Deterministic Rendezvous with Detection Using Beeps

Abstract

Two mobile agents, starting at arbitrary, possibly different times from arbitrary nodes of an unknown network, have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. Agents have different labels which are positive integers. 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 the agents get to the same node in the same round. We want to achieve 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 an agent can communicate to the other agent its presence at a node. We use two variations of the arguably weakest model of communication, called the beeping model, introduced in [8]. In each round an agent can either listen or beep. In the local beeping model, an agent hears a beep in a round 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 beep in a round if it listens in this round and if the other agent is at the same node and beeps, and it hears a soft beep in a round if it listens in this round and if the other agent is at some other 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. This is impossible for some networks of unbounded size. Hence we rephrase the question: Can bounded-energy agents always achieve rendezvous with detection in boundedsize 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.

Accelerating Amoebots via Reconfigurable Circuits

“…On a more technical level, the beeping model differs from our circuit model in a sense that nodes are aware of their neighbors in the circuit model, which allows them to exchange (constant-sized) messages with each other, whereas in the beeping model, a listening node that receives a beep is not aware of the particular neighbor that beeped in that specific round. Several problems like interval coloring [5], maximal independent set [1,35], consensus [27], rendezvous of two agents [17], leader election [16,21] or clock synchronization [15,19,24,25] have already been investigated in the beeping model.…”

Asynchronous Broadcasting with Bivalent Beeps

“…On the other hand, the number of soft beeps heard by node u by round r must be at least 1/2 of the number of loud beeps it heard by round r. This implies 8 7 · 8 z ≥ 1 2 · 8 z+1 , which is a contradiction. This completes the first part of the proof.…”

“…In [19], various distributed problems were investigated under several variations of the beeping model from [6], and randomized emulations between these models were shown. In [8], the authors studied the task of rendezvous of agents communicating by beeps. The time of synchronous broadcasting and gossiping with beeps was studied in [7].…”

Time- and Space-Optimal Clock Synchronization in the Beeping Model

“…Their algorithm runs in O(log n) rounds, which matches the lower bound for that problem. Other problems that have been considered in this model are maximal independent set [AAB + 13, SJX13], leader election [FSW14,GN15] or rendevouz of two agents [EP17].…”