Synchronous Rendezvous for Location-Aware Agents

We study rendezvous of two anonymous agents, where each agent knows its own initial position in the environment. Their task is to meet each other as quickly as possible. The time of the rendezvous is measured by the number of synchronous rounds that agents need to use in the worst case in order to meet. In each round, an agent may make a simple move or it may stay motionless. We consider two types of environments, finite or infinite graphs and Euclidean spaces. A simple move traverses a single edge (in a graph) or at most a unit distance (in Euclidean space). The rendezvous consists in visiting by both agents the same point of the environment simultaneously (in the same round). In this paper, we propose several asymptotically optimal rendezvous algorithms. In particular, we show that in the line and trees as well as in multi-dimensional Euclidean spaces and grids the agents can rendezvous in time O(d), where d is the distance between the initial positions of the agents. The problem of location-aware rendezvous was studied before in the asynchronous model for Euclidean spaces and multi-dimensional grids, where the emphasis was on the length of the adopted rendezvous trajectory. We show that, contrary to the asynchronous case, where the cost of rendezvous is dominated by the size of potentially large neighborhoods, the agents are able to meet in all graphs of at most n nodes in time almost linear in d, namely, O(d log n). We also prove that there exists an infinite family of graphs in which synchronized rendezvous requires time ω(d). Regular submission to DISC 2011 University of Liverpool, Liverpool L69 3BX, UK. E-mail: [email protected]. Département d’informatique, Université du Québec en Outaouais, Gatineau, Québec J8X 3X7, Canada. E-mail: [email protected]. Supported in part by NSERC discovery grant. University of Liverpool, Liverpool L69 3BX, UK. E-mail: [email protected]. Phone: +44 151 794 7910. Supported in part by the Royal Society IJP grant. INRIA Bordeaux Sud-Ouest, LaBRI, 33405 Talence, France. E-mail: [email protected]. University of Liverpool, Liverpool L69 3BX, UK. E-mail: [email protected].