All-Pairs Small-Stretch Paths

✍ Scribed by Edith Cohen; Uri Zwick

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 188 KB
• Volume: 38
• Category: Article
• ISSN: 0196-6774
• DOI: 10.1006/jagm.2000.1117

✦ Synopsis

Let G = (V, E) be a weighted undirected graph. A path between a,v E 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, STRETCHz, runs in d(n 3/2m1/2) time and finds stretch 2 paths. The second algorithm, STRETCH7j3, runs in d(n7i3) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH3, runs in 6(n2) 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.