Round Compression for Parallel Matching Algorithms

The last decade has witnessed the success of a number of \emph{massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. These frameworks allow for much more local memory and computation, compared to the classical distributed or PRAM models. In this context, we investigate the complexity of one of the most fundamental graph problems: \emph{Maximum Matching}. We show that if the memory per machine is $\Omega(n)$ (or even only $n/(\log n)^{O(\log \log n)}$), then for any fixed constant $\epsilon > 0$, one can compute a $(2+\epsilon)$-approximation to Maximum Matching in $O\left((\log \log n)^2\right)$ MPC rounds. This constitutes an exponential improvement over previous work---both the one designed specifically for MPC and the one obtained via connections to PRAM or distributed algorithms---which required $\Theta(\log n)$ rounds to achieve similar approximation guarantees. The starting point of our result is a (distributed) algorithm that computes an $O(1)$-approximation in $O(\log n)$ parallel phases. Its straightforward MPC implementation requires $\Theta(\log n)$ rounds. Our exponential speedup stems from compressing several phases of a modified version of this algorithm into a constant number of MPC rounds. We obtain this via a careful analysis of controlled randomness, which enables the emulation of multiple phases on separate machines without any coordination between them. We leave it as an intriguing open question whether a similar speedup can be obtained for other PRAM and distributed graph algorithms.