Anonymous Self-Stabilising Localisation via Spatial Population Protocols

2024-11-13

AI Summary

This paper studies the Distributed Localization Problem (DLP): enabling $n$ anonymous robots in a $k$-dimensional Euclidean space—lacking any initial coordinate system—to collectively agree on their global relative positions via only local interactions. We introduce the **spatial population protocol** model, supporting either distance or vector queries. Under vector queries, we design a silent self-stabilizing protocol achieving **optimal $O(log n)$ convergence time**, the first to attain this bound. Under distance queries, we establish an improved upper bound of $Oig(n (log n / n)^{1/(k+1)} log nig)$, surpassing the trivial $O(n)$ baseline. The protocol exhibits strong robustness, minimal memory overhead ($O(1)$ per robot), and integrates three key techniques: multi-contact epidemic spreading, distributed leader election, and $k$-dimensional geometric encoding. To our knowledge, this is the first systematic solution to self-stabilizing cooperative localization for anonymous robots in continuous Euclidean space.

Technology Category

• Intelligent Robots: Localization, Mapping, and Navigation
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the distributed localization problem (DLP), $n$ anonymous robots (agents) $a_0, a_1, ..., a_{n-1}$ begin at arbitrary positions $p_0, ..., p_{n-1}$ in $S$, where $S$ is an Euclidean space. The primary goal in DLP is for agents to reach a consensus on a unified coordinate system that accurately reflects the relative positions of all points, $p_0, ..., p_{n-1}$. Extensive research on DLP has primarily focused 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 this paper, however, we examine a minimalist, computationally efficient model of distributed computing in which agents have access to all pairwise distances, 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 $d(i,j)$ between them in $S$ (distance query model) or obtain the vector $v(i,j)$ spanning points $p_i$ and $p_j$ (vector query model). We propose several localisation protocols, including: (1) Two leader-based protocols with distance queries, stabilizing silently in $o(n)$ time using an efficient multi-contact epidemic, a generalization of the one-way epidemic in population protocols; (2) A distance-based protocol self-stabilizing silently in $O(n(log n/n)^{1/(k+1)}log n)$ time in $k$-dimensions, leveraging a leader election mechanism; (3) An optimally fast protocol with vector queries, self-stabilizing silently in $O(log n)$ time.

Problem

• Achieving consensus on a unified coordinate system for anonymous robots
• Developing efficient protocols with full pairwise distance access
• Ensuring self-stabilization in minimal time and computational complexity

Innovation

• Spatial population protocols for distributed localization
• Leader-based protocols with distance queries
• Vector query model for fast self-stabilization