Dynamic Programming

Dynamic programming (DP) is a problem-solving technique used to optimize decision-making by breaking complex problems into simpler subproblems, solving each subproblem only once, and storing their solutions for reuse. It leverages the properties of optimal substructure (where the optimal solution to a problem can be constructed from optimal solutions of its subproblems) and overlapping subproblems (where the same subproblems are solved multiple times) to efficiently compute results. DP can be implemented using either memoization (top-down, recursive approach with caching) or tabulation (bottom-up, iterative approach with tables), reducing time complexity from exponential to polynomial in many cases. Widely applied in fields like computer science, mathematics, and bioinformatics, DP provides a structured way to tackle problems such as sequence alignment, shortest paths, and resource allocation.