Ultrafast Distributed Coloring of High Degree Graphs

TL;DR

This paper introduces a simplified, fast randomized distributed algorithm for coloring high-degree graphs, achieving near-optimal rounds in the CONGEST model, with implications for understanding complexity dependencies.

Contribution

It presents a new simplified randomized algorithm for $ ext{Delta}+1$-list coloring that outperforms previous methods and works efficiently in the CONGEST model.

Findings

Colors all $n$-node graphs with maximum degree $ig( ext{log}^{2+ ext{Omega}(1)} nig)$ in $O( ext{log}^* n)$ rounds.

Shatters low-degree graphs into small components in $O( ext{log}^* ext{Delta})$ rounds.

Shows the randomized complexity depends on the deterministic complexity of related coloring problems.

Abstract

We give a new randomized distributed algorithm for the $\Delta+1$-list coloring problem. The algorithm and its analysis dramatically simplify the previous best result known of Chang, Li, and Pettie [SICOMP 2020]. This allows for numerous refinements, and in particular, we can color all $n$-node graphs of maximum degree $\Delta \ge \log^{2+\Omega(1)} n$ in $O(\log^* n)$ rounds. The algorithm works in the CONGEST model, i.e., it uses only $O(\log n)$ bits per message for communication. On low-degree graphs, the algorithm shatters the graph into components of size $\operatorname{poly}(\log n)$ in $O(\log^* \Delta)$ rounds, showing that the randomized complexity of $\Delta+1$-list coloring in CONGEST depends inherently on the deterministic complexity of related coloring problems.