Collaborative dispersion by silent robots

In the dispersion problem, a set of k co-located mobile robots must relocate themselves in distinct nodes of an unknown network. The network is modeled as an anonymous graph G=(V,E), where the graph's nodes are not labeled. The edges incident to a node v with degree d are labeled with port numbers in the range {0,1,…,d−1} at v. The robots have unique IDs in the range [0,L], where L≥k, and are initially placed at a source node s. The task of the dispersion was traditionally achieved based on the assumption of two types of communication abilities: (a) when some robots are at the same node, they can communicate by exchanging messages between them, and (b) any two robots in the network can exchange messages between them. This paper investigates whether this communication ability among co-located robots is absolutely necessary to achieve dispersion. We establish that even in the absence of the ability of communication, the task of the dispersion by a set of mobile robots can be achieved in a much weaker model, where a robot at a node v has access to following very restricted information at the beginning of any round: (1) am I alone at v? (2) did the number of robots at v increase or decrease compared to the previous round? We propose a deterministic distributed algorithm that achieves the dispersion on any given graph G=(V,E) in time O(klog⁡L+k²log⁡Δ), where Δ is the maximum degree of a node in G. Further, each robot uses O(log⁡L+log⁡Δ) additional memory, i.e., memory other than the memory required to store its id. We also prove that the task of the dispersion cannot be achieved by a set of mobile robots with o(log⁡L+log⁡Δ) additional memory.