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"... Abstract Let G = (V; E) be a weighted undirected graph. A path between u; v 2 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 se ..."

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Abstract Let G = (V; E) be a weighted undirected graph. A path between u; v 2 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 \Theta 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, STRETCH2, runs in ~O(n3=2m1=2) time and finds stretch 2 paths. The second algorithm, STRETCH7=3, runs in ~O(n7=3) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH3, runs in ~O(n2) and finds stretch 3 paths.