  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.
