The well-studied DISPERSION problem is a fundamental coordination problem in distributed robotics, where a set of mobile robots must relocate so that each occupies a distinct node of a network. DISPERSION assumes that a robot can settle at any node as long as no other robot settles on that node. In this work, we introduce LOCATION-AWARE DISPERSION, a novel generalization of DISPERSION that incorporates location awareness: Let $G = (V, E)$ be an anonymous, connected, undirected graph with $n = |V|$ nodes, each labeled with a color $\sf{col}(v) \in C = \{c_1, \dots, c_t\}, t\leq n$. A set $R = \{r_1, \dots, r_k\}$ of $k \leq n$ mobile robots is given, where each robot $r_i$ has an associated color $\mathsf{col}(r_i) \in C$. Initially placed arbitrarily on the graph, the goal is to relocate the robots so that each occupies a distinct node of the same color. When $|C|=1$, LOCATION-AWARE DISPERSION reduces to DISPERSION. There is a solution to DISPERSION in graphs with any $k\leq n$ without knowing $k,n$. Like DISPERSION, the goal is to solve LOCATION-AWARE DISPERSION minimizing both time and memory requirement at each agent. We develop several deterministic algorithms with guaranteed bounds on both time and memory requirement. We also give an impossibility and a lower bound for any deterministic algorithm for LOCATION-AWARE DISPERSION. To the best of our knowledge, the presented results collectively establish the algorithmic feasibility of LOCATION-AWARE DISPERSION in anonymous networks and also highlight the challenges on getting an efficient solution compared to the solutions for DISPERSION.

翻译：已有深入研究的DISPERSION问题是分布式机器人学中的一个基本协调问题，其中一组移动机器人必须重新定位，使得每个机器人占据网络的一个不同节点。DISPERSION假设只要没有其他机器人占据某个节点，机器人就可以在该节点上停留。在这项工作中，我们引入了位置感知分散问题（LOCATION-AWARE DISPERSION），这是DISPERSION的一种新颖推广，它融入了位置感知：设$G = (V, E)$为一个匿名的、连通的、无向图，具有$n = |V|$个节点，每个节点标记有一种颜色$\sf{col}(v) \in C = \{c_1, \dots, c_t\}, t\leq n$。给定一组$R = \{r_1, \dots, r_k\}$，包含$k \leq n$个移动机器人，其中每个机器人$r_i$有一个关联颜色$\mathsf{col}(r_i) \in C$。机器人最初被任意放置在图上，目标是通过重新定位机器人，使得每个机器人占据一个颜色相同的不同节点。当$|C|=1$时，位置感知分散问题简化为DISPERSION问题。对于任意$k\leq n$的图，存在DISPERSION的解决方案，且无需知道$k$和$n$。与DISPERSION类似，解决位置感知分散问题的目标是在最小化每个代理的时间和内存需求的同时达成目标。我们开发了几种确定性算法，这些算法在时间和内存需求上都有保证的界限。我们还给出了任何确定性算法解决位置感知分散问题的不可能性和下界。据我们所知，所呈现的结果共同确立了位置感知分散问题在匿名网络中的算法可行性，同时也突显了与DISPERSION的解决方案相比，获得高效解决方案所面临的挑战。