Rendezvous of Agents with Different Speeds

Most cooperative systems are, to some degree, asynchronous. This lack of synchrony is typically considered as an obstacle that makes the achievement of cooperative goals more difficult. In this work, we aim to highlight potential benefits of asynchrony, which is introduced into the system as differences between the speeds of the actions of different entities. We demonstrate the usefulness of this aspect of asynchrony, to tasks involving symmetry breaking. Specifically, in this paper, identical (except for their speeds) mobile deterministic agents are placed at arbitrary locations on a ring of length n and use their speed difference in order to rendezvous fast. We normalize the speed of the slower agent to be 1, and fix the speed of the faster agent to be some constant c > 1. (An agent does not know whether it is the slower agent or the faster one.) We present lower and upper bounds on the time of reaching rendezvous in two main cases. One case is that the agents cannot communicate in any way. For this case, we show a tight bound of cn c 2 −1 for the rendezvous time. The other case is where each agent is allowed to drop a pebble at the location it currently is (and is able to detect whether there is already a pebble there). For this case, we show an algorithm whose rendezvous time is max{ n 2(c−1) , n c }. On the negative side, we show an almost matching lower bound of max{ n 2(c−1) , n c+1}, which holds not only under the “pebble” model but also under the seemingly much stronger “white board” model.