Linear-Size Hopsets with Small Hopbound, and Distributed Routing with Low Memory

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 ǫ > 0, a (β, ǫ)-hopset of G = (V, E, ω) is a graph G′ = (V, H, ω_H) on the same vertex set, such that all distances in G are (1 +ǫ)-approximated by β-bounded distances in G∪G′ . 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 ǫ) (log _n) ^{log(3) n+O(1)} . This improves the previous bound almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [EN16a] 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 [EN16a]. We use our hopsets to devise a distributed routing scheme that exhibits near-optimal tradeoff between individual memory requirement O˜(n ^1/k) of vertices throughout preprocessing and routing phases of the algorithm, and stretch O(k), along with a near-optimal construction time ≈ D + n ^1/2+1/k, where D is the hop-diameter of the input graph. Previous distributed routing algorithms either suffered from a prohibitively large memory req