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Algorithm for finding all pairs shortest paths in graphs, allowing some edge weights to be negative

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 paths in a directed 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 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 , 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.

The Floyd–Warshall algorithm compares many possible paths through the graph between each pair of vertices. It is guaranteed to find all shortest paths and is able to do this with comparisons in a graph, During the execution of the algorithm, if there is a negative cycle, exponentially large numbers can appear, as large as , where is the largest absolute value of a negative edge in the graph. To avoid overflow/underflow problems one should check for negative numbers on the diagonal of the path matrix within the inner for loop of the algorithm. Obviously, in an undirected graph a negative edge creates a negative cycle (i. e., a closed walk) involving its incident vertices. Considering all edges of the above example graph as undirected, e. g. the vertex sequence 4 – 2 – 4 is a cycle with weight sum −2.

The Floyd–Warshall algorithm typically only provides the lengths of the paths between all pairs of vertices. With simple modifications, it is possible to create a method to reconstruct the actual path between any two endpoint vertices. While one may be inclined to store the actual path from each vertex to each other vertex, this is not necessary, and in fact, is very costly in terms of memory. Instead, we can use the shortest path tree, which can be calculated for each node in time using memory, and allows us to efficiently reconstruct a directed path between any two connected vertices.

For graphs with non negative edge weights, Dijkstra's algorithm can be used to find all shortest paths from a single vertex with running time Thus, running Dijkstra starting at each vertex takes time Since , this yields a worst case running time of repeated Dijkstra of While this matches the asymptotic worst case running time of the Floyd Warshall algorithm, the constants involved matter quite a lot. When a graph is dense (i. e., ), the Floyd Warshall algorithm tends to perform better in practice. When the graph is sparse (i. e., is significantly smaller than ), Dijkstra tends to dominate. For sparse graphs with negative edges but no negative cycles, Johnson's algorithm can be used, with the same asymptotic running time as the repeated Dijkstra approach. There are also known algorithms using fast matrix multiplication to speed up all pairs shortest path computation in dense graphs, but these typically make extra assumptions on the edge weights (such as requiring them to be small integers). In addition, because of the high constant factors in their running time, they would only provide a speedup over the Floyd–Warshall algorithm for very large graphs.