Anonymous Self-Stabilising Localisation via Spatial Population Protocols

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.