Universally Optimal Gathering Under Limited Visibility

Abstract

We consider the distributed setting of N autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles following the well-celebrated classic oblivious robots model. We study the fundamental problem of gathering N autonomous robots on a plane, which requires all robots to meet at a single point (or to position within a small area) that is not known beforehand. We consider limited visibility under which robots are only able to see other robots up to a constant Euclidean distance and focus on the time complexity of gathering by robots under limited visibility. There exists an \(\mathcal{O}(D_G)\) time algorithm for this problem in the fully synchronous setting, assuming that the robots agree on one coordinate axis (say North), where \(D_G\) is the diameter of the visibility graph of the initial configuration. In this paper, we provide the first \(\mathcal{O}(D_E)\) time algorithm for this problem in the asynchronous setting under the same assumption of robots agreement on one coordinate axis, where \(D_E\) is the Euclidean distance between farthest-pair of robots in the initial configuration. The runtime of our algorithm is a significant improvement since, for any initial configuration of \(N\ge 1\) robots, \(D_E\le D_G\), and, there exist initial configurations for which \(D_G\) can be as much as quadratic on \(D_E\), i.e., \(D_G=\varTheta (D_E^2)\). Moreover, our algorithm is universally (time) optimal since the trivial time lower bound for this problem is \(\varOmega (D_E)\).