Arbitrary Pattern Formation on Infinite Regular Tessellation Graphs

TL;DR

This paper presents a distributed algorithm for arbitrary pattern formation by robots on infinite regular tessellation graphs, including square, triangular, and hexagonal grids, under asymmetric initial configurations.

Contribution

It extends the APF problem solution to all regular tessellation graphs beyond the square grid, covering triangular and hexagonal grids.

Findings

Algorithm successfully forms arbitrary patterns on all considered tessellations.

Works for asymmetric initial configurations.

Applicable to multiple regular tessellation types.

Abstract

Given a set R of robots, each one located at different vertices 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 Look-Compute-Move model. However, it is natural to consider also the other regular tessellation graphs, that are triangular and hexagonal grids. We provide a resolution algorithm for APF when the initial configuration is asymmetric and the considered topology is any regular tessellation graph.