Linear-size hopsets with small hopbound, and constant-hopbound hopsets in RNC

For a positive parameter β, the β-bounded distance between a pair of vertices u,v in a weighted undirected graph G = (V,E,ω) is the length of the shortest u −v path in G with at most β edges, aka hops. For β as above and ϵ > 0, a (β,ϵ)-hopset of G = (V,E,ω) is a graph G_H = (V,H,ω_H ) on the same vertex set, such that all distances in G are (1 + ϵ)-approximated by β-bounded distances in G ∪ G_H . 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 Ω(n log n) edges, or with a hopbound n^Ω(1⁾. In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on ϵ) (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 n^Ω(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.