Editing Floyd–Warshall algorithm (section)

{{short description|Algorithm in graph theory}}
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{{Redirect|Floyd's algorithm|cycle detection|Floyd's cycle-finding algorithm|computer graphics|Floyd–Steinberg dithering}}
{{Infobox Algorithm
|class=[[All-pairs shortest path problem]] (for weighted graphs)
|image=
|caption =
|data=[[Graph (data structure)|Graph]]
|time=<math>\Theta (|V|^3)</math>
|average-time=<math>\Theta (|V|^3)</math>
|best-time=<math>\Theta (|V|^3)</math>
|space=<math>\Theta(|V|^2)</math>
}}

In [[computer science]], the '''Floyd–Warshall algorithm''' (also known as '''Floyd's algorithm''', the '''Roy–Warshall algorithm''', the '''Roy–Floyd algorithm''', or the '''WFI algorithm''') is an [[algorithm]] for finding [[shortest path problem|shortest paths]] in a directed [[weighted graph]] with positive or negative edge weights (but with no negative cycles).<ref name=":0">{{Introduction to Algorithms|1}} See in particular Section 26.2, "The Floyd–Warshall algorithm", pp.&nbsp;558–565 and Section 26.4, "A general framework for solving path problems in directed graphs", pp.&nbsp;570–576.</ref><ref>{{cite book | author=Kenneth H. Rosen | title=Discrete Mathematics and Its Applications, 5th Edition | publisher = Addison Wesley | year=2003 | isbn=978-0-07-119881-3 }}</ref> A single execution of the algorithm will find the lengths (summed weights) of shortest paths between all pairs of vertices. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. Versions of the algorithm can also be used for finding the [[transitive closure]] of a relation <math>R</math>, or (in connection with the [[Schulze method|Schulze voting system]]) [[widest path problem|widest paths]] between all pairs of vertices in a weighted graph.