In computer science, a problem is said to have optimal substructure if an optimal solution can be constructed efficiently from optimal solutions of its subproblems. This property is used to determine the usefulness of dynamic programming and greedy algorithms for a problem.
Typically, a greedy algorithm is used to solve a problem with optimal substructure if it can be proved by induction that this is optimal at each step (Cormen et al. pp. 381–2). Otherwise, providing the problem exhibits overlapping subproblems as well, dynamic programming is used. If there are no appropriate greedy algorithms and the problem fails to exhibit overlapping subproblems, often a lengthy but straightforward search of the solution space is the best alternative.
In the application of dynamic programming to mathematical optimization, Richard Bellman's Principle of Optimality is based on the idea that in order to solve a dynamic optimization problem from some starting period t to some ending period T, one implicitly has to solve subproblems starting from later dates s, where t
. This is an example of optimal substructure. The Principle of Optimality is used to derive the Bellman equation, which shows how the value of the problem starting from t is related to the value of the problem starting from s.
Other articles related to "optimal substructure, optimal":
... a problem must have in order for dynamic programming to be applicable optimal substructure and overlapping subproblems ... Optimal substructure means that the solution to a given optimization problem can be obtained by the combination of optimal solutions to its subproblems ... devising a dynamic programming solution is to check whether the problem exhibits such optimal substructure ...
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