Find Sum of Array Product of Magical Sequences - Video Solutions

Problem Overview: You are given constraints that define “magical sequences” and must compute the total sum of products of arrays derived from those sequences. Direct enumeration quickly explodes because the number of valid sequences grows exponentially, so the challenge is counting contributions without constructing every sequence explicitly.

Approach 1: Brute Force Enumeration (Exponential Time, O(k^n) time, O(n) space)

The most direct idea is to generate every valid magical sequence, build the corresponding array, compute its product, and accumulate the sum. This requires recursive backtracking that tries all allowed values at each position while validating the sequence constraints. Although simple to reason about, the number of candidate sequences grows exponentially with the sequence length. Even moderate input sizes make this infeasible. The brute force approach is mainly useful for understanding the structure of the problem or validating optimized solutions on small inputs.

Approach 2: Dynamic Programming with Bitmask State (O(n · 2^m) time, O(2^m) space)

A better strategy is to represent the state of a partially built sequence using a bitmask. Each bit encodes which elements or constraints have already been used. Instead of recomputing the same partial configurations repeatedly, store intermediate results in a DP table keyed by the mask and position. For each transition, update the cumulative product contribution and propagate it to the next valid state. This technique converts repeated exponential exploration into a manageable state-space traversal. Problems involving subsets and constraints often benefit from bitmask based DP combined with memoization.

Approach 3: Combinatorics + Memoized Search (O(states) time, O(states) space)

The optimal approach observes that many sequences contribute identical multiplicative patterns. Instead of enumerating each sequence individually, count how many sequences produce the same structural configuration and multiply that count by the resulting product contribution. Use combinatorial formulas to determine how many ways a configuration can be formed, then perform a memoized DFS over valid states. Each state stores the number of ways and the accumulated product contribution. Memoization avoids recomputation, while combinatorics collapses large groups of equivalent sequences into a single calculation. This hybrid technique blends combinatorics, dynamic programming, and bit-level state encoding to scale to the problem’s limits.

Recommended for interviews: Start by explaining the brute force enumeration to show you understand the definition of magical sequences. Then move quickly to the combinatorics + memoized search approach. Interviewers expect you to recognize the exponential explosion and reduce it using counting techniques and cached subproblems. Demonstrating the transition from naive enumeration to state compression with combinatorics shows strong problem-solving maturity.