Dynamic programming is a powerful algorithmic technique used to solve complex problems by breaking them down into simpler subproblems and storing the results of these subproblems to avoid redundant computations. This approach is particularly effective for problems that exhibit overlapping subproblems and optimal substructure properties. The key concepts in dynamic programming include: 1. **Overlapping Subproblems**: This occurs when a problem can be broken down into subproblems that are reused multiple times. Instead of solving the same subproblem repeatedly, dynamic programming stores the results in a table (often called a memoization table) and retrieves the result when needed. 2. **Optimal Substructure**: A problem exhibits optimal substructure if an optimal solution to the problem can be constructed from optimal solutions to its subproblems. This property allows dynamic programming to build up solutions to larger problems by combining solutions to smaller problems. There are two primary approaches to implementing dynamic programming: - **Top-Down Approach (Memoization)**: This approach involves solving the problem recursively while storing the results of subproblems in a cache. When the same subproblem is encountered again, the algorithm retrieves the stored result instead of recalculating it. - **Bottom-Up Approach (Tabulation)**: In this approach, the algorithm iteratively builds up solutions to subproblems, starting from the simplest ones and using them to solve progressively larger problems. This method usually involves filling out a table, where each entry represents the solution to a specific subproblem. Dynamic programming is widely used in various applications, including optimization problems, combinatorial problems, and algorithm design. Some classic examples include the Fibonacci sequence, the knapsack problem, and the longest common subsequence problem. Understanding dynamic programming is essential for tackling complex computational problems efficiently.