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3077. Maximum Strength of K Disjoint Subarrays

Description

You are given an array of integers nums with length n, and a positive odd integer k.

Select exactly k disjoint subarrays sub1, sub2, ..., subk from nums such that the last element of subi appears before the first element of sub{i+1} for all 1 <= i <= k-1. The goal is to maximize their combined strength.

The strength of the selected subarrays is defined as:

strength = k * sum(sub1)- (k - 1) * sum(sub2) + (k - 2) * sum(sub3) - ... - 2 * sum(sub{k-1}) + sum(subk)

where sum(subi) is the sum of the elements in the i-th subarray.

Return the maximum possible strength that can be obtained from selecting exactly k disjoint subarrays from nums.

Note that the chosen subarrays don't need to cover the entire array.

 

Example 1:

Input: nums = [1,2,3,-1,2], k = 3

Output: 22

Explanation:

The best possible way to select 3 subarrays is: nums[0..2], nums[3..3], and nums[4..4]. The strength is calculated as follows:

strength = 3 * (1 + 2 + 3) - 2 * (-1) + 2 = 22

 

Example 2:

Input: nums = [12,-2,-2,-2,-2], k = 5

Output: 64

Explanation:

The only possible way to select 5 disjoint subarrays is: nums[0..0], nums[1..1], nums[2..2], nums[3..3], and nums[4..4]. The strength is calculated as follows:

strength = 5 * 12 - 4 * (-2) + 3 * (-2) - 2 * (-2) + (-2) = 64

Example 3:

Input: nums = [-1,-2,-3], k = 1

Output: -1

Explanation:

The best possible way to select 1 subarray is: nums[0..0]. The strength is -1.

 

Constraints:

  • 1 <= n <= 104
  • -109 <= nums[i] <= 109
  • 1 <= k <= n
  • 1 <= n * k <= 106
  • k is odd.

Solutions

Solution 1: Dynamic Programming

For the $i$th number $nums[i - 1]$, if it is selected and is in the $j$th subarray, then its contribution to the answer is $nums[i - 1] \times (k - j + 1) \times (-1)^{j+1}$. We denote $(-1)^{j+1}$ as $sign$, so its contribution to the answer is $sign \times nums[i - 1] \times (k - j + 1)$.

We define $f[i][j][0]$ as the maximum energy value when selecting $j$ subarrays from the first $i$ numbers, and the $i$th number is not selected. We define $f[i][j][1]$ as the maximum energy value when selecting $j$ subarrays from the first $i$ numbers, and the $i$th number is selected. Initially, $f[0][0][1] = 0$, and the rest of the values are $-\infty$.

When $i > 0$, we consider how $f[i][j]$ transitions.

If the $i$th number is not selected, then the $i-1$th number can either be selected or not selected, so $f[i][j][0] = \max(f[i-1][j][0], f[i-1][j][1])$.

If the $i$th number is selected, if the $i-1$th number and the $i$th number are in the same subarray, then $f[i][j][1] = \max(f[i][j][1], f[i-1][j][1] + sign \times nums[i-1] \times (k - j + 1))$, otherwise $f[i][j][1] = \max(f[i][j][1], \max(f[i-1][j-1][0], f[i-1][j-1][1]) + sign \times nums[i-1] \times (k - j + 1))$.

The final answer is $\max(f[n][k][0], f[n][k][1])$.

The time complexity is $O(n \times k)$, and the space complexity is $O(n \times k)$. Where $n$ is the length of the array $nums$.

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class Solution:
    def maximumStrength(self, nums: List[int], k: int) -> int:
        n = len(nums)
        f = [[[-inf, -inf] for _ in range(k + 1)] for _ in range(n + 1)]
        f[0][0][0] = 0
        for i, x in enumerate(nums, 1):
            for j in range(k + 1):
                sign = 1 if j & 1 else -1
                f[i][j][0] = max(f[i - 1][j][0], f[i - 1][j][1])
                f[i][j][1] = max(f[i][j][1], f[i - 1][j][1] + sign * x * (k - j + 1))
                if j:
                    f[i][j][1] = max(
                        f[i][j][1], max(f[i - 1][j - 1]) + sign * x * (k - j + 1)
                    )
        return max(f[n][k])

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class Solution {
    public long maximumStrength(int[] nums, int k) {
        int n = nums.length;
        long[][][] f = new long[n + 1][k + 1][2];
        for (int i = 0; i <= n; i++) {
            for (int j = 0; j <= k; j++) {
                Arrays.fill(f[i][j], Long.MIN_VALUE / 2);
            }
        }
        f[0][0][0] = 0;
        for (int i = 1; i <= n; i++) {
            int x = nums[i - 1];
            for (int j = 0; j <= k; j++) {
                long sign = (j & 1) == 1 ? 1 : -1;
                long val = sign * x * (k - j + 1);
                f[i][j][0] = Math.max(f[i - 1][j][0], f[i - 1][j][1]);
                f[i][j][1] = Math.max(f[i][j][1], f[i - 1][j][1] + val);
                if (j > 0) {
                    long t = Math.max(f[i - 1][j - 1][0], f[i - 1][j - 1][1]) + val;
                    f[i][j][1] = Math.max(f[i][j][1], t);
                }
            }
        }
        return Math.max(f[n][k][0], f[n][k][1]);
    }
}

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class Solution {
public:
    long long maximumStrength(vector<int>& nums, int k) {
        int n = nums.size();
        long long f[n + 1][k + 1][2];
        memset(f, -0x3f3f3f3f3f3f3f3f, sizeof(f));
        f[0][0][0] = 0;
        for (int i = 1; i <= n; i++) {
            int x = nums[i - 1];
            for (int j = 0; j <= k; j++) {
                long long sign = (j & 1) == 1 ? 1 : -1;
                long long val = sign * x * (k - j + 1);
                f[i][j][0] = max(f[i - 1][j][0], f[i - 1][j][1]);
                f[i][j][1] = max(f[i][j][1], f[i - 1][j][1] + val);
                if (j > 0) {
                    long long t = max(f[i - 1][j - 1][0], f[i - 1][j - 1][1]) + val;
                    f[i][j][1] = max(f[i][j][1], t);
                }
            }
        }
        return max(f[n][k][0], f[n][k][1]);
    }
};

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func maximumStrength(nums []int, k int) int64 {
    n := len(nums)
    f := make([][][]int64, n+1)
    const inf int64 = math.MinInt64 / 2
    for i := range f {
        f[i] = make([][]int64, k+1)
        for j := range f[i] {
            f[i][j] = []int64{inf, inf}
        }
    }
    f[0][0][0] = 0
    for i := 1; i <= n; i++ {
        x := nums[i-1]
        for j := 0; j <= k; j++ {
            sign := int64(-1)
            if j&1 == 1 {
                sign = 1
            }
            val := sign * int64(x) * int64(k-j+1)
            f[i][j][0] = max(f[i-1][j][0], f[i-1][j][1])
            f[i][j][1] = max(f[i][j][1], f[i-1][j][1]+val)
            if j > 0 {
                t := max(f[i-1][j-1][0], f[i-1][j-1][1]) + val
                f[i][j][1] = max(f[i][j][1], t)
            }
        }
    }
    return max(f[n][k][0], f[n][k][1])
}

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function maximumStrength(nums: number[], k: number): number {
    const n: number = nums.length;
    const f: number[][][] = Array.from({ length: n + 1 }, () =>
        Array.from({ length: k + 1 }, () => [-Infinity, -Infinity]),
    );
    f[0][0][0] = 0;
    for (let i = 1; i <= n; i++) {
        const x: number = nums[i - 1];
        for (let j = 0; j <= k; j++) {
            const sign: number = (j & 1) === 1 ? 1 : -1;
            const val: number = sign * x * (k - j + 1);
            f[i][j][0] = Math.max(f[i - 1][j][0], f[i - 1][j][1]);
            f[i][j][1] = Math.max(f[i][j][1], f[i - 1][j][1] + val);
            if (j > 0) {
                f[i][j][1] = Math.max(f[i][j][1], Math.max(...f[i - 1][j - 1]) + val);
            }
        }
    }
    return Math.max(...f[n][k]);
}

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