Dynamic Programming

Start by clarifying the exact contract for Dynamic Programming: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is deriving state transitions and proving dependency order. 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 identify repeated subproblems; define minimal sufficient state; set base and invalid states; choose memoization or bottom-up; compress only after proof. Recursion is not DP unless repeated states are cached. 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 time is states times 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: For a specified DP with n states and constant fan-out, the baseline is O(n) time and O(n) space.