Asynchronous Gathering in a Torus

TL;DR

This paper presents a deterministic algorithm enabling asynchronous, oblivious robots with limited visibility to gather on an asymmetric torus network without explicit communication, overcoming symmetry challenges.

Contribution

It extends the gathering problem study to torus networks with weak multiplicity detection, providing a novel solution for asymmetric, unoriented environments.

Findings

Algorithm guarantees gathering from any rigid configuration.

Works on asymmetric, unoriented torus networks.

Addresses limitations of multiplicity-based landmark strategies.

Abstract

We consider the gathering problem for asynchronous and oblivious robots that cannot communicate explicitly with each other, but are endowed with visibility sensors that allow them to see the positions of the other robots. Most of the investigations on the gathering problem on the discrete universe are done on ring shaped networks due to the number of symmetric configuration. We extend in this paper the study of the gathering problem on torus shaped networks assuming robots endowed with local weak multiplicity detection. That is, robots cannot make the difference between nodes occupied by only one robot from those occupied by more than one robots unless it is their current node. As a consequence, solutions based on creating a single multiplicity node as a landmark for the gathering cannot be used. We present in this paper a deterministic algorithm that solves the gathering problem starting from any rigid configuration on an asymmetric unoriented torus shaped network.