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Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility

Raphael Gerlach, Sören von der Gracht, Christopher Hahn, Jonas Harbig, Peter Kling

TL;DR

A technique based on the theory of dynamical systems is introduced to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation and shows that a variant of Go-to-the-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters.

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).

Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility

TL;DR

A technique based on the theory of dynamical systems is introduced to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation and shows that a variant of Go-to-the-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters.

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).
Paper Structure (25 sections, 19 theorems, 19 equations, 3 figures, 3 algorithms)

This paper contains 25 sections, 19 theorems, 19 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. General Pattern Formation in the $\mathcal{OBLOT}$ Model.
  3. Symmetries in GPF.
  4. Our Contribution.
  5. Outline.
  6. Further Related Work
  7. Preliminaries & Notation
  8. Basic Notions & Notation.
  9. Characterizing Protocols via their Evolution Function
  10. Preserving Symmetries via Local Invertibility
  11. Protocol 1: Preserving Symmetries via Averaging
  12. Protocol 2: Preserving Symmetries via Contracting Waves
  13. Requirements of the algorithm.
  14. Overview.
  15. Boundary Algorithm
  16. ...and 10 more sections

Key Result

Theorem 1

Consider the dynamics of an arbitrary swarm protocol with evolution function $F\colon \mathbb{R}^{2n} \to \mathbb{R}^{2n}$. Assume that $F$ is (locally) invertible. Then, any configuration $\mathbf{z} \in \mathbb{R}^{2n}$ and its successor configuration $\mathbf{z}^+ \coloneqq F(\mathbf{z})$ have th

Figures (3)

  • Figure 1:
  • Figure 2: Graph of bump function $b(X)$. It decreases monotonically from $1$ (for $X = \|z_i - z_j\|^2 = 0$; i.e., when $j$ is at the same position as $i$) to $0$ (for $X = 1$; i.e., when $j$ is at the brink of being invisible).
  • Figure :

Theorems & Definitions (36)

  • Theorem 1: name=, restate=[name=restated]thmSymPreservSuff
  • Definition 2: Symmetricity DBLP:journals/siamcomp/FujinagaYOKY15
  • Definition 3: Symmetry of a Configuration 1
  • Definition 4: Symmetry of a Configuration (alternative)
  • Lemma 5
  • Definition 6: Boundary-Robots
  • Remark 7
  • Remark 8
  • Lemma 9: name=, restate=[name=restated]lemEgtmInvertable
  • Lemma 10: name=, restate=[name=restated]lemConvexEgtmStaysConvex
  • ...and 26 more