Dispersion of mobile robots using global communication

Abstract

The dispersion problem on graphs asks k≤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, and relocation of self-driven electric cars (robots) to recharge stations (nodes). In this paper, we consider dispersion using the global communication model where a robot can communicate with any other robot in the graph (but the graph is unknown to robots). We provide two novel deterministic algorithms for arbitrary graphs in a synchronous setting where all robots perform their actions in every time step. Our first algorithm is based on a DFS traversal and guarantees (i) O(kΔ) steps runtime using O(log⁡(k+Δ))) bits at each robot and (ii) O(min⁡(m,kΔ)) steps runtime using O(Δ+log⁡k) bits at each robot, where m is the number of edges and Δ is the maximum degree of the graph. The second algorithm is based on a BFS traversal and guarantees O((D+k)Δ(D+Δ)) steps runtime using O(log⁡D+Δlog⁡k)) bits at each robot, where D is the diameter of the graph. Our results complement the existing results established using the local communication model where a robot can communication only with other robots present at the same node.