Beeping a Maximal Independent Set Fast

We adapt a recent algorithm by Ghaffari [SODA'16] for computing a Maximal Independent Set in the LOCAL model, so that it works in the significantly weaker BEEP model. For networks with maximum degree $\Delta$, our algorithm terminates locally within time $O((\log \Delta + \log (1/\epsilon)) \cdot \log(1/\epsilon))$, with probability at least $1 - \epsilon$. The key idea of the modification is to replace explicit messages about transmission probabilities with estimates based on the number of received messages. After the successful introduction (and implicit use) of local analysis, e.g., by Barenboim et al. [JACM'16], Chung et al. [PODC'14], Ghaffari [SODA'16], and Halldorsson et al. [PODC'15], we study this concept in the BEEP model for the first time. By doing so, we improve over local bounds that are implicitly derived from previous work (that uses traditional global analysis) on computing a Maximal Independent Set in the \beep model for a large range of values of the parameter $\Delta$. At the same time, we show that our algorithm in the \beep model only needs to pay a $\log(1/\epsilon)$ factor in the runtime compared to the best known MIS algorithm in the much more powerful \local model. We demonstrate that this overhead is negligible, as communication via beeps can be implemented using significantly less resources than communication in the LOCAL model. In particular, when looking at implementing these models, one round of the \local model needs at least $O(\Delta)$ time units, while one round in the BEEP model needs $O(\log\Delta)$ time units, an improvement that diminishes the loss of a $\log(1/\epsilon)$ factor in most settings.