The Mutual Visibility Problem for Fat Robots with Lights

TL;DR

This paper introduces an efficient algorithm for mutual visibility among fat robots with lights, reducing the number of colors needed from nine to two while maintaining linear rounds, improving previous solutions.

Contribution

It presents a novel algorithm that achieves mutual visibility with only two colors and linear rounds, optimizing the luminous robots model for fat robots.

Findings

Requires only 2 colors for mutual visibility

Achieves $O(n)$ rounds complexity

Optimal number of colors needed for fat robots

Abstract

Given a set of $n\geq 1$ unit disk robots in the Euclidean plane, we consider the fundamental problem of providing mutual visibility to them: the robots must reposition themselves to reach a configuration where they all see each other. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between them. This problem was solved by Sharma et al. [ICDCN, 2018] in the luminous robots model, where each robot is equipped with an externally visible light that can assume colors from a fixed set of colors, using 9 colors and $O(n)$ rounds. In this work, we present an algorithm that requires only 2 colors and $O(n)$ rounds. The number of colors is optimal since at least two colors are required even for point robots [Di Luna et al., Information and Computation, 2017].