A fast algorithm for source-wise round-trip spanners

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Detail(s)

Original languageEnglish
Pages (from-to)34-44
Journal / PublicationTheoretical Computer Science
Volume876
Online published24 May 2021
Publication statusPublished - 12 Jul 2021

Abstract

In this paper, we study the problem of fast constructions of source-wise round-trip spanners in weighted directed graphs. For a source vertex set V in a graph G(V, E), an S-sourcewise round-trip spanner of G of stretch k is a subgraph H of G such that for every pair of vertices u, × V, their round-trip distance in H is at most k times of their round-trip distance in G. We show that for a graph G(V, E) with n vertices and m edges, an s-sized source vertex set V and an integer > 1, there exists an algorithm that in time O (ms1/log5⁡ n) constructs an S-sourcewise round-trip spanner of stretch O (log ⁡n) and O (ns1/log⁡n) edges with high probability. Compared to the fast algorithms for constructing all-pairs round-trip spanners [26,12], our algorithm improves the running time and the number of edges in the spanner when k is super-constant. Compared with the existing algorithm for constructing source-wise round-trip spanners [36], our algorithm significantly improves their construction time Ω(min⁡{ms, nω}) (where ω ∈ [2, 2.373) and 2.373 is the matrix multiplication exponent) to nearly linear O (ms1/log5n), at the expense of paying an extra (log⁡ n) in the stretch. As an important building block of the algorithm, we develop a graph partitioning algorithm to partition G into clusters of bounded radius and prove that for every u, vS × V at small round-trip distance, the probability of separating them in different clusters is small. The algorithm takes the size of S as input and does not need the knowledge of S. With the algorithm and a reachability vertex size estimation algorithm, we show that the recursive algorithm for constructing standard round-trip spanners [26] can be adapted to the source-wise setting. We rigorously prove the correctness and computational complexity of the adapted algorithms. Finally, we show how to remove the dependence on the edge weight in the source-wise case.

Research Area(s)

  • Graph algorithms, Graph partitioning, Graph spanners, Round-trip spanners

Citation Format(s)

A fast algorithm for source-wise round-trip spanners. / Zhu, Chun Jiang; Han, Song; Lam, Kam-Yiu.
In: Theoretical Computer Science, Vol. 876, 12.07.2021, p. 34-44.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review