Conditionally Optimal Parallel Coloring of Forests

TL;DR

This paper presents the first deterministic, conditionally optimal parallel algorithm for 3-coloring forests in the low-space MPC model, achieving $O(\log \log n)$ rounds with optimal space, and extends to maximal independent set and matching.

Contribution

It introduces a novel $O(\log \log n)$-round algorithm for partitioning forests into layers, enabling deterministic solutions for coloring, MIS, and matching.

Findings

Deterministic 3-coloring algorithm runs in $O(\log \log n)$ rounds.

New layered partition technique with each node having at most two neighbors in higher layers.

Algorithms for MIS and matching match the best known randomized results, but are deterministic and simpler.

Abstract

We show the first conditionally optimal deterministic algorithm for $3$-coloring forests in the low-space massively parallel computation (MPC) model. Our algorithm runs in $O(\log \log n)$ rounds and uses optimal global space. The best previous algorithm requires $4$ colors [Ghaffari, Grunau, Jin, DISC'20] and is randomized, while our algorithm are inherently deterministic. Our main technical contribution is an $O(\log \log n)$-round algorithm to compute a partition of the forest into $O(\log n)$ ordered layers such that every node has at most two neighbors in the same or higher layers. Similar decompositions are often used in the area and we believe that this result is of independent interest. Our results also immediately yield conditionally optimal deterministic algorithms for maximal independent set and maximal matching for forests, matching the state of the art [Giliberti, Fischer, Grunau, SPAA'23]. In contrast to their solution, our algorithms are not based on derandomization, and are arguably simpler.