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Breaking the Linear-Memory Barrier in \(\mathsf {MPC}\): Fast \(\mathsf {MIS}\) on Trees with Strongly Sublinear Memory

  • Sebastian Brandt
  • Manuela FischerEmail author
  • Jara Uitto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11639)

Abstract

Recently, studying fundamental graph problems in the Massively Parallel Computation (\(\mathsf {MPC}\)) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow \(\widetilde{\varOmega }(n)\) memory per machine, where n is the number of nodes in the graph and \(\widetilde{\varOmega }\) 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 \(\mathsf {MPC}\) model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory \(\mathsf {MPC}\) model and also to streaming algorithms, in this low-memory \(\mathsf {MPC}\) 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 (\(\mathsf {MIS}\)) problem can be solved efficiently, that is, in \(O(\log ^3 \log n)\) rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory \(\mathsf {MPC}\) algorithm in \(\widetilde{O}(\sqrt{\log n})\)-algorithm in a concurrent work by Ghaffari and Uitto [SODA’19] and substantially reduces the local memory from \(\widetilde{\varOmega }(n)\) required by the recent \(O(\log \log n)\)-round \(\mathsf {MIS}\) 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].

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sebastian Brandt
    • 1
  • Manuela Fischer
    • 1
    Email author
  • Jara Uitto
    • 1
    • 2
  1. 1.ETH ZurichZurichSwitzerland
  2. 2.University of FreiburgFreiburg im BreisgauGermany

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