Constrained Subsequence Sum

Constrained Subsequence Sum

What is this problem about?

The Constrained Subsequence Sum interview question is an optimization challenge that combines dynamic programming with interval constraints. You are given an integer array and an integer k. You need to find the maximum sum of a non-empty subsequence such that for every two consecutive elements in the subsequence, their indices in the original array differ by at most k.

Why is this asked in interviews?

Tech giants like Google and trading firms like Akuna Capital ask the Constrained Subsequence Sum coding problem to see if a candidate can optimize a classic O(N * K) dynamic programming solution down to O(N). It tests knowledge of advanced data structures like the Monotonic Queue and the ability to maintain sliding window maximums.

Algorithmic pattern used

This utilizes the Array, Monotonic Queue, Sliding Window, Dynamic Programming interview pattern. The standard DP approach defines dp[i] as the maximum sum ending at index i. dp[i] = nums[i] + max(0, dp[i-k ... i-1]). To find the maximum in the range [i-k, i-1] efficiently, we use a Monotonic Deque that keeps indices of the dp table in descending order of their values.

Example explanation

Array: [10, 2, -10, 5, 20], k = 2.

1. dp[0] = 10. Deque: [0].
2. dp[1] = 2 + 10 = 12. Deque: [1, 0].
3. dp[2] = -10 + 12 = 2. Deque: [1, 2]. (0 is removed as it is > k steps away).
4. dp[3] = 5 + 12 = 17. Deque: [3, 1]. (17 is bigger than 2, so 2 is popped). Final max is the maximum value in the dp array.

Common mistakes candidates make

• Using a standard Max-Heap: While O(N log N) with a heap is better than O(N * K), it is still not as optimal as the O(N) deque solution.
• Forgetting the "non-empty" rule: Not correctly handling cases where all numbers are negative.
• Off-by-one errors: Mismanaging the sliding window boundaries, leading to accessing values further than k indices away.

Interview preparation tip

Whenever you need to find the maximum or minimum value in a sliding window of size k while iterating through an array, the Monotonic Queue should be your first thought.