Location-Aware Dispersion on Anonymous Graphs

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.