We consider the problem of computing all-pairs shortest paths in a directed
graph with real weights assigned to vertices.
For an nΓn 0-1 matrix C, let KCβ be the complete weighted graph
on the rows of C where the weight of an edge between two rows is equal to
their Hamming distance. Let MWT(C) be the weight of a minimum weight spanning
tree of KCβ.
We show that the all-pairs shortest path problem for a directed graph G on
n vertices with nonnegative real weights and adjacency matrix AGβ can be
solved by a combinatorial randomized algorithm in time
O(n2n+min{MWT(AGβ),MWT(AGtβ)}β)
As a corollary, we conclude that the transitive closure of a directed graph
G can be computed by a combinatorial randomized algorithm in the
aforementioned time. O(n2n+min{MWT(AGβ),MWT(AGtβ)}β)
We also conclude that the all-pairs shortest path problem for uniform disk
graphs, with nonnegative real vertex weights, induced by point sets of bounded
density within a unit square can be solved in time O(n2.75)