Distributed Coloring and the Local Structure of Unit-Disk Graphs

Abstract

Coloring unit-disk graphs efficiently is an important problem in the global and distributed settings, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In this context it is important to bound not only the complexity of the coloring algorithms, but also the number of colors used. In this paper, we consider two natural distributed settings. In the location-aware setting (when nodes know their coordinates in the plane), we give a constant time distributed algorithm coloring any unit-disk graph G with at most $(3+\epsilon )\omega (G)+6$ colors, for any constant $\epsilon >0$, where $\omega (G)$ is the clique number of G. This improves upon a classical 3-approximation algorithm for this problem, for all unit-disk graphs whose chromatic number significantly exceeds their clique number. When nodes do not know their coordinates in the plane, we give a distributed algorithm in the LOCAL model that colors every unit-disk graph G with at most $5.68\omega (G)$ colors in $O(\log ^3 \log n)$ rounds. Moreover, when $\omega (G)=O(1)$, the algorithm runs in $O(\log ^* n)$ rounds. This algorithm is based on a study of the local structure of unit-disk graphs, which is of independent interest. We conjecture that every unit-disk graph G has average degree at most $4\omega (G)$, which would imply the existence of a $O(\log n)$ round algorithm coloring any unit-disk graph G with (approximatively) $4\omega (G)$ colors.

The authors are partially supported by ANR Projects GATO (anr-16-ce40-0009-01), GrR (anr-18-ce40-0032), MIN-MAX (anr-19-ce40-0014) and SoS (anr-17-CE40-0033), and by LabEx PERSYVAL-lab (anr-11-labx-0025). The full version of the paper is available on arXiv [7]. It contains some proofs that are omitted in this version together with an additional appendix discussing a Fourier analytical approach toward Conjecture 1.