Separating Bounded and Unbounded Asynchrony for Autonomous Robots: Point Convergence with Limited Visibility

We consider distributed computations, by identical autonomous mobile entities, that solve the Point Convergence problem: given an arbitrary initial configuration of entities, disposed in the Euclidean plane, move in such a way that, for all ϵ>0, a configuration is eventually reached and maintained in which the separation between all entities is at most ϵ. The problem has been previously studied in a variety of settings. Our study concerns the minimal assumptions under which entities, moving asynchronously with limited and unknown visibility range and subject to limited imprecision in measurements, can be guaranteed to converge in this way. We present an algorithm that solves Point Convergence, provided the degree of asynchrony is bounded by some arbitrarily large but fixed constant. This provides a strong positive answer to a decade old open question posed by Katreniak. We also prove that, in an otherwise comparable setting, Point Convergence is impossible with unbounded asynchrony. This serves to distinguish the power of bounded and unbounded asynchrony in the control of autonomous mobile entities, settling at the same time a long-standing question whether in the Euclidean plane synchronous entities are more powerful than asynchronous ones.