DP Related Questons

Start by clarifying the exact contract for DP Related Questions: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is proof-oriented dp construction despite an ambiguous label. 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 ask for inputs and objective; define state meaning; derive final-decision transition; choose invalid sentinel; test small cases. Zero is not universally the correct invalid-state sentinel. 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 actual state count and transitions. 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: Once the recurrence is known, n states with a transitions per state give O(na) time and O(n) space.