dp based question

Start by clarifying the exact contract for DP-Based Question: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is responsible clarification plus a reusable state-derivation process. 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 request objective and bounds; model brute-force choices; memoize sufficient state; establish sentinels; select safe iteration order. Do not invent a knapsack recurrence from an unspecified title. 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 omitted state dimensions. 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: For n dynamic-programming states with constant transitions, use O(n) time and O(n) space, reducing space only after proving prior layers are sufficient.