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A set of mobile robots (represented as points) is distributed in theCartesian plane. The collection contains an unknown subset of byzantine robotswhich are indistinguishable from the reliable ones. The reliable robots need togather, i.e., arrive to a configuration in which at the same time, all of themoccupy the same point on the plane. The robots are equipped with GPS devicesand at the beginning of the gathering process they communicate the Cartesiancoordinates of their respective positions to the central authority. On thebasis of this information, without the knowledge of which robots are faulty,the central authority designs a trajectory for every robot. The centralauthority aims to provide the trajectories which result in the shortestpossible gathering time of the healthy robots. The efficiency of a gatheringstrategy is measured by its competitive ratio, i.e., the maximal ratio betweenthe time required for gathering achieved by the given trajectories and theoptimal time required for gathering in the offline case, i.e., when the faultyrobots are known to the central authority in advance. The role of the byzantinerobots, controlled by the adversary, is to act so that the gathering is delayedand the resulting competitive ratio is maximized.  The objective of our paper is to propose efficient algorithms when thecentral authority is aware of an upper bound on the number of byzantine robots.We give optimal algorithms for collections of robots known to contain at mostone faulty robot. When the proportion of byzantine robots is known to be lessthan one half or one third, we provide algorithms with small constantcompetitive ratios. We also propose algorithms with bounded competitive ratioin the case where the proportion of faulty robots is arbitrary.

Gathering in the Plane of Location-Aware Robots in the Presence of Spies