All-pairs nearly 2-approximate shortest paths in O(n2polylogn) time

Authors:
Surender Baswana;Vishrut Goyal;Sandeep Sen
Affiliations:
Department of Computer Science & Engineering, I.I.T. Kanpur, India;Persistent Systems Limited, Pune, India;Department of Computer Science & Engineering, I.I.T. Delhi, India
Venue:
Theoretical Computer Science
Year:
2009

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Abstract

Let G=(V,E) be an unweighted undirected graph on |V|=n vertices and |E|=m edges. Let @d(u,v) denote the distance between vertices u,v@?V. An algorithm is said to compute all-pairs t-approximate shortest-paths/distances, for some t=1, if for each pair of vertices u,v@?V, the path/distance reported by the algorithm is not longer/greater than t@?@d(u,v). This paper presents two extremely simple randomized algorithms for computing all-pairs nearly 2-approximate distances. The first algorithm requires an expected O(m^2^/^3nlogn+n^2) time, and for any u,v@?V reports a distance no greater than 2@d(u,v)+1. Our second algorithm requires an expected O(n^2log^3^/^2n) time, and for any u,v@?V reports a distance bounded by 2@d(u,v)+3.