Gathering of oblivious robots on infinite grids with minimum traveled distance

A largely studied version of the gathering problem asks to move a set of robots initially placed at different vertices of an anonymous graph toward a common vertex, and to let them remain at such a vertex. Asynchronous 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 occupied vertices (Look), it decides whether to move toward one of its neighbors (Compute), and then it performs the computed move (Move), eventually. So far, the main goal has been to detect the minimal assumptions that allow to accomplish the task, without taking care of any cost measure. In this paper, we are interested in optimal algorithms in terms of total number of moves. We consider infinite grids, and we fully characterize when optimal gathering is achievable by providing a distributed algorithm.