Floyd–Warshall algorithm

In computer science, the Floyd–Warshall algorithm (sometimes known as the WFI Algorithm or Roy–Floyd algorithm) is a graph analysis algorithm 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 algorithm is an example of dynamic programming. It was discovered by Robert Floyd in 1962 and is basically the same as the algorithm discovered by Bernard Roy in 1959 and again by Stephen Warshall in 1962.^[1]

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 V³ comparisons. This is remarkable considering that there may be up to V² 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 G with vertices V, each numbered 1 through N. Further consider a function shortestPath(i, j, k) that returns the shortest possible path from i to j using only vertices 1 to k as intermediate points along the way. Now, given this function, our goal is to find the shortest path from each i to each j using only nodes 1 to k + 1.

There are two candidates for this path: either the true shortest path only uses nodes in the set {1, ..., k}; or there exists some path that goes from i to k + 1, then from k + 1 to j that is better. We know that the best path from i to j that only uses nodes 1 through k is defined by shortestPath(i, j, k), and it is clear that if there were a better path from i to k + 1 to j, then the length of this path would be the concatenation of the shortest path from i to k + 1 (using vertices in {1, ..., k} and the shortest path from k + 1 to j (also using vertices in {1, ..., k}).

Therefore, we can define shortestPath(i, j, k) in terms of the following recursive formula:

shortestPath(i,j,k) = min{shortestPath(i,j,k − 1),shortestPath(i,k,k − 1) + shortestPath(k,j,k − 1)},

shortestPath(i,j,0) = edgeCost(i,j).

This formula is the heart of Floyd–Warshall. The algorithm works by first computing shortestPath(i, j, k) for all (i, j) pairs for k = 1, then k = 2, etc. This process continues until k = n, and we have found the shortest path for all (i, j) pairs using any intermediate vertices.

Pseudocode

Conveniently, when calculating the kth case, one can overwrite the information saved from the computation of 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 vertices (1..k−1).  Each path[i][j] is initialized to
 9    edgeCost(i,j) or infinity if there is no edge between i and j.
10 */
11
12 procedure FloydWarshall ()
13    for k := 1 to n
14       for i := 1 to n
15          for j := 1 to n
16             path[i][j] = min ( path[i][j], path[i][k]+path[k][j] );

This algorithm can be modified slightly to also give the path from any vertex to another (as opposed to merely finding the length of the path). This requires only the small change of keeping track of the intermediate vertex k. If we know that vertex k is intermediate for two vertices i and j, the whole path from i to j is path from i to k, followed by path from k to j. These two shorter paths are determined recursively.

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 arbitrarily negative). Nevertheless, if there are negative cycles, Floyd–Warshall can be used to detect them. The intuition is as follows:

• The Floyd–Warshall algorithm iteratively revises path lengths between all pairs of verticies (i, j), including where i = j;
• Initially, the length of the path (i,i) is zero;
• A path {(i,k), (k,i)} can only improve upon this if it has length less than zero, i.e. denotes a negative cycle;
• Thus, after the algorithm, (i,i) will be negative if there exists a negative-length path from i back to i.

Hence, to detect negative cycles using the Floyd–Warshall, one can inspect the diagonal of the path matrix, and the presence of a negative number indicates that the graph contains at least one negative cycle.^[2]

Analysis

To find all n² of shortestPath(i,j,k) (for all i and j) from those of shortestPath(i,j,k−1) requires 2n² operations. Since we begin with shortestPath(i,j,0) = edgeCost(i,j) and compute the sequence of n matrices shortestPath(i,j,1), shortestPath(i,j,2), …, shortestPath(i,j,n), the total number of operations used is n · 2n² = 2n³. Therefore, the complexity of the algorithm is Θ(n³) 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.
• Fast computation of Pathfinder networks.
• Maximum Bandwidth Paths in Flow Networks

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 joshuarobinson.net
• A C++ implementation is available in the boost::graph library
• A Java implementation is in the Apache commons graph library.
• Another Java implementation is on Algowiki
• A C# implementation is in QuickGraph
• A PHP implementation and PL/pgSQL implementation are available at Microshell.
• A MATLAB implementation is available using the Matlab_bgl package.

See also

• Dijkstra's algorithm, an algorithm for finding single-source shortest paths in a more restrictive class of inputs, graphs in which all edge weights are non-negative
• Johnson's algorithm, an algorithm for solving the same problem as the Floyd–Warshall algorithm, all pairs shortest paths in graphs with some edge weights negative. Compared to the Floyd–Warshall algorithm, Johnson's algorithm is more efficient for sparse graphs.

References

• Cormen, Thomas H.; Leiserson, Charles E., Rivest, Ronald L. (1990). Introduction to Algorithms (1st ed.). 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). 
• Roy, Bernard (1959). "Transitivité et connexité.". C. R. Acad. Sci. Paris 249: 216–218. 

External links

• Analyze Floyd's algorithm in an online Javascript IDE
• Interactive animation of Floyd–Warshall algorithm
• Floyd–Warshall in C#, as part of QuickGraph