Sparsifying distributed algorithms with ramifications in massively parallel computation and centralized local computation

Abstract

We introduce a method for sparsifying distributed algorithms and exhibit how it leads to improvements that go past known barriers in two algorithmic settings of large-scale graph processing: Massively Parallel Computation (MPC), and Local Computation Algorithms (LCA).• MPC with Strongly Sublinear Memory: Recently, there has been growing interest in obtaining MPC algorithms that are faster than their classic O(log n)-round parallel (PRAM) counterparts for problems such as Maximal Independent Set (MIS), Maximal Matching, 2-Approximation of Minimum Vertex Cover, and (1 + e)-Approximation of Maximum Matching. Currently, all such MPC algorithms require memory of [MATH HERE] per machine: Czumaj et al. [STOC'18] were the first to handle [MATH HERE] memory, running in O((log log n)2) rounds, who improved on the n1+Ω(1) memory requirement of the O(1)-round algorithm of Lattanzi et al [SPAA'11]. We obtain [MATH HERE]-round MPC algorithms for all these four problems that work even when each machine has strongly sublinear memory, e.g., nα for any constant α ϵ (0, 1). Here, Δ denotes the maximum degree. These are the first sublogarithmic-time MPC algorithms for (the general case of) these problems that break the linear memory barrier.• LCAs with Query Complexity Below the Parnas-Ron Paradigm: Currently, the best known LCA for MIS has query complexity ΔO(log Δ) poly(log n), by Ghaffari [SODA'16], which improved over the ΔO(log2 Δ) poly(log n) bound of Levi et al. [Algorithmica'17]. As pointed out by Rubinfeld, obtaining a query complexity of poly(Δ log n) remains a central open question. Ghaffari's bound almost reaches a [MATH HERE] barrier common to all known MIS LCAs, which simulate a distributed algorithm by learning the full local topology, a la Parnas-Ron [TCS'07]. There is a barrier because the distributed complexity of MIS has a lower bound of [MATH HERE], by results of Kuhn, et al. [JACM'16], which means this methodology cannot go below query complexity [MATH HERE]. We break this barrier and obtain an LCA for MIS that has a query complexity ΔO(log log Δ) poly(log n).