All-pairs small-stretch paths

Let G = (V, E) be a weighted undirected graph. A path between u, v is an element of V is said to be of stretch t if its length is at most t times the distance between u, and v in the graph. We consider the problem of finding small-stretch paths between all pairs of vertices in the graph G. It is easy to see that finding paths of stretch less than 2 between all pairs of vertices in an undirected graph with n vertices is at least as hard as the Boolean multiplication of two n x n matrices. We describe three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges. The first algorithm, STRETCH(2), runs in (O) over tilde(n(3)/(2)m(1/2)) time and finds stretch 2 paths. The second algorithm, STRETCH(7/3), runs in (O) over tilde(n(7/3)) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH(3), runs in (O) over tilde(n(2)) and finds stretch 3 paths. Our algorithms are simpler, more efficient and more accurate than the previously best algorithms for finding small-stretch paths. Unlike all previous algorithms, our algorithms are not based on the construction of sparse spanners or sparse neighbourhood covers.