What does "dynamic programming" refer to?

Dynamic programming is a method used to solve complex problems by breaking them down into simpler, overlapping subproblems, which are then solved just once and stored for future reference. This approach is particularly effective for optimization problems, where the solution can be constructed efficiently from previously computed solutions to smaller instances of the same problem.

In dynamic programming, the key steps involve identifying the recursive structure of the problem, solving each of the subproblems, and storing their solutions in a table (either top-down with memoization or bottom-up). This prevents the redundant calculation of the same subproblems, which significantly reduces the time complexity compared to naive recursive solutions.

For example, in solving a problem like the Fibonacci sequence, using a straightforward recursive approach would result in an exponential time complexity due to repeated calculations of the same Fibonacci numbers. However, by employing dynamic programming, we can compute the Fibonacci numbers iteratively or recursively with memoization, resulting in linear time complexity.

Thus, the essence of dynamic programming lies in its ability to efficiently solve problems by leveraging the solutions of simpler subproblems, making it a fundamental technique in algorithm design.