Optimal and competitive runtime bounds for continuous, local gathering of mobile robots

We consider a scenario in which n mobile robots with a limited viewing range are distributed arbitrarily in the plane, such that the visibility graph of the robots is connected. The goal is to gather the robots in one (not predefined) point. Each robot may base its decision where to move only on the current relative positions of the robots which are in its viewing range. That is, besides having a limited viewing range, the robots are oblivious (they do not use information from the past), they do not have IDs, and they do not have a common sense of direction. On the other hand side, we assume that they are points, i.e., have no extent. Variants of this problem have been studied extensively in different discrete time models. In this paper, we study the gathering problem in a continuous time model. That is, the robots continuously sense the positions of their neighboring robots within their viewing range, and continuously adapt speed and direction, according to a local rule. We assume a speed limit normalized to 1, so that the maximum distance traveled by any robot is smaller or equal to the runtime. Gordon, Wagner and Bruckstein have proposed a simple and intuitive continuous algorithm in ANTS '04, and they showed that their algorithm gathers the robots in finite time. But the runtime of this algorithm has been open since then. We present a runtime analysis for their algorithm and show two runtime bounds. The first one is an optimal worst case bound O(n), the second one shows the log(OPT)-competitiveness in the sense that if OPT is the runtime of an optimal global algorithm, the local algorithm is at most by a factor of log(OPT) slower than the global algorithm. Best previous bounds on the distance traveled by the robots are obtained for discrete time models and are O(n^2) in the worst case and only O(n) competitive.