Coloring Trees in Massively Parallel Computation

TL;DR

This paper introduces deterministic algorithms for 3-coloring, maximal independent set, and maximal matching in trees within the low-space MPC model, achieving faster runtimes than previous randomized methods.

Contribution

It presents the first deterministic low-space MPC algorithms for 3-coloring and related problems on trees, improving runtime and applicability over prior randomized algorithms.

Findings

3-coloring algorithm runs in $O( ext{log}^2 ext{log} n)$ time.

Maximal independent set and matching algorithms run in $O(1)$ time after coloring.

For constant-degree trees, runtime improves to $O( ext{log} ext{log} n)$.

Abstract

We present $O(\log^2 \log n)$ time 3-coloring, maximal independent set and maximal matching algorithms for trees in the Massively Parallel Computation (MPC) model. Our algorithms are deterministic, apply to arbitrary-degree trees and work in the low-space MPC model, where local memory is $O(n^\delta)$ for $\delta \in (0,1)$ and global memory is $O(m)$. Our main result is the 3-coloring algorithm, which contrasts the randomized, state-of-the-art 4-coloring algorithm of Ghaffari, Grunau and Jin [DISC'20]. The maximal independent set and maximal matching algorithms follow in $O(1)$ time after obtaining the coloring. The key ingredient of our 3-coloring algorithm is an $O(\log^2 \log n)$ time adaptation of the rake-and-compress tree decomposition used by Chang and Pettie [FOCS'17], and established by Miller and Reif. When restricting our attention to trees of constant degree, we bring the runtime down to $O(\log \log n)$.