The.Algorithm.Design.Manual.Springer-Verlag.1998

All-Pairs Shortest Path
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All-Pairs Shortest Path
If we want to find the length of the shortest path between all pairs of vertices, we could run Dijkstra's algorithm n
times, once from each possible starting vertex. This yields a cubic time algorithm for all-pairs shortest path, since
.
Can we do better? Significantly improving the complexity is still an open question, but there is a superslick dynamic
programming algorithm that also runs in .
There are several ways to characterize the shortest path between two nodes in a graph. The Floyd-Warshall algorithm
starts by numbering the vertices of the graph from 1 to n. We use these numbers here not to label the vertices, but to
order them. Define to be the length of the shortest path from i to j using only vertices numbered from 1, 2,..., k
as possible intermediate vertices.
What does this mean? When k = 0, we are allowed no intermediate vertices, so that every path consists of at most one
edge. Thus . In general, adding a new vertex k+1 as a possible intermediary helps only if there is a short
path that goes through it, so
This recurrence performs only a constant amount of work per cell. The following dynamic programming algorithm
implements the recurrence:
Floyd(G)
Let , the weight matrix of G
for k=1 to n
file:///E|/BOOK/BOOK2/NODE78.HTM (1 of 2) [19/1/2003 1:29:18]
for i=1 to n
for j=1 to n