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
O~(n3/2), 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
ℓ-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