689. Maximum Sum of 3 Non-Overlapping Subarrays

In a given array nums of positive integers, find three non-overlapping subarrays with maximum sum.

Each subarray will be of size k, and we want to maximize the sum of all 3*k entries.

Return the result as a list of indices representing the starting position of each interval (0-indexed). If there are multiple answers, return the lexicographically smallest one.

Example:

Input: [1,2,1,2,6,7,5,1], 2
Output: [0, 3, 5]
Explanation: Subarrays [1, 2], [2, 6], [7, 5] correspond to the starting indices [0, 3, 5].
We could have also taken [2, 1], but an answer of [1, 3, 5] would be lexicographically larger.

Note:

• nums.length will be between 1 and 20000.
• nums[i] will be between 1 and 65535.
• k will be between 1 and floor(nums.length / 3).

Approach #1: DP. [C++]

class Solution {
public:
    vector<int> maxSumOfThreeSubarrays(vector<int>& nums, int k) {
        int len = nums.size();
        vector<int> sum = {0}, posLeft(len, 0), posRight(len, len-k);
        for (int i : nums) sum.push_back(sum.back()+i);
        for (int i = k, total = sum[k] - sum[0]; i < len; ++i) {
            if (sum[i+1] - sum[i+1-k] > total) {
                total = sum[i+1] - sum[i+1-k];
                posLeft[i] = i + 1 -k;
            } else 
                posLeft[i] = posLeft[i-1];
        }
        
        for (int i = len-k-1, total = sum[len] - sum[len-k]; i >= 0; --i) {
            if (sum[i+k] - sum[i] > total) {
                total = sum[i+k] - sum[i];
                posRight[i] = i;
            } else 
                posRight[i] = posRight[i+1];
        }
        
        int maxsum = 0;
        vector<int> ans;
        for (int i = k; i <= len-2*k; ++i) {
            int l = posLeft[i-1], r = posRight[i+k];
            int tot = (sum[i+k] - sum[i]) + (sum[l+k] - sum[l]) + (sum[r+k] - sum[r]);
            if (tot > maxsum) {
                maxsum = tot;
                ans = {l, i, r};
            }
        }
        
        return ans;
    }
};

　　

Analysis:

The question asks for three non-overlapping intervals with maximum sum of all 3 intervals. If the middle interval is [i, i+k-1], where k <= i <= n-2k, the left intervals. If the middle interval is [i, i+k-1], where k <= i <= n - 2k, the left interval has to be in subrange [0, i-1], ans the right interval is from subrange [i+k, n-1].

So the following solution is based on DP.

posLeft[i] is the starting index for the left interval in range [0, i];

posRight[i] is the strating index for the right interval in range [i, n-1];

Then we test every possible strating index og middle interval, i.e. k <= i <= n-2k, ans we can get the corresponding left and right max sum intervals easily from DP. and the run time is O(n).

Caution. In order to get lexicgraphical smallest order, when there are tow intervals with equal max sum, always select the left most one. So in the code. the is condition is ">=" for right interval due to backward searching, and ">" for left interval. 

Reference:

https://leetcode.com/problems/maximum-sum-of-3-non-overlapping-subarrays/discuss/108231/C%2B%2BJava-DP-with-explanation-O(n)