Dynamic Programming Question on Grids

Start by clarifying the exact contract for Grid Dynamic Programming: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is coordinate state and predecessor-aware iteration. 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 clarify moves and obstacles; define dp at each cell; initialize boundaries; process after predecessors; compress rows when dependency permits. Upward moves or cycles can invalidate a one-pass recurrence. 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 O(mn) time and O(n) or O(mn) space. 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.