DOI: 10.1007/s00453-006-0074-2 Algorithmica (2006) 46: 69–96 Algorithmica © 2006 Springer Science+Business Media, Inc. Deterministic Rendezvous in Graphs 1 Anders Dessmark, 2 Pierre Fraigniaud, 3 Dariusz R. Kowalski, 4 and Andrzej Pelc 5 Abstract. Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l ) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is ( D log l ), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l , where τ is the difference between startup times of the agents. We also show a lower bound (n 2 ) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded. Key Words. Algorithm, Deterministic, Graph, Mobile agent, Rendezvous. 1. Introduction. Two mobile agents located in nodes of a network, modeled as an undirected connected graph, have to meet at some node of the graph. This task is known in the literature as the rendezvous problem in graphs [6], and in this paper we seek efficient deterministic algorithms to solve it. If nodes of the graph are labeled then agents can decide to meet at a predetermined node and the rendezvous problem reduces to graph 1 This work was done during the authors’ visits at the Research Chair in Distributed Computing of the Universit´ e du Qu´ ebec en Outaouais. The results of this paper appeared in preliminary version in two conference papers: A. Dessmark, P. Fraigniaud and A. Pelc, Deterministic rendezvous in graphs, Proc. 11th Annual European Symposium on Algorithms (ESA 2003), September 2003, Budapest, Hungary, LNCS 2832, pp. 184– 195, and D. Kowalski and A. Pelc, Polynomial deterministic rendezvous in arbitrary graphs, Proc. 15th Annual Symposium on Algorithms and Computation (ISAAC 2004), December 2004, Hong Kong, LNCS 3341, pp. 644–656. 2 Department of Computer Science, Lund University, Box 118, S-22100 Lund, Sweden. andersd@cs.lth.se. 3 CNRS, LRI, Universit´ e Paris-Sud, 91405 Orsay, France. Pierre.Fraigniaud@lri.fr. Additional support from INRIA Project “Grand Large.” 4 Department of Computer Science, The University of Liverpool, Chadwick Building, Peach Street, Liverpool L69 7ZF, England. darek@csc.liv.ac.uk. Supported in part by NSF-NATO Award 0209588. 5 D´ epartement d’informatique, Universit´ e du Qu´ ebec en Outaouais, Gatineau, Qu´ ebec, Canada J8X 3X7. pelc@uqo.ca. Supported in part by NSERC discovery grant and by the Research Chair in Distributed Computing of the Universit´ e du Qu´ ebec en Outaouais. Received September 27, 2004; revised August 4, 2005. Communicated by R. Fleischer. Online publication June 19, 2006.