Location-Aware Dispersion on Anonymous Graphs

TL;DR

The paper defines Location-Aware Dispersion, a color-constrained generalization of Dispersion on anonymous graphs, and studies its algorithmic feasibility under various knowledge and initial-configuration assumptions. It introduces a multi-phase, deterministic approach that uses group-based exploration, inter-group connectivity, and memory-efficient dispersion, with rigorous analysis showing both lower bounds and upper bounds on time and memory. A central contribution is a phase-structured algorithm that builds an inter-group tree to coordinate dispersion while respecting color constraints, plus impossibility results (e.g., $k=1$ with unknown $n$) and tight lower bounds. The results illustrate substantial differences from standard Dispersion, including cases that become unsolvable and otherwise achievable with higher resource guarantees, and they map out near-tight performance across rooted, dispersed, general, and unknown configurations, highlighting both theoretical and practical implications for cooperative multi-robot coordination in color-constrained anonymous networks.

Abstract

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.

Figures (1)

• Figure 1: Graph $G$, where $hollow$ nodes represent unvisited and $colored$ nodes with labels ($parent$ port) represent visited nodes. Robots with $yellow$ headed denote active, $solid$ headed denote oscillating, and $hollow$ headed denote leader. (a) Initial configuration, and begin DFS traversal with gr_size between 2 to 4. (b) During traversal, $r_a$ becomes leader and expands its group ($a_i$ filled blue, $a_i\in F_{r_a}, i\leq4$), where the black arrow denotes the exit port (including $r_a$, exits its group to store inter-group edges). (c) At $b_1$, leader$r_b$ is elected, where red solid line represents inter-group edges with direction from parent to child group. (d) leader$r_d$ with $\textit{gr\_size}<2$, visits an visited node $c_1$, where dotted lines indicate the traversal. (e) $r_d$ expands its group including nodes of subtree at $c_1$, while $r_c$ shrinks, switches to leader as $\textit{gr\_size}<2$. (f) $r_c$ includes a helper node $b_4$ and expands its group. (g) active robot traverses back to $S$. (h) End of DFS traversal.

Theorems & Definitions (30)

• Theorem 1: Impossibility
• proof
• Theorem 2: Lower bound
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof