Recently, studying fundamental graph problems in the Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. A standard assumption, common to most traditional approaches, is to allow Ω(n) memory per machine, where n is the number of nodes in the graph and Ω hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [arXiv:1707.03478], it might be unrealistic for a single machine to have linear or only slightly sublinear memory. In this paper, we propose the study of a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the standard MPC model and also streaming, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new 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(log 3 log n) rounds, when the input graph is a tree. This substantially reduces the local memory from n poly log n required by the recent O(log log n)-round MIS algorithm of Ghaffari et al. [PODC'18] to n ε , 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 exponentially improve on the corresponding bound in the LOCAL/PRAM models by Lenzen and Wattenhofer [PODC'11].
In REF , Fischer and Uitto presented an O((log log n) 2 ) rounds MPC algorithm for the maximal independent set problem (MIS) on trees.
Breaking the Linear-Memory Barrier in MPC: Fast MIS on Trees with n ε Memory per Machine.
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Mathematics Computer Science