Global Synchronization and Consensus Using Beeps in a Fault-Prone MAC

Consensus is one of the fundamental tasks studied in distributed computing. Processors have input values from some set $V$ and they have to decide the same value from this set. If all processors have the same input value, then they must all decide this value. We study the task of consensus in a Multiple Access Channel (MAC) prone to faults, under a very weak communication model called the $\mathit{beeping\ model}$. Communication proceeds in synchronous rounds. Some processors wake up spontaneously, in possibly different rounds decided by an adversary. In each round, an awake processor can either listen, i.e., stay silent, or beep, i.e., emit a signal. In each round, a fault can occur in the channel independently with constant probability $0<p<1$. In a fault-free round, an awake processor hears a beep if it listens in this round and if one or more other processors beep in this round. A processor still dormant in a fault-free round in which some other processor beeps is woken up by this beep and hears it. In a faulty round nothing is heard, regardless of the behaviour of the processors. An algorithm working with error probability at most $\epsilon$, for a given $\epsilon>0$, is called $\epsilon$-$\mathit{safe}$. Our main result is the design and analysis, for any constant $\epsilon>0$, of a deterministic $\epsilon$-safe consensus algorithm that works in time $O(\log w)$ in a fault-prone MAC, where $w$ is the smallest input value of all participating processors. We show that this time cannot be improved, even when the MAC is fault-free. The main algorithmic tool that we develop to achieve our goal, and that might be of independent interest, is a deterministic algorithm that, with arbitrarily small constant error probability, establishes a global clock in a fault-prone MAC in constant time

Distributed Voting in Beep Model

We consider the problem of distributed multi-choice voting in a setting that each node can communicate with its neighbors merely by sending beep signals. Given its simplicity, the beep communication model is of practical importance in different applications such as system biology and wireless sensor networks. Yet, the distributed majority voting has not been resolved in this setting. In this paper, we propose two algorithms, named Distributed Voting with Beeps, to resolve this problem. In the first proposed algorithm, the adjacent nodes having the same value form a set called spot. Afterwards, the spots with majority value try to corrode the spots with non-majority values. The second proposed algorithm is based on pairwise interactions between nodes. The proposed algorithms have a termination detection procedure to check whether voting is achieved. We establish theoretical guarantees for the convergence of these algorithms. In particular, we show that the success probability of the first algorithm tends to one as the percentage of the initial votes in majority increases. For the second algorithm, we show that it returns the correct output with high probability. Our experiments show that the algorithms are fairly invariant to the network topology, initial distribution of values, and network size.Comment: Published in Signal Processing journal, Elsevier, 202