In the context of dynamic programming, what is a subproblem?

A subproblem in dynamic programming refers to a smaller instance of a larger problem. In dynamic programming, complex problems are broken down into simpler subproblems, which are easier to solve. The solutions to these subproblems are stored and reused (memorized), allowing the overall problem to be solved more efficiently without redundant calculations.

This technique is particularly useful because many optimal problems exhibit the principle of overlapping subproblems, where the same subproblems are solved multiple times. By recognizing these repeated instances and solving each subproblem only once, dynamic programming optimizes performance by reducing computational redundancy.

In contrast, independent problems that cannot be solved do not align with the concept of subproblems, as subproblems must contribute to solving the original problem. A repeated instance of the same problem refers to the same problem being solved multiple times without reuse of previously computed solutions, which isn't the goal of dynamic programming. Lastly, a problem that requires a unique solution each time contradicts the fundamental approach of dynamic programming, where the focus is on solving and optimizing through reuse of previously calculated results. The identification of subproblems is essential for effectively applying dynamic programming techniques.