  In this work, we present a constant-round algorithm for the $2$-ruling set
problem in the Congested Clique model. As a direct consequence, we obtain a
constant round algorithm in the MPC model with linear space-per-machine and
optimal total space. Our results improve on the $O(\log \log \log n)$-round
algorithm by [HPS, DISC'14] and the $O(\log \log \Delta)$-round algorithm by
[GGKMR, PODC'18]. Our techniques can also be applied to the semi-streaming
model to obtain an $O(1)$-pass algorithm.
  Our main technical contribution is a novel sampling procedure that returns a
small subgraph such that almost all nodes in the input graph are adjacent to
the sampled subgraph. An MIS on the sampled subgraph provides a $2$-ruling set
for a large fraction of the input graph. As a technical challenge, we must
handle the remaining part of the graph, which might still be relatively large.
We overcome this challenge by showing useful structural properties of the
remaining graph and show that running our process twice yields a $2$-ruling set
of the original input graph with high probability.
