Floyd–Warshall algorithm

Floyd–Warshall algorithm

Class	All-pairs shortest path problem (for weighted graphs)
Data structure	Graph
Worst-case performance	${\displaystyle \Theta (|V|^{3})}$
Best-case performance	${\displaystyle \Theta (|V|^{3})}$
Average performance	${\displaystyle \Theta (|V|^{3})}$
Worst-case space complexity	${\displaystyle \Theta (|V|^{2})}$

In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles). A single execution of the algorithm will find the lengths (summed weights) of the 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 ${\displaystyle R}$, or (in connection with the Schulze voting system) widest paths between all pairs of vertices in a weighted graph.

The Floyd–Warshall algorithm is an example of dynamic programming, and was published in its currently recognized form by Robert Floyd in 1962. However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 and also by Stephen Warshall in 1962 for finding the transitive closure of a graph, and is closely related to Kleene's algorithm (published in 1956) for converting a deterministic finite automaton into a regular expression. The modern formulation of the algorithm as three nested for-loops was first described by Peter Ingerman, also in 1962.

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