Breaking symmetries on tessellation graphs via asynchronous robots

We consider the coordination of autonomous mobile robots operating in the standard Look-Compute-Move cycles. Robots are assumed to be very weak computational units, since they are asynchronous, oblivious, anonymous, silent and execute the same distributed algorithm. In this area, the main focus has been on the important class of Pattern Formation problems, where the robots are required to arrange themselves to form a given geometric shape. This class of problems has been extensively studied in the Euclidean plane, whereas few results exist when robots move on a discretization of the plane, like infinite grids. In infinite grids, in order to form any pattern, the problem of breaking symmetries clearly emerges. Breaking the symmetry by moving some leader robot is not a straightforward task due to the movement restrictions as all the adjacent nodes of the leader may be occupied. Due to the asynchrony of robots, this fact greatly increases the difficulty of the problem. We assume regular tessellation graphs as discretization of the Euclidean plane, and we devise an algorithm able to solve the Symmetry Breaking problem on both the square and triangular grids. The algorithm is proposed so that it can be also combined with other modules.