One medium-hard DP problem

Derive DP, do not guess it

The phrase does not provide a recurrence. Ask what decisions are made, whether the answer is a count, optimum, or feasibility result, and which input limits rule out brute force. Define dp[...] in a full sentence so every dimension has a purpose. List legal transitions, explain why they cover all possibilities without overlap, and set base cases before choosing iteration order.

Start with the complete table and calculate time as the number of states times transitions per state. If a state reads only a prior row or small neighborhood, compress space afterward while preserving the dependency order. Test smallest inputs, unreachable states, repeated values, negative values if allowed, and modulo behavior for counting. A generic answer cannot honestly promise a particular Big-O until the actual problem is known; for example, a two-dimensional DP is often O(n*m) time and space, but only if those are real state dimensions. Ask for the exact prompt rather than inventing one.