Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

• Main Authors: Gokarna Sharma, Ramachandran Vaidyanathan, Jerry L. Trahan
• Format: Article
• Language: English
• Published: MDPI AG 2021-02-01
• Series: Algorithms
• Subjects: distributed algorithms; autonomous mobile robots; robots with lights; complete visibility; collisions; convex hull
• Online Access: https://www.mdpi.com/1999-4893/14/2/56

We consider the distributed setting of <i>N</i> autonomous mobile robots that operate in <i>Look-Compute-Move</i> (LCM) cycles and use colored lights (the <i>robots with lights</i> model). We assume <i>obstructed visibility</i> where a robot cannot see another robot if a third robot is positioned between them on the straight line segment connecting them. In this paper, we consider the problem of positioning <i>N</i> autonomous robots on a plane so that every robot is visible to all others (this is called the <span style="font-variant: small-caps">Complete Visibility</span> problem). This problem is fundamental, as it provides a basis to solve many other problems under obstructed visibility. In this paper, we provide the first, asymptotically optimal, <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> time, <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> color algorithm for <span style="font-variant: small-caps">Complete Visibility</span> in the asynchronous setting. This significantly improves on an <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>-time translation of the existing <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> time, <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> color semi-synchronous algorithm to the asynchronous setting. The proposed algorithm is <i>collision-free</i>, i.e., robots do not share positions, and their paths do not cross. We also introduce a new technique for moving robots in an asynchronous setting that may be of independent interest, called Beacon-Directed Curve Positioning.