A Fast Algorithm for Source-wise Round-trip Spanners release_bj2go2mo2fdybbrvkndf7su6pu

by Chun Jiang Zhu, Song Han, Kam-Yiu Lam

Released as a article .

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 S⊆ 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∈ S× 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 S⊆ V and an integer k>1, there exists an algorithm that in time O(ms^1/klog^5n) constructs an S-sourcewise round-trip spanner of stretch O(klog n) and O(ns^1/klog^2n) edges with high probability. Compared to the fast algorithms for constructing all-pairs round-trip spanners <cit.>, our algorithm improve 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 <cit.>, our algorithm significantly improves their construction time Ω(min{ms,n^ω}) (where ω∈ [2,2.373) and 2.373 is the matrix multiplication exponent) to nearly linear O(ms^1/klog^5n), at the expense of paying an extra O(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,v∈ S× 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 <cit.> can be adapted to the source-wise setting.
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