Closest Subsequence Sum

Closest Subsequence Sum

What is this problem about?

The Closest Subsequence Sum interview question is a high-difficulty variation of the subset-sum problem. You are given an array of integers and a target value. You need to find the sum of any subsequence that is closest to the target. Because the array can have up to 40 elements, a simple 2^40 recursion is impossible (1 trillion operations).

Why is this asked in interviews?

Google and Sprinklr ask the Closest Subsequence Sum coding problem to test your knowledge of the Meet-in-the-Middle algorithm. This is a specialized optimization for problems where the search space is exactly 2^N. It demonstrates a candidate's ability to break a large, unsolvable problem into two smaller, solvable ones.

Algorithmic pattern used

The Meet-in-the-Middle pattern:

1. Split the array into two halves (20 elements each).
2. Generate all possible subsequence sums for each half (2^20 \approx 1 million sums each).
3. Sort the sums of the second half.
4. For each sum S1 in the first half, we need S2 from the second half such that S1 + S2 is closest to target.
5. Use Binary Search to find the ideal S2 in the sorted list of sums from the second half.

Example explanation

Array: [1, 3, 5, 7], target: 10

1. Split: Left = [1, 3], Right = [5, 7].
2. Left Sums: {0, 1, 3, 4}.
3. Right Sums: {0, 5, 7, 12}.
4. For L=3, we want R close to 10 - 3 = 7.
5. Binary search in {0, 5, 7, 12} finds 7.
6. 3 + 7 = 10. Difference is 0.

Common mistakes candidates make

• Brute Force: Trying to use backtracking on all 40 elements, which will time out.
• Standard DP: Trying to use a boolean DP array. If the target or elements are very large (or negative), the DP array size will be massive.
• Memory Management: Not realizing that 2^20 integers take about 4MB, which is fine, but 2^40 would be petabytes.

Interview preparation tip

Whenever you see a problem with a constraint around N=40 and a search space of 2^N, "Meet-in-the-Middle" is almost certainly the intended solution.