Dispersion of Mobile Robots in the Global Communication Model

TL;DR

This paper introduces three deterministic algorithms for dispersing mobile robots in unknown graphs and trees under a global communication model, improving efficiency over local communication methods.

Contribution

It presents novel algorithms for robot dispersion with optimal or near-optimal time and memory bounds in the global communication setting.

Findings

DFS-based algorithm is optimal for memory and time in constant-degree graphs.

BFS-based algorithm for trees is optimal when number of robots is at most proportional to graph diameter.

Algorithms significantly outperform local communication models in efficiency.

Abstract

The dispersion problem on graphs asks $k\leq n$ robots placed initially arbitrarily on the nodes of an $n$-node anonymous graph to reposition autonomously to reach a configuration in which each robot is on a distinct node of the graph. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing etc. In this paper, we consider dispersion in the {\em global communication} model where a robot can communicate with any other robot in the graph (but the graph is unknown to robots). We provide three novel deterministic algorithms, two for arbitrary graphs and one for arbitrary trees, in a synchronous setting where all robots perform their actions in every time step. For arbitrary graphs, our first algorithm is based on a DFS traversal and guarantees $O(\min(m,k\Delta))$ steps runtime using $\Theta(\log (\max(k,\Delta)))$ bits at each robot, where $m$ is the number of edges and $\Delta$ is the maximum degree of the graph. The second algorithm for arbitrary graphs is based on a BFS traversal and guarantees $O( \max(D,k) \Delta (D+\Delta))$ steps runtime using $O(\max(D,\Delta \log k))$ bits at each robot, where $D$ is the diameter of the graph. The algorithm for arbitrary trees is also based on a BFS travesal and guarantees $O(D\max(D,k))$ steps runtime using $O(\max(D,\Delta \log k))$ bits at each robot. Our results are significant improvements compared to the existing results established in the {\em local communication} model where a robot can communication only with other robots present at the same node. Particularly, the DFS-based algorithm is optimal for both memory and time in constant-degree arbitrary graphs. The BFS-based algorithm for arbitrary trees is optimal with respect to runtime when $k\leq O(D)$.