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 $\varOmega (D_E)$.