121 = idx1;
                    idx122 = i - k * 2 + 1;
                }
                if (mx12 + s3 > s) {
                    s = mx12 + s3;
                    ans = new int[] {idx121, idx122, i - k + 1};
                }
                s1 -= nums[i - k * 3 + 1];
                s2 -= nums[i - k * 2 + 1];
                s3 -= nums[i - k + 1];
            }
        }
        return ans;
    }
}
class Solution {
public:
    vector<int> maxSumOfThreeSubarrays(vector<int>& nums, int k) {
        vector<int> ans(3);
        int s = 0, s1 = 0, s2 = 0, s3 = 0;
        int mx1 = 0, mx12 = 0;
        int idx1 = 0, idx121 = 0, idx122 = 0;
        for (int i = k * 2; i < nums.size(); ++i) {
            s1 += nums[i - k * 2];
            s2 += nums[i - k];
            s3 += nums[i];
            if (i >= k * 3 - 1) {
                if (s1 > mx1) {
                    mx1 = s1;
                    idx1 = i - k * 3 + 1;
                }
                if (mx1 + s2 > mx12) {
                    mx12 = mx1 + s2;
                    idx121 = idx1;
                    idx122 = i - k * 2 + 1;
                }
                if (mx12 + s3 > s) {
                    s = mx12 + s3;
                    ans = {idx121, idx122, i - k + 1};
                }
                s1 -= nums[i - k * 3 + 1];
                s2 -= nums[i - k * 2 + 1];
                s3 -= nums[i - k + 1];
            }
        }
        return ans;
    }
};
func maxSumOfThreeSubarrays(nums []int, k int) []int {
	ans := make([]int, 3)
	s, s1, s2, s3 := 0, 0, 0, 0
	mx1, mx12 := 0, 0
	idx1, idx121, idx122 := 0, 0, 0
	for i := k * 2; i < len(nums); i++ {
		s1 += nums[i-k*2]
		s2 += nums[i-k]
		s3 += nums[i]
		if i >= k*3-1 {
			if s1 > mx1 {
				mx1 = s1
				idx1 = i - k*3 + 1
			}
			if mx1+s2 > mx12 {
				mx12 = mx1 + s2
				idx121 = idx1
				idx122 = i - k*2 + 1
			}
			if mx12+s3 > s {
				s = mx12 + s3
				ans = []int{idx121, idx122, i - k + 1}
			}
			s1 -= nums[i-k*3+1]
			s2 -= nums[i-k*2+1]
			s3 -= nums[i-k+1]
		}
	}
	return ans
}
function maxSumOfThreeSubarrays(nums: number[], k: number): number[] {
    const n: number = nums.length;
    const s: number[] = Array(n + 1).fill(0);

    for (let i = 0; i < n; ++i) {
        s[i + 1] = s[i] + nums[i];
    }

    const pre: number[][] = Array(n)
        .fill([])
        .map(() => new Array(2).fill(0));
    const suf: number[][] = Array(n)
        .fill([])
        .map(() => new Array(2).fill(0));

    for (let i = 0, t = 0, idx = 0; i < n - k + 1; ++i) {
        const cur: number = s[i + k] - s[i];
        if (cur > t) {
            pre[i + k - 1] = [cur, i];
            t = cur;
            idx = i;
        } else {
            pre[i + k - 1] = [t, idx];
        }
    }

    for (let i = n - k, t = 0, idx = 0; i >= 0; --i) {
        const cur: number = s[i + k] - s[i];
        if (cur >= t) {
            suf[i] = [cur, i];
            t = cur;
            idx = i;
        } else {
            suf[i] = [t, idx];
        }
    }

    let ans: number[] = [];
    for (let i = k, t = 0; i < n - 2 * k + 1; ++i) {
        const cur: number = s[i + k] - s[i] + pre[i - 1][0] + suf[i + k][0];
        if (cur > t) {
            ans = [pre[i - 1][1], i, suf[i + k][1]];
            t = cur;
        }
    }

    return ans;
}

方法二：预处理前后缀 + 枚举中间子数组

我们可以预处理得到数组 $nums$ 的前缀和数组 $s$，其中 $s[i] = \sum_{j=0}^{i-1} nums[j]$，那么对于任意的 $i$，$j$，$s[j] - s[i]$ 就是子数组 $[i, j)$ 的和。

接下来，我们使用动态规划的方法，维护两个长度为 $n$ 的数组 $pre$ 和 $suf$，其中 $pre[i]$ 表示 $[0, i]$ 范围内长度为 $k$ 的子数组的最大和及其起始位置，$suf[i]$ 表示 $[i, n)$ 范围内长度为 $k$ 的子数组的最大和及其起始位置。

然后，我们枚举中间子数组的起始位置 $i$，那么三个子数组的和就是 $pre[i-1][0] + suf[i+k][0] + (s[i+k] - s[i])$，其中 $pre[i-1][0]$ 表示 $[0, i-1]$ 范围内长度为 $k$ 的子数组的最大和，$suf[i+k][0]$ 表示 $[i+k, n)$ 范围内长度为 $k$ 的子数组的最大和，$(s[i+k] - s[i])$ 表示 $[i, i+k)$ 范围内长度为 $k$ 的子数组的和。我们找出和的最大值对应的三个子数组的起始位置即可。

时间复杂度 $O(n)$，空间复杂度 $O(n)$。其中 $n$ 是数组 $nums$ 的长度。

class Solution:
    def maxSumOfThreeSubarrays(self, nums: List[int], k: int) -> List[int]:
        n = len(nums)
        s = list(accumulate(nums, initial=0))
        pre = [[] for _ in range(n)]
        suf = [[] for _ in range(n)]
        t = idx = 0
        for i in range(n - k + 1):
            if (cur := s[i + k] - s[i]) > t:
                pre[i + k - 1] = [cur, i]
                t, idx = pre[i + k - 1]
            else:
                pre[i + k - 1] = [t, idx]
        t = idx = 0
        for i in range(n - k, -1, -1):
            if (cur := s[i + k] - s[i]) >= t:
                suf[i] = [cur, i]
                t, idx = suf[i]
            else:
                suf[i] = [t, idx]
        t = 0
        ans = []
        for i in range(k, n - 2 * k + 1):
            if (cur := s[i + k] - s[i] + pre[i - 1][0] + suf[i + k][0]) > t:
                ans = [pre[i - 1][1], i, suf[i + k][1]]
                t = cur
        return ans
class Solution {
    public int[] maxSumOfThreeSubarrays(int[] nums, int k) {
        int n = nums.length;
        int[] s = new int[n + 1];
        for (int i = 0; i < n; ++i) {
            s[i + 1] = s[i] + nums[i];
        }
        int[][] pre = new int[n][0];
        int[][] suf = new int[n][0];
        for (int i = 0, t = 0, idx = 0; i < n - k + 1; ++i) {
            int cur = s[i + k] - s[i];
            if (cur > t) {
                pre[i + k - 1] = new int[] {cur, i};
                t = cur;
                idx = i;
            } else {
                pre[i + k - 1] = new int[] {t, idx};
            }
        }
        for (int i = n - k, t = 0, idx = 0; i >= 0; --i) {
            int cur = s[i + k] - s[i];
            if (cur >= t) {
                suf[i] = new int[] {cur, i};
                t = cur;
                idx = i;
            } else {
                suf[i] = new int[] {t, idx};
            }
        }
        int[] ans = new int[0];
        for (int i = k, t = 0; i < n - 2 * k + 1; ++i) {
            int cur = s[i + k] - s[i] + pre[i - 1][0] + suf[i + k][0];
            if (cur > t) {
                ans = new int[] {pre[i - 1][1], i, suf[i + k][1]};
                t = cur;
            }
        }
        return ans;
    }
}
class Solution {
public:
    vector<int> maxSumOfThreeSubarrays(vector<int>& nums, int k) {
        int n = nums.size();
        vector<int> s(n + 1, 0);
        for (int i = 0; i < n; ++i) {
            s[i + 1] = s[i] + nums[i];
        }

        vector<vector<int>> pre(n, vector<int>(2, 0));
        vector<vector<int>> suf(n, vector<int>(2, 0));

        for (int i = 0, t = 0, idx = 0; i < n - k + 1; ++i) {
            int cur = s[i + k] - s[i];
            if (cur > t) {
                pre[i + k - 1] = {cur, i};
                t = cur;
                idx = i;
            } else {
                pre[i + k - 1] = {t, idx};
            }
        }

        for (int i = n - k, t = 0, idx = 0; i >= 0; --i) {
            int cur = s[i + k] - s[i];
            if (cur >= t) {
                suf[i] = {cur, i};
                t = cur;
                idx = i;
            } else {
                suf[i] = {t, idx};
            }
        }

        vector<int> ans;
        for (int i = k, t = 0; i < n - 2 * k + 1; ++i) {
            int cur = s[i + k] - s[i] + pre[i - 1][0] + suf[i + k][0];
            if (cur > t) {
                ans = {pre[i - 1][1], i, suf[i + k][1]};
                t = cur;
            }
        }

        return ans;
    }
};
func maxSumOfThreeSubarrays(nums []int, k int) (ans []int) {
	n := len(nums)
	s := make([]int, n+1)
	for i := 0; i < n; i++ {
		s[i+1] = s[i] + nums[i]
	}

	pre := make([][]int, n)
	suf := make([][]int, n)

	for i, t, idx := 0, 0, 0; i < n-k+1; i++ {
		cur := s[i+k] - s[i]
		if cur > t {
			pre[i+k-1] = []int{cur, i}
			t, idx = cur, i
		} else {
			pre[i+k-1] = []int{t, idx}
		}
	}

	for i, t, idx := n-k, 0, 0; i >= 0; i-- {
		cur := s[i+k] - s[i]
		if cur >= t {
			suf[i] = []int{cur, i}
			t, idx = cur, i
		} else {
			suf[i] = []int{t, idx}
		}
	}

	for i, t := k, 0; i < n-2*k+1; i++ {
		cur := s[i+k] - s[i] + pre[i-1][0] + suf[i+k][0]
		if cur > t {
			ans = []int{pre[i-1][1], i, suf[i+k][1]}
			t = cur
		}
	}

	return
}