Floyd–Warshall algorithm

Floyd–Warshall algorithm
Class All-pairs shortest path problem (for weighted graphs)
Data structure Graph
Worst case performance O(|V|3)
Best case performance Ω(|V|3)
Worst case space complexity Θ(|V|2)
Graph and tree search algorithms
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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 graph (with positive or negative edge weights). A single execution of the algorithm will find the lengths (summed weights) of the shortest paths between all pairs of vertices though it does not return details of the paths themselves. The algorithm is an example of dynamic programming. It 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.[1]

Contents

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 Θ(V3) comparisons in a graph. This is remarkable considering that there may be up to Ω(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 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 vertices 1 to k + 1.

There are two candidates for each of these paths: either the true shortest path only uses vertices 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 vertices 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:

\begin{align} & {} \quad \textrm{shortestPath}(i,j,k) \\ = & \min\left\{\textrm{shortestPath}(i,j,k-1), \right. \\ & \qquad \qquad \left. \textrm{shortestPath}(i,k,k-1) + \textrm{shortestPath}(k,j,k-1)\right\}, \\[12pt] & {} \quad \textrm{shortestPath}(i,j,0) = \textrm{edgeCost}(i,j). \end{align}

This formula is the heart of the Floyd–Warshall algorithm. 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] );

Behaviour with negative cycles

For numerically meaningful output, the Floyd–Warshall algorithm 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, the Floyd–Warshall algorithm can be used to detect them. The intuition is as follows:

Hence, to detect negative cycles using the Floyd–Warshall algorithm, 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] Obviously, in an undirected graph a negative edge creates a negative cycle (i.e., a closed walk) involving its incident vertices.

Path reconstruction

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. For each vertex, one need only store the information about which vertex one has to go through if one wishes to end up at any given vertex. Therefore, information to reconstruct all paths can be stored in an single N×N matrix 'next' where next[i][j] represents the vertex one must travel through if one intends to take the shortest path from i to j. Implementing such a scheme is trivial as when a new shortest path is found between two vertices, the matrix containing the paths is updated. The next matrix is updated along with the path matrix such that at completion both tables are complete and accurate, and any entries which are infinite in the path table will be null in the next table. The path from i to j is then path from i to next[i][j], followed by path from next[i][j] to j. These two shorter paths are determined recursively. This modified algorithm runs with the same time and space complexity as the unmodified algorithm.

 1 procedure FloydWarshallWithPathReconstruction ()
 2    for k := 1 to n
 3       for i := 1 to n
 4          for j := 1 to n
 5             if path[i][k] + path[k][j] < path[i][j] then
 6                path[i][j] := path[i][k]+path[k][j];
 7                next[i][j] := k;
 8
 9 procedure GetPath (i,j)
10    if path[i][j] equals infinity then
11      return "no path";
12    int intermediate := next[i][j];
13    if intermediate equals 'null' then
14      return " ";   /* there is an edge from i to j, with no vertices between */
15   else
16      return GetPath(i,intermediate) + intermediate + GetPath(intermediate,j);

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:

Implementations

See also

References

  1. ↑ Weisstein, Eric. "Floyd-Warshall Algorithm". Wolfram MathWorld. http://mathworld.wolfram.com/Floyd-WarshallAlgorithm.html. Retrieved 13 November 2009. 
  2. ↑ "Lecture 12: Shortest paths (continued)" (PDF). Network Flows and Graphs. Department of Industrial Engineering and Operations Research, University of California, Berkeley. 7 October 2008. http://www.ieor.berkeley.edu/~ieor266/Lecture12.pdf. 

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