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1191. K-Concatenation Maximum Sum

Description

Given an integer array arr and an integer k, modify the array by repeating it k times.

For example, if arr = [1, 2] and k = 3 then the modified array will be [1, 2, 1, 2, 1, 2].

Return the maximum sub-array sum in the modified array. Note that the length of the sub-array can be 0 and its sum in that case is 0.

As the answer can be very large, return the answer modulo 109 + 7.

 

Example 1:

Input: arr = [1,2], k = 3
Output: 9

Example 2:

Input: arr = [1,-2,1], k = 5
Output: 2

Example 3:

Input: arr = [-1,-2], k = 7
Output: 0

 

Constraints:

  • 1 <= arr.length <= 105
  • 1 <= k <= 105
  • -104 <= arr[i] <= 104

Solutions

Solution 1: Prefix Sum + Case Discussion

We denote the sum of all elements in the array $arr$ as $s$, the maximum prefix sum as $mxPre$, the minimum prefix sum as $miPre$, and the maximum subarray sum as $mxSub$.

We traverse the array $arr$. For each element $x$, we update $s = s + x$, $mxPre = \max(mxPre, s)$, $miPre = \min(miPre, s)$, $mxSub = \max(mxSub, s - miPre)$.

Next, we consider the value of $k$:

  • When $k = 1$, the answer is $mxSub$.
  • When $k \ge 2$, if the maximum subarray spans two $arr$, then the answer is $mxPre + mxSuf$, where $mxSuf = s - miPre$.
  • When $k \ge 2$ and $s > 0$, if the maximum subarray spans three $arr$, then the answer is $(k - 2) \times s + mxPre + mxSuf$.

Finally, we return the result of the answer modulo $10^9 + 7$.

The time complexity is $O(n)$, and the space complexity is $O(1)$. Here, $n$ is the length of the array $arr$.

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class Solution:
    def kConcatenationMaxSum(self, arr: List[int], k: int) -> int:
        s = mx_pre = mi_pre = mx_sub = 0
        for x in arr:
            s += x
            mx_pre = max(mx_pre, s)
            mi_pre = min(mi_pre, s)
            mx_sub = max(mx_sub, s - mi_pre)
        ans = mx_sub
        mod = 10**9 + 7
        if k == 1:
            return ans % mod
        mx_suf = s - mi_pre
        ans = max(ans, mx_pre + mx_suf)
        if s > 0:
            ans = max(ans, (k - 2) * s + mx_pre + mx_suf)
        return ans % mod

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class Solution {
    public int kConcatenationMaxSum(int[] arr, int k) {
        long s = 0, mxPre = 0, miPre = 0, mxSub = 0;
        for (int x : arr) {
            s += x;
            mxPre = Math.max(mxPre, s);
            miPre = Math.min(miPre, s);
            mxSub = Math.max(mxSub, s - miPre);
        }
        long ans = mxSub;
        final int mod = (int) 1e9 + 7;
        if (k == 1) {
            return (int) (ans % mod);
        }
        long mxSuf = s - miPre;
        ans = Math.max(ans, mxPre + mxSuf);
        if (s > 0) {
            ans = Math.max(ans, (k - 2) * s + mxPre + mxSuf);
        }
        return (int) (ans % mod);
    }
}

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class Solution {
public:
    int kConcatenationMaxSum(vector<int>& arr, int k) {
        long s = 0, mxPre = 0, miPre = 0, mxSub = 0;
        for (int x : arr) {
            s += x;
            mxPre = max(mxPre, s);
            miPre = min(miPre, s);
            mxSub = max(mxSub, s - miPre);
        }
        long ans = mxSub;
        const int mod = 1e9 + 7;
        if (k == 1) {
            return ans % mod;
        }
        long mxSuf = s - miPre;
        ans = max(ans, mxPre + mxSuf);
        if (s > 0) {
            ans = max(ans, mxPre + (k - 2) * s + mxSuf);
        }
        return ans % mod;
    }
};

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func kConcatenationMaxSum(arr []int, k int) int {
    var s, mxPre, miPre, mxSub int
    for _, x := range arr {
        s += x
        mxPre = max(mxPre, s)
        miPre = min(miPre, s)
        mxSub = max(mxSub, s-miPre)
    }
    const mod = 1e9 + 7
    ans := mxSub
    if k == 1 {
        return ans % mod
    }
    mxSuf := s - miPre
    ans = max(ans, mxSuf+mxPre)
    if s > 0 {
        ans = max(ans, mxSuf+(k-2)*s+mxPre)
    }
    return ans % mod
}

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