FLOYD–WARSHALL ALGORITHM

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In computer science, the 'Floyd–Warshall algorithm' (sometimes known as the 'Roy–Floyd 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. It does so in $V^3$ time, where ''V'' is the number of vertices in the graph. The 'Floyd–Warshall algorithm' is an example of dynamic programming.

Contents

Algorithm

Pseudocode

Drawbacks

Analysis

Applications and generalizations

Implementations

References

See Also

External links

Algorithm

The Floyd-Warshall algorithm compares all possible paths through the graph between each pair of vertices. Amazingly, it is able to do this with only $V^3$ comparisons (this is remarkable considering that there may be up to $V^2$ 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 $G$ 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 through $k$ 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 through $k+1$.
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 $shortestPath(i,j,1)$ for all (i,j) pairs, then using that to find $shortestPath(i,j,2)$ for all $(i,j)$ pairs, etc. This process continues until k=n, and we have found the shortest path for all $(i,j)$ pairs using any number of intermediate vertices.

Pseudocode

Conveniently, when calculating the $k$th case, one can overwrite the information saved from the computation of $k-1$. This means the algorithm uses linear 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' $mathit\{k\}\; :=\; 1$ 'to' $mathit\{n\}$
14 'begin'
15 'for each' $mathit\{(i,j)\}$ 'in' $(1..n)$
16 'begin'
17 path[i][j] = min ( path[i][j], path[i][k]+path[k][j] );
18 'end'
19 'end'
20 'endproc'

Drawbacks

While Floyd–Warshall works on graphs with negative edges, negative cycles will cause it to fail. In this case, the solution may not be optimal. If this happens, the matrix will end with a negative diagonal.

Analysis

To find all $mathit\{n\}^2$ of $mathcal\{W\}\_k$ from those of $mathcal\{W\}\_\{mathit\{k\}-1\}$ requires $2mathit\{n\}^2$ bit operations. Since we begin with $mathcal\{W\}\_0\; =\; mathcal\{W\}\_mathcal\{R\}$ and compute the sequence of $mathit\{n\}$ zero-one matrices $mathcal\{W\}\_1$, $mathcal\{W\}\_2$, $...$, $mathcal\{W\}\_mathit\{n\}\; =\; mathcal\{M\}\_\{mathcal\{R\}^$
★ }, the total number of bit operations used is $mathit\{n\}\; imes\; 2mathit\{n\}^2\; =\; 2mathit\{n\}^3$.

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

★ Introduction to Algorithms, , Thomas H., Cormen, 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.

★

★ Automata Studies, , S. C., Kleene, Princeton University Press, ,

★

★ Discrete Mathematics and Its Applications, 5th Edition., Kenneth H. Rosen, , , Addison Wesley, 2003,

See Also

★ Robert Floyd

★ Stephen Warshall

External links

★ Analyze Floyd's algorithm in an online Javascript IDE

★ Interactive animation of Floyd–Warshall algorithm