Breaking the Linear-Memory Barrier in MPC: Fast MIS on Trees with Strongly Sublinear Memory

Recently, studying fundamental graph problems in the \emph{Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow $\widetilde{\Omega}(n)$ memory per machine, where $n$ is the number of nodes in the graph and $\widetilde{\Omega}$ hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [STOC'18], it might be unrealistic for a single machine to have linear or only slightly sublinear memory. In this paper, we thus study a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory MPC model and also to streaming algorithms, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new and strikingly simple technique to cope with this imposed locality. In particular, we show that the Maximal Independent Set (MIS) problem can be solved efficiently, that is, in $O(\log³ \log n)$ rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory MPC algorithm in $O(\sqrt{\log n})$-algorithm in a concurrent work by Ghaffari and Uitto [SODA'19] and substantially reduces the local memory from $\widetilde{\Omega}(n)$ required by the recent $O(\log \log n)$-round MIS algorithm of Ghaffari et al. [PODC'18] to $n^{\alpha}$ for any $\alpha>0$, without incurring a significant loss in the round complexity. Moreover, it demonstrates how to make use of the all-to-all communication in the MPC model to almost exponentially improve on the corresponding bound in the $\mathsf{LOCAL}$ and $\mathsf{PRAM}$ models by Lenzen and Wattenhofer [PODC'11].