Fast Distributed Brooks' Theorem

TL;DR

This paper presents a fast randomized distributed algorithm for graph coloring that operates in polylog log n rounds, matching known lower bounds for high-degree graphs, and introduces a novel reduction technique for high-degree graph coloring.

Contribution

It introduces a new reduction method for high-degree graphs, enabling polylog log log n round algorithms for distributed coloring, and provides the first CONGEST algorithm for non-constant degree graphs.

Findings

Achieves polylog log n round complexity for Δ-coloring in the LOCAL model.

Provides a new proof of Brooks' theorem for high degree graphs.

Develops the first CONGEST algorithm for Δ-coloring in non-constant degree graphs.

Abstract

We give a randomized $\Delta$-coloring algorithm in the LOCAL model that runs in $\text{poly} \log \log n$ rounds, where $n$ is the number of nodes of the input graph and $\Delta$ is its maximum degree. This means that randomized $\Delta$-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, $\text{poly}\log\log n$, given the known $\Omega(\log_\Delta\log n)$ lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of $(\text{deg}+1)$-list coloring instances, for $\Delta = \omega(\log^4 n)$, resulting in a $\text{poly} \log\log n$-round CONGEST algorithm for such graphs. This reduction is of independent interest for other settings, including providing a new proof of Brooks' theorem for high degree graphs, and leading to a constant-round Congested Clique algorithm in such graphs. When $\Delta=\omega(\log^{21} n)$, our algorithm even runs in $O(\log^* n)$ rounds, showing that the base in the $\Omega(\log_\Delta\log n)$ lower bound is unavoidable. Previously, the best LOCAL algorithm for all considered settings used a logarithmic number of rounds. Our result is the first CONGEST algorithm for $\Delta$-coloring non-constant degree graphs.