Lessons from the Congested Clique applied to MapReduce

The main result of this paper is a simulation algorithm which, under quite general constraints, transforms algorithms running in the Congested Clique model into algorithms running in the MapReduce model. As a case study of applying this simulation algorithm, we first present a distributed O(Δ)O(Δ)-coloring algorithm running on the Congested Clique in O(1)O(1) rounds, if Δ≥log2⁡nΔ≥log2⁡n, and O(log⁡log⁡log⁡n)O(log⁡log⁡log⁡n) rounds otherwise. Applying the simulation theorem to this Congested Clique O(Δ)O(Δ)-coloring algorithm yields an O(1)O(1)-round O(Δ)O(Δ)-coloring algorithm in the MapReduce model. We apply our simulation algorithm to other Congested Clique algorithms including the 2-ruling set and metric facility location algorithms of Berns et al. (ICALP 2012).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 CONGESTCONGEST model), the major impediment to constructing fast algorithms is the O(log⁡n)O(log⁡n) 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.