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Article Dans Une Revue Theoretical Computer Science Année : 2023

Distributed coloring and the local structure of unit-disk graphs

Résumé

Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, 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 $4\omega(G)$ colors, 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 \textsf{LOCAL} model that colors every unit-disk graph $G$ with at most $5.68\,\omega(G)+1$ colors 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 (approximately) $4\omega(G)$ colors in the \textsf{LOCAL} model. We provide partial results towards this conjecture using Fourier-analytical tools.

Dates et versions

hal-03269425 , version 1 (24-06-2021)

Identifiants

Citer

Louis Esperet, Sébastien Julliot, Arnaud de Mesmay. Distributed coloring and the local structure of unit-disk graphs. Theoretical Computer Science, 2023, 944, pp.113674. ⟨10.1016/j.tcs.2022.12.024⟩. ⟨hal-03269425⟩
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