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.