All-pairs nearly 2-approximate shortest-paths in O(n2 polylog n) time

Authors:
Surender Baswana;Vishrut Goyal;Sandeep Sen
Affiliations:
Max-Planck-Institut für Informatik, Saarbrücken, Germany;Persistent Systems Private Limited, Pune, India;Department of Computer Science and Engineering, I.I.T., Kharagpur, India
Venue:
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Year:
2005

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Abstract

Let G(V,E) be an unweighted undirected graph on |V | = n vertices. Let δ(u,v) denote the shortest 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 · δ(u,v). This paper presents two randomized algorithms for computing all-pairs nearly 2-approximate distances. The first algorithm takes expected O(m2/3n log n + n2) time, and for any u,v ∈ V reports distance no greater than 2δ(u,v) + 1. Our second algorithm requires expected O(n2 log3/2) time, and for any u,v ∈ V reports distance bounded by 2δ(u,v)+3. This paper also presents the first expected O(n2) time algorithm to compute all-pairs 3-approximate distances.