dynamic-programming

To identify overlapping subproblems and optimal substructure for dynamic programming, verify if the main optimization problem can be broken into smaller subproblems whose optimal solutions combine into the overall optimal solution, and check if these subproblems are solved repeatedly.

Use bottom-up tabulation in dynamic programming when you need strict control over the computation order and want to avoid recursion limits, whereas memoization is preferable when only a subset of subproblems actually needs evaluation during execution.

The standard dynamic programming workflow involves defining problem states, proving the optimal substructure, formulating the recurrence relation, computing results once using bottom-up or memoized approaches, and finally reconstructing the exact solution from the cached data.

To reconstruct the optimal solution after computing dynamic programming states, you store predecessor pointers or decisions alongside your tabulated values, then trace backwards from the final state to the base case to assemble the complete sequence.

Dynamic programming can be directly applied to matrix-chain multiplication and longest common subsequence problems, as both exhibit optimal substructure and overlapping subproblems suitable for caching intermediate results to avoid exponential blowups.