Counting in one-hop beeping networks

Abstract

We consider networks of processes which interact with beeps. In the basic model defined by Cornejo and Kuhn [7], which we refer to as the BL variant, processes can choose in each round either to beep or to listen. Those who beep are unable to detect simultaneous beeps. Those who listen can only distinguish between silence and the presence of at least one beep. Beeping models are weak in essence and even simple tasks may become difficult or unfeasible with them. In this paper, we address the problem of computing how many participants there are in a one-hop network: the counting problem. We first observe that no algorithm can compute this number with certainty in BL, whether the algorithm be deterministic or even randomised (Las Vegas). We thus consider the stronger variant where beeping nodes are able to detect simultaneous beeps, referred to as B cd L (for collision detection). We prove that at least n rounds are necessary in B cd L, and we present an algorithm whose running time is O(n) rounds with high probability. Further experimental results show that its expected running time is less than 10n. Finally, we discuss how this algorithm can be adapted in other beeping models. In particular, we show that it can be emulated in BL, at the cost of a logarithmic slowdown and of trading its Las Vegas nature (result certain, time uncertain) against Monte Carlo (time certain, result uncertain).

Contention Resolution with Constant Throughput and Log-Logstar Channel Accesses

“…This aligns with the beeping model introduced by Cornejo and Kuhn [34]. Under this model, both leader election [42] and counting [24,21] have been investigated.…”

Communication Complexity of Wait-Free Computability in Dynamic Networks

“…Many interesting avenues are open to explore in future work. It would be interesting to consider wait-free solvability in beeping models like in [5], which are weaker than ours, because in those models, when A beeps at the same round than B, neither knows if the other beeped. Our results for two processes apply directly to the iterated shared memory model [18], as in this case the models are essentially the same.…”

Design patterns in beeping algorithms: Examples, emulation, and analysis

“…A similar doubling/halving pattern was used in [42]. One could also increment or decrement the denominator of p as done in [9]. The consequences of choosing one over the other are not discussed in this paper.…”

“…Other general strategies exist, such as incrementing or decrementing the denominator of p (see e.g. [9]), they are not considered here. In the third analysis (which is less generic), we use part of the input as an ingredient for the update of the adaptive probability.…”

“…The basic model introduced by Cornejo and Kuhn in [13] is BL; the model considered by Afek et al in [1] (Section 6) and Jeavons et al in [22] is B cd L; and the model considered in [41] and in Section 4 of [1] is BL cd . In [9], the authors consider the B cd L cd model along with all others and their impact on the counting problem. While some of these variants are stronger than others, all of them remain weak in essence.…”