Sparse hopsets in congested clique

We give the first Congested Clique algorithm that computes a sparse hopset with polylogarithmic hopbound in polylogarithmic time. Given a graph G = (V, E), a (β, ε)-hopset H with “hopbound” β, is a set of edges added to G such that for any pair of nodes u and v in G there is a path with at most β hops in G ∪ H with length within (1 + ε) of the shortest path between u and v in G. Our hopsets are significantly sparser than the recent construction of Censor-Hillel et al. [6], that constructs a hopset of size Õ(n3/2)1, but with a smaller polylogarithmic hopbound. On the other hand, the previously known construction of sparse hopsets with polylogarithmic hopbound in the Congested Clique model, proposed by Elkin and Neiman [10, 11, 12], all require polynomial rounds. One tool that we use is an efficient algorithm that constructs an `-limited neighborhood cover, that may be of independent interest. Finally, as a side result, we also give a hopset construction in a variant of the low-memory Massively Parallel Computation model, with improved running time over existing algorithms.