The main results of this paper are (I) a simulation algorithm which, under
quite general constraints, transforms algorithms running on the Congested
Clique into algorithms running in the MapReduce model, and (II) a distributed
O(Δ)-coloring algorithm running on the Congested Clique which has an
expected running time of (i) O(1) rounds, if Δ≥Θ(log4n);
and (ii) O(loglogn) rounds otherwise. Applying the simulation theorem to
the Congested-Clique O(Δ)-coloring algorithm yields an O(1)-round
O(Δ)-coloring algorithm in the MapReduce model.
Our simulation algorithm illustrates a natural correspondence between
per-node bandwidth in the Congested Clique model and memory per machine in the
MapReduce model. In the Congested Clique (and more generally, any network in
the CONGEST model), the major impediment to constructing fast
algorithms is the O(logn) restriction on message sizes. Similarly, in the
MapReduce model, the combined restrictions on memory per machine and total
system memory have a dominant effect on algorithm design. In showing a fairly
general simulation algorithm, we highlight the similarities and differences
between these models.Comment: 15 page