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Recently, studying fundamental graph problems in the \emph{Massively ParallelComputation (MPC) framework, inspired by the MapReduce paradigm, has gained alot of attention. An assumption common to a vast majority of approaches is toallow $\widetilde{\Omega}(n)$ memory per machine, where $n$ is the number ofnodes 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 onlyslightly sublinear memory.  In this paper, we thus study a more practical variant of the MPC model whichonly requires substantially sublinear or even subpolynomial memory per machine.In contrast to the linear-memory MPC model and also to streaming algorithms, inthis low-memory MPC setting, a single machine will only see a small number ofnodes in the graph. We introduce a new and strikingly simple technique to copewith this imposed locality.  In particular, we show that the Maximal Independent Set (MIS) problem can besolved efficiently, that is, in $O(\log^3 \log n)$ rounds, when the input graphis a tree. This constitutes an almost exponential speed-up over the low-memoryMPC algorithm in $O(\sqrt{\log n})$-algorithm in a concurrent work by Ghaffariand Uitto [SODA'19] and substantially reduces the local memory from$\widetilde{\Omega}(n)$ required by the recent $O(\log \log n)$-round MISalgorithm of Ghaffari et al. [PODC'18] to $n^{\alpha}$ for any $\alpha>0$,without incurring a significant loss in the round complexity. Moreover, itdemonstrates how to make use of the all-to-all communication in the MPC modelto almost exponentially improve on the corresponding bound in the$\mathsf{LOCAL}$ and $\mathsf{PRAM}$ models by Lenzen and Wattenhofer[PODC'11].

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