Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space

TL;DR

This paper introduces a deterministic graph sparsification method to derandomize parallel algorithms, enabling efficient solutions for Maximal Matching and Independent Set in low-space MPC models, advancing the state-of-the-art in distributed graph algorithms.

Contribution

It presents a novel deterministic sparsification technique that derandomizes existing randomized MPC algorithms for fundamental graph problems under low space constraints.

Findings

Deterministic MPC algorithms for Maximal Matching and Independent Set with O(log Δ + log log n) rounds.

The algorithms operate with O(n^ε) space per machine for any constant ε > 0.

Improved round complexity over previous randomized and deterministic algorithms, matching conditional lower bounds.

Abstract

The Massively Parallel Computation (MPC) model is an emerging model which distills core aspects of distributed and parallel computation. It has been developed as a tool to solve (typically graph) problems in systems where the input is distributed over many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in $n$, the number of nodes in the input graph) memory, with randomized algorithms presented for fundamental graph problems of Maximal Matching and Maximal Independent Set. However, there have been no prior corresponding \emph{deterministic} algorithms. A major challenge underlying the sublinear space setting is that the local space of each machine might be too small to store all the edges incident to a single node. This poses a considerable obstacle compared to the classical models in which each node is assumed to know and have easy access to its incident edges. To overcome this barrier we introduce a new \emph{graph sparsification technique} that \emph{deterministically} computes a low-degree subgraph with additional desired properties. Using this framework to derandomize the well-known randomized algorithm of Luby [SICOMP'86], we obtain $O(\log \Delta+\log\log n)$-round \emph{deterministic} MPC algorithms for solving the fundamental problems of \emph{Maximal Matching} and \emph{Maximal Independent Set} with $O(n^{\epsilon})$ space on each machine for any constant $\epsilon > 0$. Based on the recent work of Ghaffari et al. [FOCS'18], this additive $O(\log\log n)$ factor is \emph{conditionally} essential. These algorithms can also be shown to run in $O(\log \Delta)$ rounds in the closely related model of \congc, improving upon the state-of-the-art bound of $O(\log^2 \Delta)$ rounds by Censor-Hillel et al. [DISC'17].