Linear-Size Hopsets with Small Hopbound, and Constant-Hopbound Hopsets in RNC

Abstract

For a positive parameter β, the β-bounded distance between a pair of vertices u,v in a weighted undirected graph G = (V,E,omega) is the length of the shortest u-v path in G with at most β edges, aka hops. For β as above and e > 0, a (β,e)-hopset of G = (V,E,omega) is a graph GH =(V,H,omegaH) on the same vertex set, such that all distances in G are (1+e)-approximated by β-bounded distances in G ∪ GH. Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with Omega(n log n) edges, or with a hopbound nOmega(1). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on e) (log log n)log log n + O(1). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [11] for computing hopsets with a constant (i.e., independent of n) hopbound requires nOmega(1) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [11]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.