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

TL;DR

This paper introduces an algorithm for autonomous robots to achieve point convergence under limited visibility and bounded asynchrony, and proves impossibility results for unbounded asynchrony, clarifying the impact of asynchrony bounds.

Contribution

It provides the first algorithm guaranteeing convergence with bounded asynchrony and limited visibility, and establishes that unbounded asynchrony prevents convergence under natural connectivity assumptions.

Findings

Bounded asynchrony allows point convergence with limited visibility.

Unbounded asynchrony makes convergence impossible under connectivity constraints.

The work distinguishes the computational power of bounded versus unbounded asynchrony.

Abstract

Among fundamental problems in the context of distributed computing by autonomous mobile entities, one of the most representative and well studied is {\sc Point Convergence}: given an arbitrary initial configuration of identical entities, disposed in the Euclidean plane, move in such a way that, for all $\eps>0$, a configuration in which the separation between all entities is at most $\eps$ is eventually reached and maintained. The problem has been previously studied in a variety of settings, including full visibility, exact measurements (like distances and angles), and synchronous activation of entities. 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 {\sc Point Convergence}, for entities in the plane, in such a setting, provided the degree of asynchrony is bounded: while any one entity is active, any other entity can be activated at most $k$ times, for some arbitrarily large but fixed $k$. This provides a strong positive answer to a decade old open question posed by Katreniak. We also prove that in a comparable setting that permits unbounded asynchrony, {\sc Point Convergence} in the plane is impossible, contingent on the natural assumption that algorithms maintain the (visible) connectivity among entities present in the initial configuration. This variant, that we call {\sc Cohesive Convergence}, 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 synchronously scheduled entities are more powerful than asynchronously scheduled entities.