Overcoming Congestion in Distributed Coloring

TL;DR

This paper introduces a novel distributed sampling technique that significantly improves the efficiency of graph coloring algorithms and local property testing in distributed networks, achieving near-optimal round complexities.

Contribution

The authors develop a new sampling and communication method that enhances distributed coloring algorithms and property testing, reducing round complexity in the CONGEST model.

Findings

D1LC can be solved in $O(\log^5 \log n)$ CONGEST rounds.

D1LC can be solved in $O(\log^* n)$ rounds for graphs with high minimum degree.

The technique enables constant-round local property testing and density estimation.

Abstract

We present a new technique to efficiently sample and communicate a large number of elements from a distributed sampling space. When used in the context of a recent LOCAL algorithm for $(\operatorname{degree}+1)$-list-coloring (D1LC), this allows us to solve D1LC in $O(\log^5 \log n)$ CONGEST rounds, and in only $O(\log^* n)$ rounds when the graph has minimum degree $\Omega(\log^7 n)$, w.h.p. The technique also has immediate applications in testing some graph properties locally, and for estimating the sparsity/density of local subgraphs in $O(1)$ CONGEST rounds, w.h.p.