Gathering a swarm of robots through shortest paths

The gathering problem has been largely studied in the last years with respect to different environments. The requirement is to move a team of robots ini- tially placed at different locations toward a common point. Robots move based on the so called Look-Compute-Move model. Each time a robot wakes up, it perceives the current configuration in terms of robots’ positions (Look), it decides whether and where to move (Compute), and makes the computed move (Move) in the case that the decision was affirmative. All the phases are performed asynchronously for each robot. Robots are oblivious, anonymous, silent, and execute the same distributed and deterministic algorithm. So far, the goal has been mainly to detect the mini- mal assumptions that allow to accomplish the gathering task, without taking care of any cost measure of the provided solutions. We provide an overview of recent results that first extend the classic notion of optimization problem to the context of robot-based computing systems, and then show that the gathering problem can be optimally solved. As cost measure, the overall traveled distance performed by all robots is considered. This implies that the provided optimal algorithms must be able to solve the gathering by moving robots through shortest paths. The presented op- timal algorithms refer to robots moving on either the plane or graphs. In the latter case, different topologies are considered, like trees, rings, and infinite grids.