Maximum Product of Subsequences With an Alternating Sum Equal to K - Solution & Explanation

Problem Overview: You are given an array and a target K. Choose a subsequence whose alternating sum (add the first element, subtract the second, add the third, and so on) equals K. Among all valid subsequences, return the maximum possible product of its elements.

Approach 1: Brute Force Subsequence Enumeration (O(2^n) time, O(n) space)

The direct method generates every subsequence and computes its alternating sum and product. For each subset, iterate through elements in order, flipping the sign based on index parity (+ - + -). If the final alternating sum equals K, update the maximum product. This approach uses recursion or bitmask enumeration and requires storing the current product and alternating sum during traversal. With 2^n possible subsequences, the method becomes impractical once n grows beyond small limits.

Approach 2: Dynamic Programming with Hash Map States (O(n * S) time, O(S) space)

A more practical solution tracks reachable alternating sums while scanning the array once. Maintain DP states keyed by (alternating_sum, parity), where parity indicates whether the next element contributes positively or negatively. Each state stores the maximum product achievable for that configuration. For every number, iterate through existing states and update two possibilities: include the element (update sum based on parity and multiply the product) or skip it. A hash map efficiently stores only reachable sums, which avoids scanning large unused ranges. This technique combines ideas from dynamic programming and hash tables while iterating through the array.

The key insight is that the alternating sign depends only on how many elements were already chosen. Tracking parity makes the alternating calculation constant time, and storing the best product per state avoids recomputing dominated results.

Approach 3: Pruned DP with State Compression (O(n * S) time, O(S) space)

The DP map can grow if many sums are reachable. You can prune dominated states by keeping only the maximum product for each (sum, parity). Any state producing a smaller product for the same configuration can be discarded because future multiplications will never outperform the larger one. This keeps the state space compact and improves practical performance while maintaining the same asymptotic complexity.

Recommended for interviews: Interviewers expect the dynamic programming approach with a hash map. Starting with the brute force method shows you understand the subsequence search space. Transitioning to DP demonstrates the key optimization: storing intermediate alternating sums and reusing them instead of recomputing all subsequences.