Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility

TL;DR

This paper investigates how to preserve symmetry during near-gathering in swarms of simple, oblivious robots with limited visibility, introducing new algorithms and analysis techniques based on dynamical systems theory.

Contribution

It provides the first conditions under which symmetry can be preserved in such robot swarms and introduces algorithms that achieve near-gathering without symmetry disruption.

Findings

A variant of Go-to-the-Average preserves symmetry but may create multiple clusters.

A new symmetry-preserving near-gathering algorithm works on convex, hole-free swarms.

Dynamical systems theory helps analyze symmetry effects in robot algorithms.

Abstract

In the general pattern formation (GPF) problem, a swarm of simple autonomous, disoriented robots must form a given pattern. The robots' simplicity imply a strong limitation: When the initial configuration is rotationally symmetric, only patterns with a similar symmetry can be formed [Yamashita, Suzyuki; TCS 2010]. The only known algorithm to form large patterns with limited visibility and without memory requires the robots to start in a near-gathering (a swarm of constant diameter) [Hahn et al.; SAND 2024]. However, not only do we not know any near-gathering algorithm guaranteed to preserve symmetry but most natural gathering strategies trivially increase symmetries [Castenow et al.; OPODIS 2022]. Thus, we study near-gathering without changing the swarm's rotational symmetry for disoriented, oblivious robots with limited visibility (the OBLOT-model, see [Flocchini et al.; 2019]). We introduce a technique based on the theory of dynamical systems to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation. Until now, it was unknown whether the considered OBLOT-model allows for any non-trivial algorithm that always preserves symmetry. Our first result shows that a variant of Go-to-the-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters. Our second result is a symmetry-preserving near-gathering algorithm that works on swarms with a convex boundary (the outer boundary of the unit disc graph) and without holes (circles of diameter 1 inside the boundary without any robots).