What is the optimal substructure property in dynamic programming?

The optimal substructure property is a key characteristic of problems that can be effectively solved using dynamic programming. This property essentially states that an optimal solution to a problem can be constructed from optimal solutions to its subproblems. In other words, when tackling a problem that exhibits optimal substructure, it is possible to break it down into smaller, manageable subproblems, solve each of those optimally, and then use those solutions to build up to the optimal solution for the original, larger problem.

This principle is crucial in dynamic programming because it allows for the storing of results of subproblems (often using a table) to avoid redundant calculations and thereby improve efficiency. Problems such as the shortest path in a graph or calculating Fibonacci numbers illustrate this well; the optimal solution to the larger problem directly relies on combining optimal solutions from smaller, related subproblems.

This contrasts with other properties mentioned in the other choices. Notably, the assertion that a problem has no smaller subproblems does not align with the essence of dynamic programming, which is fundamentally about breaking problems into smaller parts. Likewise, stating that all subproblems are independent is not accurate, as many optimal solutions depend on the interrelationships between subproblems. Furthermore, claiming that only the largest