Efficient Randomized Distributed Coloring in CONGEST

TL;DR

This paper introduces a new randomized distributed vertex coloring algorithm in the CONGEST model that significantly improves the round complexity for graphs with large maximum degree, achieving near-logarithmic bounds.

Contribution

It presents the first efficient CONGEST algorithm for $( ext{max degree}+1)$-list coloring with polylogarithmic rounds, surpassing previous results limited to the LOCAL model.

Findings

Achieves $( ext{max degree}+1)$-list coloring in $O( ext{log}^5 ext{log} n)$ rounds in CONGEST.

Improves upon the previous $O( ext{log} ext{max degree} + ext{log}^6 ext{log} n)$ rounds algorithm.

Demonstrates exponential speedup for large maximum degree graphs.

Abstract

Distributed vertex coloring is one of the classic problems and probably also the most widely studied problems in the area of distributed graph algorithms. We present a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an $n$-node graph $G$, and where the nodes of $G$ operate in synchronous communication rounds in which they can exchange $O(\log n)$-bit messages over all the edges of $G$. For graphs with maximum degree $\Delta$, we show that the $(\Delta+1)$-list coloring problem (and therefore also the standard $(\Delta+1)$-coloring problem) can be solved in $O(\log^5\log n)$ rounds. Previously such a result was only known for the significantly more powerful LOCAL model, where in each round, neighboring nodes can exchange messages of arbitrary size. The best previous $(\Delta+1)$-coloring algorithm in the CONGEST model had a running time of $O(\log\Delta + \log^6\log n)$ rounds. As a function of $n$ alone, the best previous algorithm therefore had a round complexity of $O(\log n)$, which is a bound that can also be achieved by a na\"{i}ve folklore algorithm. For large maximum degree $\Delta$, our algorithm hence is an exponential improvement over the previous state of the art.