Deterministic Rendezvous with Detection Using Beeps

Abstract

Two mobile agents, starting at possibly different times from arbitrary nodes of an unknown network, have to meet at some node. Agents move in synchronous rounds. They have different positive integer labels. Each agent knows its own label but not the label of the other agent. In traditional formulations of the rendezvous problem, meeting is accomplished when agents get to the same node in the same round. We seek a more demanding goal, called rendezvous with detection: agents must become aware that the meeting is accomplished, simultaneously declare this and stop. This awareness depends on how agents communicate. We use two variations of a very weak communication model, called the beeping model, introduced in [8]. In each round an agent either listens or beeps. In the local beeping model, an agent hears a beep if it listens in this round and if the other agent is at the same node and beeps. In the global beeping model, an agent hears a loud (resp. a soft) beep if it listens in this round and if the other agent is at the same node (resp. at another node) and beeps.

We first present a deterministic algorithm of rendezvous with detection working, even for the local beeping model, in an arbitrary unknown network in time polynomial in the size of the network and in the length of the smaller label (i.e., in the logarithm of this label). However, in this algorithm, agents spend a lot of energy: the number of moves that an agent must make, is proportional to the time of rendezvous. It is thus natural to ask if bounded-energy agents, i.e., agents that can make at most c moves, for some integer c, can always achieve rendezvous with detection as well, in bounded size networks. We prove that the answer to this question is positive, even in the local beeping model but, perhaps surprisingly, this ability comes at a steep price of time: the meeting time of bounded-energy agents is exponentially larger than that of unrestricted agents. By contrast, we show an algorithm for rendezvous with detection in the global beeping model that works for bounded-energy agents (in bounded-size networks) as fast as for unrestricted agents.

A. Pelc—Supported in part by NSERC discovery grant 8136 – 2013 and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais.