Kth Smallest Subarray Sum

Kth Smallest Subarray Sum

1. What is this problem about?

The Kth Smallest Subarray Sum interview question is a sophisticated array manipulation challenge that requires finding a specific value from a virtual collection of sums. Given an array of positive integers and an integer k, the goal is to identify the $k$-th smallest value among all possible subarray sums. A subarray is a contiguous part of an array. For an array of size $n$, there are $n(n+1)/2$ such subarrays. Instead of calculating and sorting all these sums—which would be computationally expensive—the problem asks for a more optimized approach to pinpoint the exact sum that would occupy the $k$-th position in a sorted list of all subarray sums.

2. Why is this asked in interviews?

This problem is a favorite at top-tier tech companies like Google because it tests a candidate's ability to combine multiple algorithmic techniques. It moves beyond simple array traversal and requires a deep understanding of how to optimize search spaces. Interviewers use the Kth Smallest Subarray Sum coding problem to evaluate if a candidate can recognize when a problem can be framed as a search over a range of values rather than a search over the input elements themselves. It also assesses proficiency in handling "counting" subproblems efficiently.

3. Algorithmic pattern used

The most effective Array, Sliding Window, Binary Search interview pattern for this problem involves Binary Search on the Answer. Since all elements are positive, the minimum possible subarray sum is the minimum element (or 0), and the maximum is the sum of the entire array. By performing a binary search on this range [0, total_sum], we can check how many subarrays have a sum less than or equal to a midpoint M. To count these subarrays in linear time, we use the Sliding Window (or Two Pointers) technique. If the count is at least k, then the $k$-th smallest sum might be M or smaller; otherwise, it must be larger.

4. Example explanation

Consider an array [1, 2, 3] and k = 3. The possible subarrays are: [1], [2], [3], [1,2], [2,3], [1,2,3]. Their sums are: 1, 2, 3, 3, 5, 6. Sorted sums: 1, 2, 3, 3, 5, 6. The 3rd smallest sum is 3. Using binary search, we might check if the sum is 3. The sliding window counts subarrays with sum $\le 3$:

• [1] (sum 1)
• [1, 2] (sum 3)
• [2] (sum 2)
• [3] (sum 3) Total count is 4. Since $4 \ge k$, we know the answer is $\le 3$.

5. Common mistakes candidates make

• Naive Brute Force: Attempting to generate all $O(n^2)$ subarray sums, storing them in a list, and sorting them. This results in $O(n^2 \log n)$ time and $O(n^2)$ space, which fails for large arrays.
• Incorrect Binary Search Range: Setting the upper bound of the binary search to the maximum element instead of the sum of all elements.
• Off-by-one errors: Mismanaging the sliding window pointers or the binary search boundary updates (low = mid + 1 vs high = mid).
• Ignoring constraints: Failing to realize that the array contains only positive integers, which is crucial for the sliding window logic to work correctly.

6. Interview preparation tip

When you see a problem asking for the "Kth something," always consider if you can "Binary Search on the result." If you can efficiently count how many items are less than or equal to a value $X$, then binary search is likely the way to go. Practice similar problems like "Kth Smallest Element in a Sorted Matrix" to master this powerful paradigm.