What type of problems can dynamic programming be applied to?

Dynamic programming can be applied to problems that can be divided into overlapping subproblems. This method is particularly effective for solving complex problems by breaking them down into simpler subproblems, which can be solved independently. The key feature of dynamic programming is that it stores the results of these subproblems to avoid redundant calculations. This efficiency is achieved through the principle of optimality, which states that the optimal solution to a problem can be constructed from optimal solutions to its subproblems.

In practice, dynamic programming is used for various kinds of problems, including but not limited to optimization tasks, such as finding the shortest path in a graph, calculating the maximum profit in a knapsack problem, or solving the longest common subsequence issue. Problems that exhibit both overlapping subproblems and optimal substructure, where the solution can be expressed in terms of the solutions to its subproblems, are ideally suited for this approach.

While sorting, optimization, or graph-related problems can potentially involve dynamic programming, the distinguishing factor is the ability to decompose the problem into smaller subproblems that share solutions. This characteristic makes the adaptability of dynamic programming broad, applicable to a wide range of computational scenarios beyond just those explicitly listed in the incorrect choices.