Dynamic Programming Problem

Start by clarifying the exact contract for Dynamic Programming Problem: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is validation of state sufficiency and reconstruction needs. Work a tiny example aloud and name the invariant after each step, because that is the proof that the approach stays correct as the input grows.

A dependable implementation plan is to get full constraints; write brute-force choices; merge equivalent subproblems; verify recurrence by hand; add parent choices if reconstruction is needed. Choose sentinels based on min max count or feasibility semantics. Keep the code aligned with that invariant instead of adding special cases that silently change the contract. Test the smallest input, a normal case, duplicates or ties where relevant, an adversarial boundary, and a case that exercises the failure path. The expected complexity is depends on the actual state graph. Explain both time and auxiliary memory, including recursion or helper structures; do not hide sorting, copying, or repeated scans. If the title omits critical constraints, say what you need to know before committing to a more specialized solution.

Complexity: State the recurrence first: a DP with n states and a candidate transitions per state costs O(na) time and O(n) space.