## 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).

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Intuition

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It is natural to consider an array W of each interval\'s sum, where each interval is the given length K. To create W, we can either use prefix sums, or manage the sum of the interval as a window slides along the array.

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From there, we approach the reduced problem: Given some array W and an integer K, what is the lexicographically smallest tuple of indices (i, j, k) with i + K <= j and j + K <= k that maximizes W[i] + W[j] + W[k]?

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Algorithm

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Suppose we fixed j. We would like to know on the intervals and , where the largest value of (and respectively ) occurs first. (Here, first means the smaller index.)

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We can solve these problems with dynamic programming. For example, if we know that is where the largest value of occurs first on , then on the first occurrence of the largest must be either or . If say, is better, then we set best = 6.

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At the end, left[z] will be the first occurrence of the largest value of W[i] on the interval , and right[z] will be the same but on the interval . This means that for some choice j, the candidate answer must be (left[j-K], j, right[j+K]). We take the candidate that produces the maximum W[i] + W[j] + W[k].

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Complexity Analysis

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Time Complexity: , where is the length of the array. Every loop is bounded in the number of steps by , and does work.

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Space complexity: . W, left, and right all take memory.

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Analysis written by: @awice

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