A set of mobile robots is placed at points of an infinite line. The robots are equipped with GPS devices and they may communicate their positions on the line to a central authority. The collection contains an unknown subset of "spies", i.e., byzantine robots, which are indistinguishable from the non-faulty ones. The set of the non-faulty robots need to rendezvous in the shortest possible time in order to perform some task, while the byzantine robots may try to delay their rendezvous for as long as possible. The problem facing a central authority is to determine trajectories for all robots so as to minimize the time until the non-faulty robots have rendezvoused. The trajectories must be determined without knowledge of which robots are faulty. Our goal is to minimize the competitive ratio between the time required to achieve the first rendezvous of the non-faulty robots and the time required for such a rendezvous to occur under the assumption that the faulty robots are known at the start. We provide a bounded competitive ratio algorithm, where the central authority is informed only of the set of initial robot positions, without knowing which ones or how many of them are faulty. When an upper bound on the number of byzantine robots is known to the central authority, we provide algorithms with better competitive ratios. In some instances we are able to show these algorithms are optimal.
Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. 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 edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O(D|Lmin|2) when D is known and O((D + |Lmax|)3) if D is unknown, where |Lmin| and |Lmax| are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost O(n|Lmin|), if the size n of the ring is known, and of cost O(n|Lmax|), if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|Lmin|) if the topology of the graph and the initial positions are known to agents. The author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time. This time is called the rendezvous value of the region. It is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise. A general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players. These concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network. The discrete rendezvous problem on a cycle graph for players restricted to symmetric Markovian strategies is then solved. Finally, the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other. The rendezvous-search problem was posed by the author 25years ago. In its basic form, it asks how two unit speed players can find each other in least expected time, when randomly placed in a known dark region. The problem received little attention until about 10 years ago. This article surveys the rapid progress that has been made since then, and also presents some new results. 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 present fast deterministic algorithms for this rendezvous problem.  Two mobile agents, modeled as points starting at different locations of an unknown terrain, have to meet. The terrain is a polygon with polygonal holes. We consider two versions of this rendezvous problem: exact RV, when the points representing the agents have to coincide at some time, and ?-RV, when these points have to get at distance less than ? in the terrain. In any terrain, each agent chooses its trajectory, but the movements of the agent on this trajectory are controlled by an adversary that may, e.g., speed up or slow down the agent. Agents have bounded memory: their computational power is that of finite state machines. Our aim is to compare the feasibility of exact and of ?-RV when agents are anonymous vs. when they are labeled. We show classes of polygonal terrains which distinguish all the studied scenarios from the point of view of feasibility of rendezvous. The features which influence the feasibility of rendezvous include symmetries present in the terrains, boundedness of their diameter, and the number of vertices of polygons in the terrains. 