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The Mutual Visibility Problem for Fat Robots with Lights

Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen

TL;DR

An algorithm that requires only 2 colors and $O(n) rounds is presented, which is optimal since at least two colors are required even for point 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].

The Mutual Visibility Problem for Fat Robots with Lights

TL;DR

An algorithm that requires only 2 colors and $O(n) rounds is presented, which is optimal since at least two colors are required even for point robots.

Abstract

Given a set of 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 rounds. In this work, we present an algorithm that requires only 2 colors and 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].
Paper Structure (11 sections, 16 theorems, 8 figures, 4 algorithms)

This paper contains 11 sections, 16 theorems, 8 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Preliminaries
  5. The Mutual Visibility Algorithm
  6. The Side Depletion Phase
  7. The Interior Depletion Phase
  8. Special Cases
  9. Analysis
  10. Concluding Remarks
  11. Pseudocode

Key Result

lemma thmcounterlemma

Given a configuration ${\mathbb{C}}_{k}$ and an edge $e=\overline{v_1v_2}$ of ${\mathbb{H}}_{k}$, if a robot $r_i\in e$ moves away from $e$, it will move into the safe zone $S(e)$.

Figures (8)

  • Figure 1: An example of an initial instance (a) and an end configuration (b).
  • Figure 2: (a) The safe zone of $e=\overline{v_1v_2}$. (b) The safe zone of a side robot $r_2$ on $e$.
  • Figure 3: Robot $r_1$ cannot determine whether robot $r_2$ blocks visibility to a corner robot. In either case the line parallel to $\overline{r_1 r_2}$ is used to compute the safe zone. (a) Robot $r_2$ hides a corner robot. (b) Robot $r_2$ does not hide a corner robot.
  • Figure 4: The different colors of the robots: corner robots (red), side robots (off), and interior robots (off).
  • Figure 5: One side robot $r_1$ on an edge $e=\overline{v_1v_2}$ moves to become a corner robot.
  • ...and 3 more figures

Theorems & Definitions (31)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 21 more