Arbitrary pattern formation on infinite regular tessellation graphs

Abstract

Given a set R of robots, each one located at a different vertex of an infinite regular tessellation graph, we aim to explore the Arbitrary Pattern Formation (APF) problem. Given a multiset F of grid vertices such that |R|=|F|, APF asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections.So far, as possible graph discretizing the Euclidean plane only the standard square grid has been considered in the context of the classical OBLOT model. However, it is natural to consider also the other regular tessellation graphs, that are triangular and hexagonal grids. In particular, the former can be considered as the most general in terms of possible symmetries and trajectories.We provide a resolution algorithm for APF when the initial configuration is asymmetric and the considered topology is any regular tessellation graph. The algorithm and its correctness are based on a rigorous methodology.