Editing Optimal substructure (section)

{{Short description|Property of a computational problem}}
[[Image:Shortest path optimal substructure.svg|200px|thumb|'''Figure 1'''. Finding the shortest path using optimal substructure.  Numbers represent the length of the path; straight lines indicate single [[Edge (graph theory)|edges]], wavy lines indicate shortest [[Path (graph theory)|paths]], i.e., there might be other vertices that are not shown here.]]
In [[computer science]], a problem is said to have '''optimal substructure''' if an optimal solution can be constructed from optimal solutions of its subproblems. This property is used to determine the usefulness of greedy algorithms for a problem.<ref name=cormen>{{cite book|title=Introduction to Algorithms |edition=3rd|last1=Cormen|first1=Thomas H. |last2=Leiserson |first2=Charles E. |last3=Rivest|first3=Ronald L. |last4= Stein |first4=Clifford|date=2009 |isbn=978-0-262-03384-8|publisher=[[MIT Press]]|authorlink1=Thomas H. Cormen |authorlink2=Charles E. Leiserson|authorlink3=Ron Rivest |authorlink4=Clifford Stein}}</ref>  

Typically, a [[greedy algorithm]] is used to solve a problem with optimal substructure if it can be proven by induction that this is optimal at each step.<ref name=cormen /> Otherwise, provided the problem exhibits [[overlapping subproblems]] as well,  [[Divide and conquer algorithm|divide-and-conquer]] methods or [[dynamic programming]] may be 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.
<!-- A special case of optimal substructure is the case where a subproblem S<sub>ab</sub> has an activity P<sub>y</sub>, then it should contain optimal solutions to subproblems S<sub>ay</sub> and S<sub>yb</sub>. --> <!-- *TODO: Add Recursion, misc. -->

In the application of [[dynamic programming]] to [[Optimization (mathematics)|mathematical optimization]], [[Richard Bellman]]'s [[Bellman equation#Bellman's principle of optimality|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<s<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''.