Communicating with Beeps

The beep model is a very weak communications model in which devices in a network can communicate only via beeps and silence. As a result of its weak assumptions, it has broad applicability to many different implementations of communications networks. This comes at the cost of a restrictive environment for algorithm design. Despite being only recently introduced, the beep model has received considerable attention, in part due to its relationship with other communication models such as that of ad-hoc radio networks. However, there has been no definitive published result for several fundamental tasks in the model. We aim to rectify this with our paper. We present algorithms for the tasks of broadcast, gossiping, and multi-broadcast. Our $O(D+\log M)$-time algorithm for broadcasting is a simple formalization of a concept known as beep waves, and is asymptotically optimal. We give an $O(n \log L)$-time depth-first search procedure, and show how this can be used as the basis for an $O(n \log LM)$-time gossiping algorithm. Finally, we present almost optimal algorithms for the more general problem of multi-broadcast. When message provenance is required, we give an $O(k \log \frac{LM}{k}+D \log L)$-time algorithm and a corresponding $\Omega(k \log \frac{LM}{k}+D)$ lower bound. When provenance is not required, we give an algorithm taking $O(k \log \frac {M}{k}+D \log L)$ time when $M>k$ and $O(M+D \log L)$ otherwise, and a corresponding lower bound of $\Omega(k \log \frac{M}{k}+D)$ when $M>k$ and $\Omega(M+D)$ otherwise. Our algorithms are all explicit, deterministic, and practical, and give efficient means of communication while making arguably the minimum possible assumptions about the network.