Global Synchronization and Consensus Using Beeps in a Fault-Prone Multiple Access Channel

Abstract

Global synchronization is an important prerequisite to many distributed tasks. Communication between processors proceeds in synchronous rounds. Processors are woken up in possibly different rounds. The clock of each processor starts in its wakeup round showing local round 0, and ticks once per round, incrementing the value of the local clock by one. The global round 0, unknown to processors, is the wakeup round of the earliest processor. Global synchronization (or establishing a global clock) means that each processor chooses a local clock round such that their chosen rounds all correspond to the same global round t.We study the task of global synchronization in a Multiple Access Channel (MAC) prone to faults, under a very weak communication model called the beeping model. 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 ϵ, for a given ϵ>0, is called ϵ-safe. Our main result is the design and analysis, for any ϵ>0, of a deterministic ϵ-safe global synchronization algorithm that works in O(logp2⁡ϵ) time in any fault-prone MAC using beeps.As an application, we solve the consensus problem in a fault-prone MAC using beeps. 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. Using global synchronization, we give a deterministic ϵ-safe consensus algorithm that works in time O((log⁡w)(logp⁡ϵ)) in a fault-prone MAC, where w is the smallest input value of all participating processors. We show that this time cannot be significantly improved, even when the MAC is fault-free.