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

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 counterparts for problems such as MIS, Maximal Matching, 2-Approximation of Minimum Vertex Cover, and $(1+ϵ)$-Approximation of Maximum Matching. Currently, all such MPC algorithms require $\tilde{\mathrm{\Omega }}(n)$ memory per machine. Czumaj et al. [STOC’18] were the first to handle $\tilde{\mathrm{\Omega }}(n)$ memory, running in $O((\log \log n{)}^2)$ rounds. We obtain $\tilde{O}(\sqrt{\log \mathrm{\Delta }})$-round MPC algorithms for all these four problems that work even when each machine has memory $n^{\alpha }$ for any constant $\alpha \in (0,1)$. Here, $\mathrm{\Delta }$ denotes the maximum degree. These are the first sublogarithmic-time algorithms for 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 $\Delta^{O(\log \Delta)} poly(\log n)$, by Ghaffari [SODA’16]. As pointed out by Rubinfeld, obtaining a query complexity of $poly(\mathrm{\Delta }\log n)$ remains a central open question. Ghaffari’s bound almost reaches a $\Delta^{\Omega\left(\frac{\log \Delta}{\log\log \Delta}\right)}$ barrier common to all known MIS LCAs, which simulate distributed algorithms by learning the local topology, `{a} la Parnas-Ron [TCS’07]. This barrier follows from the $\Omega(\frac{\log \Delta}{\log\log \Delta})$ distributed lower bound of Kuhn, et al. [JACM’16]. We break this barrier and obtain an MIS LCA with query complexity $\Delta^{O(\log\log \Delta)} poly(\log n)$.