Split Array with Equal Sum

Split Array with Equal Sum

What is this problem about?

The Split Array with Equal Sum coding problem asks you to determine if a given array can be split into four contiguous subarrays by removing three elements at specific indices ($i, j, k$). The four resulting parts must all have the same sum. Specifically, if you remove elements at $i, j,$ and $k$ (where $0 < i, i+1 < j, j+1 < k < n-1$), you get four subarrays: [0, $i-1$], [$i+1, j-1$], [$j+1, k-1$], and [$k+1, n-1$]. The goal is to find if such indices exist.

Why is this asked in interviews?

This problem is asked at companies like Alibaba to test a candidate's ability to optimize a brute-force approach. A naive $O(N^3)$ or $O(N^4)$ search for the three indices will be too slow for large arrays. The problem requires clever use of prefix sums and hash tables to reduce the time complexity to $O(N^2)$, making it a great test of Array, Hash Table, Prefix Sum interview pattern knowledge.

Algorithmic pattern used

The core pattern is Prefix Sums for fast range sum calculation, combined with a Hash Table to store intermediate results. By fixing the middle index $j$, the problem effectively splits into two similar halves. For a fixed $j$, you find all possible $i$ that split the first half into two equal sums and store those sums in a set. Then you find all possible $k$ that split the second half into two equal sums. If any sum found in the second half exists in the set from the first half, you've found a valid split.

Example explanation (use your own example)

Consider the array [1, 2, 1, 2, 1, 2, 1]. Let's try to remove three elements. If we pick $i=2$ (value 1), $j=4$ (value 1), and $k=6$ (value 1), we are left with the elements at indices: [0, 1] which is (1, 2), [3] which is (2), [5] which is (2), and... wait, that doesn't work. Let's try again. If we remove elements such that we get four parts: [1, 2], [1, 2], [1, 2], and [1, 2], each sum is 3. The elements removed would be the separators between these groups. Finding these separators efficiently is the crux of the problem.

Common mistakes candidates make

• Inefficient sum calculation: Recomputing sums of subarrays repeatedly instead of using prefix sums.
• Missing index constraints: Ignoring the requirement that there must be at least one element between the removed indices.
• Poor complexity: Attempting to solve it with three nested loops instead of splitting the logic around the middle index.
• Empty subarrays: Not ensuring that the resulting subarrays are non-empty as implied by the index constraints.

Interview preparation tip

When dealing with "split into equal parts" problems, always look for ways to divide the problem. Fixing a "pivot" point (like the middle index in this case) and solving the left and right sides independently is a common technique to turn an $O(N^3)$ problem into an $O(N^2)$ one.