Minimum Size Subarray in Infinite Array

Minimum Size Subarray in Infinite Array

What is this problem about?

The Minimum Size Subarray in Infinite Array problem extends the classic subarray sum problem to an infinite context. You are given an array that conceptually repeats infinitely, and a target sum. Find the minimum length subarray (within this infinite repetition) that sums to the target. This Minimum Size Subarray in Infinite Array coding problem cleverly combines modular arithmetic with the sliding window and prefix sum interview pattern.

Why is this asked in interviews?

DE Shaw asks this medium problem to test mathematical extension of known algorithms. It requires recognizing that working within the infinite repetition can be reduced to working within at most 2 full copies of the array, plus prefix sum modular reasoning. The array, hash table, sliding window, and prefix sum interview pattern is directly demonstrated.

Algorithmic pattern used

The strategy has two steps:

1. Find the minimum subarray summing to target % totalSum using a prefix sum + hash map on the array.
2. Determine how many full copies of the array are needed: (target / totalSum) * n positions.
3. Combine: minimum length = length from step 1 + full array repetitions × n.

If target % totalSum == 0, only full-copy repetitions are needed. If totalSum doesn't divide target evenly into a reachable sum, return -1.

Example explanation

Array: [1, 2, 3], target = 5. TotalSum = 6.

• target % totalSum = 5. Find min subarray summing to 5 in [1,2,3,1,2,3]: [2,3]=5, length 2.
• Full copies needed: 5/6 = 0. Length contribution = 0.
• Answer = 2.

Target = 7: 7 % 6 = 1. Min subarray summing to 1: [1], length 1. Full copies = 1 (adds 3). Total = 1 + 3 = 4.

Common mistakes candidates make

• Not handling the case where target % totalSum == 0 separately.
• Forgetting to search across at least 2 copies of the array for the remainder subarray.
• Returning -1 prematurely when target > totalSum but is still achievable.
• Off-by-one in counting repetitions needed.

Interview preparation tip

Problems involving "infinite" or "circular" arrays almost always reduce to working on 1 or 2 copies. Recognizing the modular structure is the critical first step. Practice prefix sum with hash maps for minimum-length subarray problems, then extend to circular/infinite variants. The combination of modular reasoning and sliding windows is a high-value pattern that appears in several medium-to-hard problems involving repeated structures.