Decode Ways

Start by clarifying the exact contract for Decode Ways: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is prefix dp with careful zero handling. 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 define dp for decoded prefixes; add one-digit transition for 1..9; add pair transition for 10..26; compress to two prior counts. 0 is invalid alone but valid in 10 and 20. 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(n) time and O(1) extra 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.