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 a very 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, multi-broadcast and gossiping, and also, as intermediary results, means of diameter estimation and depth-first search. Our $O(D+m)$-time algorithm for broadcasting an $m$-length message is a simple formalization of a concept known as beep waves, and is asymptotically optimal. We give an $O(D\log L\cdot (1 + \frac{k}{\log D})+M)$-time algorithm for $k$-multi-broadcast, that is, broadcast performed from $k$ different sources. This takes more time than $k$ single broadcasts because we must agree on an order in which to perform them, so as to avoid interference. Finally, we present our most sophisticated result, an $O(n\log L + M)$-time gossiping algorithm. This algorithm is optimal in all cases where messages are large enough to contain node labels. In these running-time expressions, $D$ represents network diameter, $L$ represents range of node labels, and $M$ is the total sum of message lengths (in bits). Our algorithms are all deterministic, explicit, and practical, and give efficient means of communication while making arguably the minimum possible assumptions about the network.