Sparse Hopsets in Congested Clique

TL;DR

This paper introduces a new algorithm in the Congested Clique model that efficiently computes sparse hopsets with polylogarithmic hopbound in polylogarithmic time, improving over previous methods in sparsity and round complexity.

Contribution

It presents the first Congested Clique algorithm for sparse hopsets with polylogarithmic hopbound and time, and introduces an efficient neighborhood cover construction of independent interest.

Findings

Constructs sparse hopsets with polylogarithmic hopbound in polylogarithmic rounds.

Achieves sparser hopsets than recent constructions with similar hopbound.

Provides an improved hopset construction in a low-memory MPC model.

Abstract

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 $(\beta,\epsilon)$-hopset $H$ with "hopbound" $\beta$, 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 $\beta$ hops in $G \cup H$ with length within $(1+\epsilon)$ 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 $\tilde{O}(n^{3/2})$, but with a smaller polylogarithmic hopbound. On the other hand, the previously known constructions 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 $\ell$-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.