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Geometric Freeze-Tag Problem

Sharareh Alipour, Kajal Baghestani, Mahdis Mirzaei, Soroush Sahraei

TL;DR

This work advances the geometric Freeze-Tag Problem by deriving new upper bounds for wake-up times in low-dimensional Euclidean spaces under $l_2$ and $l_1$ norms. It introduces two core strategies in 2D, Arc-Strategy and Ring-Strategy, and combines them to achieve a new bound of $5.4162$ for $(\mathbb{R}^2, l_2)$, improving prior results. In 3D, it provides the first bounds for makespan under these norms, proving $13$ for $(\mathbb{R}^3, l_1)$ (implying $22.52$ for $(\mathbb{R}^3, l_2)$), and then extends to boundary and surface variants with a $12.37$ bound for boundary-FTP and $11.65$ for surface-FTP. A key methodological feature is a boundary-to-disk mapping that preserves or increases pairwise distances, enabling the transfer of disk-based strategies to hemisphere and surface problems, and yielding practical benchmarks for multi-robot wake-up tasks.

Abstract

We study the Freeze-Tag Problem (FTP), introduced by Arkin et al. (SODA'02), where the goal is to wake up a group of $n$ robots, starting from a single active robot. Our focus is on the geometric version of the problem, where robots are positioned in $\mathbb{R}^d$, and once activated, a robot can move at a constant speed to wake up others. The objective is to minimize the time it takes to activate the last robot, also known as the makespan. We present new upper bounds for the $l_1$ and $l_2$ norms in $\mathbb{R}^2$ and $\mathbb{R}^3$. For $(\mathbb{R}^2, l_2)$, we achieve a makespan of at most $5.4162r$, improving on the previous bound of $7.07r$ by Bonichon et al. (DISC'24). In $(\mathbb{R}^3, l_1)$, we establish an upper bound of $13r$, which leads to a bound of $22.52r$ for $(\mathbb{R}^3, l_2)$. Here, $r$ denotes the maximum distance of a robot from the initially active robot under the given norm. To the best of our knowledge, these are the first known bounds for the makespan in $\mathbb{R}^3$ under these norms. We also explore the FTP in $(\mathbb{R}^3, l_2)$ for specific instances where robots are positioned on a boundary, providing further insights into practical scenarios.

Geometric Freeze-Tag Problem

TL;DR

This work advances the geometric Freeze-Tag Problem by deriving new upper bounds for wake-up times in low-dimensional Euclidean spaces under and norms. It introduces two core strategies in 2D, Arc-Strategy and Ring-Strategy, and combines them to achieve a new bound of for , improving prior results. In 3D, it provides the first bounds for makespan under these norms, proving for (implying for ), and then extends to boundary and surface variants with a bound for boundary-FTP and for surface-FTP. A key methodological feature is a boundary-to-disk mapping that preserves or increases pairwise distances, enabling the transfer of disk-based strategies to hemisphere and surface problems, and yielding practical benchmarks for multi-robot wake-up tasks.

Abstract

We study the Freeze-Tag Problem (FTP), introduced by Arkin et al. (SODA'02), where the goal is to wake up a group of robots, starting from a single active robot. Our focus is on the geometric version of the problem, where robots are positioned in , and once activated, a robot can move at a constant speed to wake up others. The objective is to minimize the time it takes to activate the last robot, also known as the makespan. We present new upper bounds for the and norms in and . For , we achieve a makespan of at most , improving on the previous bound of by Bonichon et al. (DISC'24). In , we establish an upper bound of , which leads to a bound of for . Here, denotes the maximum distance of a robot from the initially active robot under the given norm. To the best of our knowledge, these are the first known bounds for the makespan in under these norms. We also explore the FTP in for specific instances where robots are positioned on a boundary, providing further insights into practical scenarios.
Paper Structure (14 sections, 10 theorems, 7 equations, 9 figures)

This paper contains 14 sections, 10 theorems, 7 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. FTP in R2, l2
  4. The wake-up ratio is at most 5.4162 in R2, l2
  5. Arc-Strategy
  6. Ring-Strategy
  7. Our combined strategy for FTP in $(\mathbb{R}^2,l_2)$
  8. Some examples
  9. FTP in R3, l1
  10. Our strategy for FTP in R3, l1
  11. Wake-up ratio for robots on the boundary of the unit l2-ball in R3
  12. Mapping
  13. Our strategy, when asleep robots are on the boundary on the unit l2-ball
  14. Our strategy for surface-FTP

Key Result

theorem 1

A robot at the origin can wake up any set of $n$ asleep robots in the unit $l_2$-disk with a makespan of at most $5.4162$.

Figures (9)

  • Figure 1: Representation of the unit $l_1$-ball in $\mathbb{R}^3$
  • Figure 2: An FTP instance and its wake-up tree. The left diagram shows the positions and movements of the robots inside the $l_2$-disk, while the right diagram displays the corresponding wake-up tree. Red arrows indicate the path from the root to a leaf.
  • Figure 3: A path in the Arc-Strategy
  • Figure 4: A path in the Ring-Strategy.
  • Figure 5: Two instances of FTP in $(\mathbb{R}^2,l_2)$ showing that the wake-up ratio is not attained for points equally distributed on the unit circle where $\theta$ is the angle from the positive $x$-axis in degree.
  • ...and 4 more figures

Theorems & Definitions (10)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • lemma 6