Given a palette $P$ of at most $\xi$ colors, and a parameter $d$, a $(d,\xi)$-coloring of a graph is an assignment of a color from the palette $P$ to every node in the graph such that any two nodes at distance at most $d$ have different colors. We prove that for every $n$-node unit disk graph with maximum degree $\Delta$, there exists a distributed algorithm computing a $(1,O(\Delta))$-coloring under the SINR (Signal-to-Interference-plus-Noise Ratio) physical model in at most $O(\Delta \log n)$ time slots, which is optimal up to a logarithmic factor. Our result is based on revisiting a previous coloring algorithm, due to T. Moscibroda and R. Wattenhofer, described in the so called graph-based model~\cite{MW08}. We also prove that, for a well defined constant $d$, a $(d, O(\Delta))$-coloring allows us to schedule an interference free TDMA-like MAC protocol under the physical SINR constraints. As a corollary, any uniform interference-free message passing algorithm with running time $\tau$ can be simulated in the SINR model in $O(\Delta (\log n + \tau))$ time slots. The latter generic result provides new insights into the distributed scheduling of radio network tasks under the harsh SINR constraints.
['Bilel Derbel', 'El-Ghazali Talbi']
Distributed Node Coloring in the SINR Model