Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It stores the results of these subproblems to avoid redundant calculations.

Dynamic programming is a powerful algorithmic technique used to solve complex problems by breaking them down into simpler subproblems. It is particularly useful for optimization problems where the solution can be constructed efficiently from solutions to smaller instances of the same problem. The core idea behind dynamic programming is to store the results of subproblems in a table (usually an array or a matrix) to avoid redundant calculations, thus improving efficiency. This approach is particularly effective for problems that exhibit overlapping subproblems and optimal substructure properties. Overlapping subproblems mean that the same subproblems are solved multiple times, while optimal substructure indicates that an optimal solution to the problem can be constructed from optimal solutions of its subproblems. Classic examples of problems that can be solved using dynamic programming include the Fibonacci sequence, the knapsack problem, and the longest common subsequence problem. For instance, calculating Fibonacci numbers naively with recursion can lead to exponential time complexity, as the same values are computed repeatedly. However, by using dynamic programming, you can compute the Fibonacci sequence in linear time by storing previously computed values. Similarly, the knapsack problem can be solved efficiently using dynamic programming by creating a table to store solutions for smaller capacities and weights, allowing you to build up to the optimal solution. Dynamic programming can be implemented in two primary ways: top-down (memoization) and bottom-up (tabulation). In the top-down approach, you recursively solve the problem while storing results, while in the bottom-up approach, you iteratively build the solution by filling up a table. Mastering dynamic programming can significantly enhance your problem-solving skills and prepare you for competitive programming and technical interviews, where these concepts are often tested.