Maximum Sum of 3 Non-Overlapping Subarrays

Maximum Sum of 3 Non-Overlapping Subarrays

1. What is this problem about?

The Maximum Sum of 3 Non-Overlapping Subarrays problem asks you to find three contiguous subarrays, each of a fixed length k, such that they do not overlap and their total sum is maximized. You need to return the starting indices of these three subarrays. For example, if you have an array and k=2, you are looking for three distinct blocks of two elements that together give you the biggest possible value.

2. Why is this asked in interviews?

This "Maximum Sum of 3 Non-Overlapping Subarrays interview question" is a classic "Hard" problem found in Meta, Microsoft, and Google interviews. It tests a candidate's ability to break down a complex global optimization problem into smaller, manageable parts using dynamic programming or pre-computation. It evaluates how well you can handle multiple dependencies and maintain optimal "left" and "right" states while iterating through the middle possibilities.

3. Algorithmic pattern used

The "Array, Sliding Window, Dynamic Programming, Prefix Sum interview pattern" involves several steps:

1. Use a sliding window (or prefix sums) to calculate the sum of every possible subarray of length k.
2. Pre-compute a left array where left[i] is the index of the best subarray of length k in the range [0, i].
3. Pre-compute a right array where right[i] is the index of the best subarray of length k in the range [i, end].
4. Iterate through every possible "middle" subarray starting at index j. The total sum is then sum(left_block) + sum(middle_block) + sum(right_block). We track the maximum of this total sum.

4. Example explanation

Array: [1, 2, 1, 2, 6, 7, 5, 1], k = 2.

• Possible sums of length 2: [3, 3, 3, 8, 13, 12, 6].
• If we fix the middle subarray as [6, 7] (sum 13), we look to the left (elements before [6, 7]) for the best subarray of length 2, and to the right for the best one there.
• Left of [6, 7] is [1, 2, 1, 2]. Best sum of length 2 is 3.
• Right of [6, 7] is [5, 1]. Best sum of length 2 is 6.
• Total sum = 3 + 13 + 6 = 22. The algorithm repeats this for all valid middle positions.

5. Common mistakes candidates make

One frequent error in the "Maximum Sum of 3 Non-Overlapping Subarrays coding problem" is having "off-by-one" errors in the indexing, especially since we have to ensure there is enough space for three subarrays of length k. Another mistake is failing to store the index of the best subarray in the pre-computed left and right arrays, returning only the sum instead. If multiple sets of indices give the same maximum sum, the problem usually requires the lexicographically smallest set, which must be handled with careful comparison logic.

6. Interview preparation tip

Practice the concept of "splitting the problem at the middle." This technique of pre-calculating optimal values from the left and right side is a standard way to solve many problems involving three parts or two parts with a gap. Once you master this "Left-Middle-Right" strategy, you'll find that many complex-looking array problems become much more approachable.