Collaborative Dispersion by Silent Robots

TL;DR

This paper demonstrates that mobile robots can achieve dispersion in unknown graphs without inter-robot communication, using only minimal local information, and provides a deterministic algorithm with proven memory bounds.

Contribution

It introduces a novel dispersion algorithm that operates without communication among robots, relying solely on limited local information, and establishes lower bounds on memory requirements.

Findings

Dispersion achieved in $O(k \, \log L + k^2 \log \Delta)$ time.

Robots require $O(\log L + \log \Delta)$ memory, which is proven to be optimal.

The model reduces communication assumptions to minimal local information.

Abstract

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 nodes of the graph are not labeled. The edges incident to a node $v$ with degree $d$ are labeled with port numbers in the range $0,1, \cdots, d-1$ at $v$. The robots have unique ids in the range $[0,L]$, where $L \ge k$, and are initially placed at a source node $s$. Each robot knows only its own id but does not know the ids of the other robots or the values of $L,k$. The task of dispersion was traditionally achieved with 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 (b) any two robots in the network can exchange messages between them. In this paper, we ask whether this ability of communication among co-located robots is necessary to achieve dispersion. We show that even if the ability of communication is not available, the task of dispersion by a set of mobile robots can be achieved in a much weaker model where a robot at a node $v$ has the access of following very restricted information at the beginning of any round: (1) am I alone at $v$? (2) the number of robots at $v$ increased or decreased compare to the previous round? We propose a deterministic algorithm that achieves dispersion on any given graph $G=(V,E)$ in time $O\left( k\log L+k^2 \log \Delta\right)$, where $\Delta$ is the maximum degree of a node in $G$. Each robot uses $O(\log L+ \log \Delta)$ additional memory. We also prove that the task of dispersion cannot be achieved by a set of mobile robots with $o(\log L + \log \Delta)$ additional memory.