Linear and Competitive Strategies for Continuous Robot Formation Problems

We study a scenario in which n mobile robots with a limited viewing range are distributed in the Euclidean plane and have to solve a formation problem. The formation problems we consider are the G athering problem and the C hain-Formation problem. In the G athering problem, the robots have to gather in one (not predefined) point, while in the C hain -F ormation problem they have to form a connected communication chain of minimal length between two stationary base stations. Each robot may base its decisions where to move only on the current relative positions of neighboring robots (that are within its viewing range); that is, besides having a limited viewing range, the robots are oblivious (they do not use information from the past), have none or only very limited identities, and they do not have a common sense of direction. Variants of these problems (especially for the G athering problem) have been studied extensively in different discrete time models. In contrast, our work focuses on a continuous time model; that is, the robots continuously sense the positions of other robots within their viewing range and continuously adapt their speed and direction according to some simple, local rules. Hereby, we assume that the robots have a maximum movement speed of one. We show that this idealized idea of continuous sensing allows us to solve the mentioned formation problems in linear time O( n ) (which, given the maximum speed of one, immediately yields a maximum traveled distance of O( n )). Note that in the more classical discrete time models, the best known strategies need at least Θ( n 2 ) or even Θ( n 2 log n ) timesteps to solve these problems. For the G athering problem, our analysis solves a problem left open by Gordon et al. [2004], where the authors could prove that gathering in a continuous model is possible in finite time, but were not able to give runtime bounds. Apart from these linear bounds, we also provide runtime bounds for both formation problems that relate the runtime of our strategies to the runtime of an optimal, global algorithm. Specifically, we show that our strategy for the G athering problem is log OPT-competitive and the strategy for the C hain -F ormation problem is log  n -competitive. Here, by c -competitive, we mean that our (local) strategy is asymptotically by at most a factor of c slower than an optimal, global strategy.