In computer science
, the Floyd–Warshall algorithm
(sometimes known as the WFI Algorithm
or Roy–Floyd algorithm
, since Bernard Roy
described this algorithm in 1959
) is a graph
for finding shortest paths
in a weighted, directed graph. A single execution of the algorithm will find the shortest paths between all pairs of vertices. The Floyd–Warshall algorithm
is an example of dynamic programming
The Floyd-Warshall algorithm compares all possible paths through the graph between each pair of vertices. It is able to do this with only
comparisons. This is remarkable considering that there may be up to
edges in the graph, and every combination of edges is tested. It does so by incrementally improving an estimate on the shortest path between two vertices, until the estimate is known to be optimal.
Consider a graph with vertices , each numbered 1 through N. Further consider a function that returns the shortest possible path from to using only vertices 1 through as intermediate points along the way. Now, given this function, our goal is to find the shortest path from each to each using only nodes 1 through .
There are two candidates for this path: either the true shortest path only uses nodes in the set ; or there exists some path that goes from to , then from to that is better. We know that the best path from to that only uses nodes 1 through is defined by , and it is clear that if there were a better path from to to , then the length of this path would be the concatenation of the shortest path from to (using vertices in ) and the shortest path from to (also using vertices in ).
Therefore, we can define in terms of the following recursive formula:
This formula is the heart of Floyd Warshall. The algorithm works by first computing for all (i,j) pairs, then using that to find for all pairs, etc. This process continues until k=n, and we have found the shortest path for all pairs using any intermediate vertices.
Conveniently, when calculating the
th case, one can overwrite the information saved from the computation of
. This means the algorithm uses quadratic memory. Be careful to note the initialization conditions:
1 /* Assume a function edgeCost(i,j) which returns the cost of the edge from i to j
2 (infinity if there is none).
3 Also assume that n is the number of vertices and edgeCost(i,i)=0
6 int path;
7 /* A 2-dimensional matrix. At each step in the algorithm, path[i][j] is the shortest path
8 from i to j using intermediate values in (1..k-1). Each path[i][j] is initialized to
12 procedure FloydWarshall ()
13 for to
14 for each in
15 path[i][j] = min (path[i][j], path[i][k]+path[k][j] );
Behaviour with negative cycles
For numerically meaningful output, Floyd-Warshall assumes that there are no negative cycles (in fact, between any pair of vertices which form part of a negative cycle, the shortest path is not well-defined because the path can be infinitely small). Nevertheless, if there are negative cycles, Floyd–Warshall can be used to detect them. A negative cycle can be detected if the path
matrix contains a negative number along the diagonal. If path[i][i] is negative for some vertex i, then this vertex belongs to at least one negative cycle.
To find all
from those of
bit operations. Since we begin with
and compute the sequence of
, the total number of bit operations used is
. Therefore, the complexity
of the algorithm is
and can be solved by a deterministic machine
in polynomial time
Applications and generalizations
The Floyd–Warshall algorithm can be used to solve the following problems, among others:
- Shortest paths in directed graphs (Floyd's algorithm).
- Transitive closure of directed graphs (Warshall's algorithm). In Warshall's original formulation of the algorithm, the graph is unweighted and represented by a Boolean adjacency matrix. Then the addition operation is replaced by logical conjunction (AND) and the minimum operation by logical disjunction (OR).
- Finding a regular expression denoting the regular language accepted by a finite automaton (Kleene's algorithm)
- Inversion of real matrices (Gauss-Jordan algorithm).
- Optimal routing. In this application one is interested in finding the path with the maximum flow between two vertices. This means that, rather than taking minima as in the pseudocode above, one instead takes maxima. The edge weights represent fixed constraints on flow. Path weights represent bottlenecks; so the addition operation above is replaced by the minimum operation.
- Testing whether an undirected graph is bipartite.
Floyd, Robert W. "Algorithm 97: Shortest Path". Communications of the ACM 5 (6): 345.
Kleene, S. C. Automata Studies. Princeton University Press.
Warshall, Stephen "A theorem on Boolean matrices". Journal of the ACM 9 (1): 11–12.
Kenneth H. Rosen (2003). Discrete Mathematics and Its Applications, 5th Edition. Addison Wesley.
- Section 26.2, "The Floyd–Warshall algorithm", pp. 558–565;
- Section 26.4, "A general framework for solving path problems in directed graphs", pp. 570–576.