In the distributed localization problem (DLP), n anonymous robots (agents) A0, A1, ..., A(n-1) begin at arbitrary positions p0, ..., p(n-1) in S, where S is a 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, p0, ... , p(n-1), in S. 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 examine three DLP scenarios: - Self-stabilising localisation protocol with distance queries We propose and analyse self-stabilising localisation protocol based on pairwise distance adjustment. We also discuss several hard instances in this scenario, and suggest possible improvements for the considered protocol, - Leader-based localisation protocol with distance queries We propose and analyse several leader-based protocols which stabilise in o(n) parallel time. These protocols rely on efficient solution to multi-contact epidemic, and - Self-stabilising localisation protocol with vector queries We propose and analyse superfast self-stabilising DLP protocol which stabilises in O(log n) parallel time.

翻译：在分布式定位问题中，n个匿名机器人（智能体）A0, A1, ..., A(n-1)起始于欧几里得空间S中的任意位置p0, ..., p(n-1)。DLP的主要目标是使智能体就一个统一的坐标系达成共识，该坐标系需精确反映S中所有点p0, ..., p(n-1)的相对位置。现有关于DLP的大量研究主要关注当智能体因数据缺失或不精确而仅能获取有限智能体间距离时，达成共识的可行性与复杂度。然而，本文研究一种极简且计算高效的分布式计算模型，其中智能体在需要时可获取所有成对距离。具体而言，我们引入一种新的群体协议变体，称为空间群体协议模型。在此变体中，每个智能体可记忆一个或固定数量的坐标，当智能体A(i)与A(j)交互时，它们不仅能交换当前知识，还能确定它们在S中的距离d(i,j)（距离查询模型）或获取连接点p(i)与p(j)的向量v(i,j)（向量查询模型）。我们研究了三种DLP场景：- 基于距离查询的自稳定定位协议：我们提出并分析了一种基于成对距离调整的自稳定定位协议，同时讨论了此场景下的若干困难实例，并针对所考虑协议提出了可能的改进方案；- 基于距离查询的领导者定位协议：我们提出并分析了若干在o(n)并行时间内稳定的领导者协议，这些协议依赖于多接触传播问题的有效解决方案；- 基于向量查询的自稳定定位协议：我们提出并分析了一种超快速自稳定DLP协议，该协议在O(log n)并行时间内稳定。