Improved MPC Algorithms for MIS, Matching, and Coloring on Trees and Beyond

TL;DR

This paper introduces faster scalable MPC algorithms for maximal independent set, matching, and coloring on trees and graphs of bounded arboricity, improving previous results and approaching optimal bounds.

Contribution

It presents $O(\log\log n)$ round algorithms for these problems, significantly improving prior $O(\log^2\log n)$ algorithms, and matching known lower bounds for matching.

Findings

Achieved $O(\log\log n)$ round algorithms for MIS, matching, and coloring.

Improved previous algorithms' complexity bounds.

Matching algorithm's complexity is likely optimal.

Abstract

We present $O(\log\log n)$ round scalable Massively Parallel Computation algorithms for maximal independent set and maximal matching, in trees and more generally graphs of bounded arboricity, as well as for constant coloring trees. Following the standards, by a scalable MPC algorithm, we mean that these algorithms can work on machines that have capacity/memory as small as $n^{\delta}$ for any positive constant $\delta<1$. Our results improve over the $O(\log^2\log n)$ round algorithms of Behnezhad et al. [PODC'19]. Moreover, our matching algorithm is presumably optimal as its bound matches an $\Omega(\log\log n)$ conditional lower bound of Ghaffari, Kuhn, and Uitto [FOCS'19].