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

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 ) (loglog n)^loglog 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 [ 19 ] 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 [ 19 ]. 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.