Find the K-Sum of an Array

Find the K-Sum of an Array

What is this problem about?

The Find the K-Sum of an Array interview question is an advanced optimization problem. You are given an array of integers and an integer k. You need to find the $k^{th}$ largest sum among all possible subsequences of the array. Since an array of size $N$ has $2^N$ subsequences, you cannot generate all of them for large $N$.

Why is this asked in interviews?

Amazon and Google use the Find the K-Sum of an Array coding problem to test a candidate's mastery of Heap (Priority Queue) interview patterns and the ability to transform a complex search space. It evaluations whether you can convert a problem involving negative numbers into a purely positive one and then use a greedy search to find the top $k$ sums.

Algorithmic pattern used

This problem relies on Greedy Search with a Min-Heap.

1. Transformation:

• Calculate the maximum possible sum ($MaxSum$) by adding all positive integers.
• Replace every element in the array with its absolute value abs(x).
• The problem now becomes finding the $k^{th}$ smallest "reduction" from $MaxSum$.

1. Heap Search:

• Sort the absolute values.
• Use a Min-Heap to store pairs (sum_reduction, index).
• Pop the smallest reduction, and push two new candidates: adding the next element or replacing the current element with the next one.

1. Result: The $k^{th}$ pop gives the $k^{th}$ smallest reduction. The final answer is MaxSum - reduction.

Example explanation

Array: [2, 4, -2], $k = 5$

1. $MaxSum = 2 + 4 = 6$.
2. Absolute values: [2, 4, 2]. Sorted: [2, 2, 4].
3. Smallest reductions:

• 0 (empty set)
• 2 (first 2)
• 2 (second 2)
• 4 (sum of 2+2)
• 4 (the 4)

1. $5^{th}$ smallest reduction is 4.
2. $5^{th}$ largest sum: $6 - 4 = 2$.

Common mistakes candidates make

• Exponential Search: Trying to use backtracking to find all sums ($O(2^N)$).
• Negative numbers: Failing to handle negative values correctly by not using the absolute value transformation.
• Heap logic: Errors in the state transitions within the priority queue.

Interview preparation tip

Learn how to use a Heap to explore "Next Best" states. This pattern is common in problems like "Kth Smallest Element in a Sorted Matrix." It turns an exponential search into a controlled $O(K \log K)$ exploration.