On the power of bounded asynchrony: convergence by autonomous robots with limited visibility

Abstract

A distributed algorithm A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} solves the Point Convergence task if an arbitrarily large collection of entities, starting in an arbitrary configuration, move under the control of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} to eventually form and thereafter maintain configurations in which the separation between all entities is arbitrarily small. This fundamental task in the standard OBLOT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {OBLOT}$$\\end{document} model of autonomous mobile entities has been previously studied in a variety of settings, including full visibility, exact measurements (including 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 operating under these constraints that solves Point Convergence, for entities moving in two or three dimensional space, with any bounded degree of asynchrony. We also prove that under similar realistic constraints, but unbounded asynchrony, Point Convergence in the plane is not possible in general, contingent on the natural assumption that algorithms maintain the (visible) connectivity among entities present in the initial configuration. This variant, that we call Cohesive Convergence, serves to distinguish the power of bounded and unbounded asynchrony in the control of autonomous mobile entities, settling a long-standing question whether in the Euclidean plane synchronously scheduled entities are more powerful than asynchronously scheduled entities.