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 discretization of the Euclidean plane only the standard square grid has been considered in the context of the classical Look-Compute-Move model. However, it is natural to consider the other regular tessellation graphs, that are triangular and hexagonal grids. For any regular tessellation graph, we provide a resolution algorithm for APF when the initial configuration is asymmetric.