Floyd–Warshall algorithm

In computer science, the Floyd–Warshall algorithm (sometimes known as the Roy–Floyd algorithm or WFI Algorithm, since Bernard Roy described this algorithm in 1959) is a graph analysis algorithm for finding shortest paths in a weighted, directed graph. A single execution of the algorithm will find the shortest path between all pairs of vertices. The Floyd–Warshall algorithm is an example of dynamic programming.

Algorithm

The Floyd-Warshall algorithm compares all possible paths through the graph between each pair of vertices. It is able to do this with only Failed to parse (Missing texvc executable; please see math/README to configure.): V^3

comparisons (this is remarkable considering that there may be up to Failed to parse (Missing texvc executable; please see math/README to configure.): V^2
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 Failed to parse (Missing texvc executable; please see math/README to configure.): G

with vertices Failed to parse (Missing texvc executable; please see math/README to configure.): V

, each numbered 1 through N. Further consider a function Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{shortestPath}(i,j,k)

that returns the shortest possible path from Failed to parse (Missing texvc executable; please see math/README to configure.): i
to Failed to parse (Missing texvc executable; please see math/README to configure.): j
using only vertices 1 through Failed to parse (Missing texvc executable; please see math/README to configure.): k
as intermediate points along the way.  Now, given this function, our goal is to find the shortest path from each Failed to parse (Missing texvc executable; please see math/README to configure.): i
to each Failed to parse (Missing texvc executable; please see math/README to configure.): j
using only nodes 1 through Failed to parse (Missing texvc executable; please see math/README to configure.): k+1

.

There are two candidates for this path: either the true shortest path only uses nodes in the set Failed to parse (Missing texvc executable; please see math/README to configure.): (1...k)

or there exists some path that goes from Failed to parse (Missing texvc executable; please see math/README to configure.): i

to Failed to parse (Missing texvc executable; please see math/README to configure.): k+1

, then from Failed to parse (Missing texvc executable; please see math/README to configure.): k+1

to Failed to parse (Missing texvc executable; please see math/README to configure.): j
that is better.  We know that the best path from Failed to parse (Missing texvc executable; please see math/README to configure.): i
to Failed to parse (Missing texvc executable; please see math/README to configure.): j
that only uses nodes 1 through Failed to parse (Missing texvc executable; please see math/README to configure.): k
is defined by Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{shortestPath}(i,j,k)

, and it is clear that if there were a better path from Failed to parse (Missing texvc executable; please see math/README to configure.): i

to Failed to parse (Missing texvc executable; please see math/README to configure.): k+1
to Failed to parse (Missing texvc executable; please see math/README to configure.): j

, then the length of this path would be the concatenation of the shortest path from Failed to parse (Missing texvc executable; please see math/README to configure.): i

to Failed to parse (Missing texvc executable; please see math/README to configure.): k+1
(using vertices in Failed to parse (Missing texvc executable; please see math/README to configure.):  (1...k) 

) and the shortest path from Failed to parse (Missing texvc executable; please see math/README to configure.): k+1

to Failed to parse (Missing texvc executable; please see math/README to configure.): j
(also using vertices in Failed to parse (Missing texvc executable; please see math/README to configure.):  (1...k) 

).

Therefore, we can define Failed to parse (Missing texvc executable; please see math/README to configure.): shortestPath(i,j,k)

in terms of the following recursive formula:

Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{shortestPath}(i,j,k) = \min(\textrm{shortestPath}(i,j,k-1),\textrm{shortestPath}(i,k,k-1)\,+\,\textrm{shortestPath}(k,j,k-1));\,\!

Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{shortestPath}(i,j,0) = \textrm{edgeCost}(i,j);\,\!

This formula is the heart of Floyd Warshall. The algorithm works by first computing Failed to parse (Missing texvc executable; please see math/README to configure.): shortestPath(i,j,1)

for all (i,j) pairs, then using that to find Failed to parse (Missing texvc executable; please see math/README to configure.): shortestPath(i,j,2)
for all Failed to parse (Missing texvc executable; please see math/README to configure.): (i,j)
pairs, etc.  This process continues until k=n, and we have found the shortest path for all Failed to parse (Missing texvc executable; please see math/README to configure.): (i,j)
pairs using any intermediate vertices.

Pseudocode

Conveniently, when calculating the Failed to parse (Missing texvc executable; please see math/README to configure.): k th case, one can overwrite the information saved from the computation of Failed to parse (Missing texvc executable; please see math/README to configure.): k-1 . 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
 4 */
 5
 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
 9    edgeCost(i,j).
10 */
11
12 procedure FloydWarshall ()
13    for Failed to parse (Missing texvc executable; please see math/README to configure.): \mathit{k} := 1
to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathit{n}

14    begin
15       for each Failed to parse (Missing texvc executable; please see math/README to configure.): \mathit{(i,j)}
in Failed to parse (Missing texvc executable; please see math/README to configure.): (1..n)

16       begin
17          path[i][j] = min ( path[i][j], path[i][k]+path[k][j] );
18       end
19    end
20 endproc

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). Nevertheless, if there are negative cycles, Floyd–Warshall can be used to detect them: either run one more iteration and see if there are any changes, or look for negative values in the diagonal.

This short section requires expansion.

Analysis

To find all Failed to parse (Missing texvc executable; please see math/README to configure.): \mathit{n}^2

of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{W}_k
from those of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{W}_{\mathit{k}-1}
requires Failed to parse (Missing texvc executable; please see math/README to configure.): 2\mathit{n}^2
bit operations. Since we begin with Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{W}_0 = \mathcal{W}_\mathcal{R}
and compute the sequence of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathit{n}
zero-one matrices Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{W}_1

, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{W}_2 , Failed to parse (Missing texvc executable; please see math/README to configure.): ... , Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{W}_\mathit{n} = \mathcal{M}_{\mathcal{R}^*} , the total number of bit operations used is Failed to parse (Missing texvc executable; please see math/README to configure.): \mathit{n} \times 2\mathit{n}^2 = 2\mathit{n}^3 . Therefore, the complexity of the algorithm is Failed to parse (Missing texvc executable; please see math/README to configure.): \Theta({n}^3) 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.

Implementations

• A Perl implementation is available in the Graph module
• A Javascript implementation is available at Alex Le's Blog
• A Python implementation is available in the NetworkX package
• A C++ implementation is available at ece.rice.edu

References

• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. (1990). Introduction to Algorithms, first edition, MIT Press and McGraw-Hill. ISBN 0-262-03141-8. 
  • 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.
• Floyd, Robert W. (June 1962). "Algorithm 97: Shortest Path". Communications of the ACM 5 (6): 345. doi:10.1145/367766.368168. 
• Kleene, S. C. (1956). "Representation of events in nerve nets and finite automata", in C. E. Shannon and J. McCarthy: Automata Studies. Princeton University Press, pp. 3–42. 
• Warshall, Stephen (January 1962). "A theorem on Boolean matrices". Journal of the ACM 9 (1): 11–12. doi:10.1145/321105.321107. 
• Kenneth H. Rosen (2003). Discrete Mathematics and Its Applications, 5th Edition.. Addison Wesley. ISBN 0-07-119881-4 (ISE). 

See also

• Robert Floyd
• Stephen Warshall
• Dijkstra's algorithm

External links

• Analyze Floyd's algorithm in an online Javascript IDE
• Interactive animation of Floyd–Warshall algorithmcs:Floyd-Warshallův algoritmus

de:Algorithmus von Floyd und Warshall es:Algoritmo de Floyd-Warshall fr:Algorithme de Floyd-Warshall ko:플로이드-와셜 알고리즘 id:Algoritma Floyd-Warshall it:Algoritmo di Floyd-Warshall he:אלגוריתם פלויד-וורשאל ja:ワーシャル-フロイド法 pl:Algorytm Floyda-Warshalla pt:Algoritmo de Floyd-Warshall sr:Флојд-Воршалов алгоритам