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Anonymous Self-Stabilising Localisation via Spatial Population Protocols

Authors Leszek Gąsieniec , Łukasz Kuszner , Ehsan Latif , Ramviyas Parasuraman , Paul Spirakis , Grzegorz Stachowiak

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ACM Subject Classification
  • Theory of computation → Distributed computing models
Keywords
  • Population Protocols
  • Distributed Localisation
  • Spacial Queries
  • Self-Stabilisation

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Abstract

In the distributed localisation problem (DLP), n anonymous robots (agents) A_0, ..., A_{n-1} are located at arbitrary points p_0, ..., p_{n-1} ∈ S, where S is a Euclidean space. Initially, each agent A_i operates within its own coordinate system in S, which may be inconsistent with those of other agents. The primary goal in DLP is for agents to reach a consensus on a unified (jointly agreed) coordinate system, in which all agents receive unique labels (coordinates) that accurately reflect the relative distances between all points p_0, ..., p_{n-1} in S. Extensive research on DLP has primarily focus on the feasibility and complexity of achieving consensus when agents have limited access to inter-agent distances, often due to missing or imprecise data. In contrast, this paper proposes a minimalist, computationally efficient distributed computing model where agents can query any pairwise relative positions, if needed. Specifically, we introduce a novel variant of population protocols, referred to as the spatial population protocols model. In this variant each agent can memorise one or a fixed number of coordinates, and when agents A_i and A_j interact, they can not only exchange their current knowledge but also either determine the distance d_{ij} between them in S (distance query model) or obtain the vector v_{ij} spanning points p_i and p_j (vector query model).
We propose and analyse several distributed localisation protocols, including: 
1) Leader-based localisation protocol with distance queries We propose and analyse two leader-based localisation protocols that stabilise silently in o(n) time. These protocols leverage an efficient solution to the novel concept of multi-contact epidemic, a natural generalisation of the core communication tool in population protocols, known as the one-way epidemic.
2) Self-stabilising leader localisation protocol with distance queries We show how to effectively utilise a leader election mechanism within the leader-based localisation protocol to get a DLP protocol that self-stabilises silently in time O(n(log n/n)^{1/(k+1)}log n) in k-dimensions. 
3) Self-stabilising localisation protocol with vector queries We propose and analyse an optimally fast DLP protocol which self-stabilises silently in O(log n) time.

Cite As Get BibTex

Leszek Gąsieniec, Łukasz Kuszner, Ehsan Latif, Ramviyas Parasuraman, Paul Spirakis, and Grzegorz Stachowiak. Anonymous Self-Stabilising Localisation via Spatial Population Protocols. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.35

Author Details

Leszek Gąsieniec
  • Department of Computer Science, University of Liverpool, UK
Łukasz Kuszner
  • Institute of Informatics, University of Gdańsk, Poland
Ehsan Latif
  • AI4STEM Education Center, University of Georgia, Athens, GA, USA
Ramviyas Parasuraman
  • School of Computing, University of Georgia, Athens, GA, USA
Paul Spirakis
  • Department of Computer Science, University of Liverpool, UK
Grzegorz Stachowiak
  • Institute of Computer Science, University of Wrocław, Poland

Funding

  • Spirakis, Paul: Supported by the EPSRC grant EP/P02002X/1.
  • Stachowiak, Grzegorz: Supported by the NCN grant 2020/39/B/ST6/03288.

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