Sparsifying Distributed Algorithms with Ramifications in Massively Parallel Computation and Centralized Local Computation

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 + ∊)-Approximation of Maximum Matching. Currently, all such MPC algorithms require memory of per machine: Czumaj et al. [STOC'18] were the first to handle memory, running in O((log log n)²) rounds, who improved on the n^1+Ω(1) memory requirement of the O(1)-round algorithm of Lattanzi et al [SPAA'11]. We obtain -round MPC algorithms for all these four problems that work even when each machine has strongly sublinear memory, e.g., n^α for any constant α ∊ (0, 1). Here, Δ denotes the maximum degree. These are the first sublogarithmictime 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 Δ^{O(log Δ)} poly(log n), by Ghaffari [SODA'16], which improved over the Δ^{O(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 barrier common to all known MIS LCAs, which sim-ulate a distributed algorithm by learning the full local topology, à la Parnas-Ron [TCS'07]. There is a barrier because the distributed complexity of MIS has a lower bound of , by results of Kuhn, et al. [JACM'16], which means this methodology cannot go below query complexity . We break this barrier and obtain an LCA for MIS that has a query complexity Δ^{O(log log Δ)} poly(log n).