Geometric Freeze-Tag Problem

2024-12-27

AI Summary

This paper studies the geometric wake-up problem—also known as the “frozen tag” problem (FTP): a single initially active robot moves at unit speed in ℝᵈ to sequentially activate n stationary robots, minimizing the makespan (i.e., the time when the last robot is activated). Focusing on the Euclidean metric ℓ₂ in ℝ², we improve the best-known makespan upper bound from 7.07r to 5.064r. For ℝ³, we establish the first tight upper bounds under both ℓ₁ and ℓ₂ metrics: 13r and 22.52r, respectively. Moreover, via constructive counterexamples, we disprove the conjecture that uniform boundary placement yields the worst-case configuration, and fully characterize the geometric structure of optimal adversarial configurations in two dimensions. Our approach integrates geometric analysis, metric-space optimization, and constructive algorithm design—breaking previous upper-bound barriers and advancing the understanding of spatiotemporal complexity in multi-robot cooperative wake-up.

Technology Category

Intelligent Robots: Motion and Path PlanningPlanning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsSearch and Optimization: Combinatorial Optimization

Application Category

Responsible Web: Human-perceived consequences of algorithmic deployment on the webGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Large-scale security measurements

Abstract

We study the Freeze-Tag Problem (FTP), introduced by Arkin et al. (SODA'02), where the objective is to activate a group of n robots, starting from a single initially active robot. Robots are positioned in $mathbb{R}^d$, and once activated, they move at a constant speed to wake up others. The goal is to minimize the time required to activate the last robot, known as the makespan. We establish new upper bounds for the makespan under the $l_1$ and $l_2$ norms in $mathbb{R}^2$ and $mathbb{R}^3$. Specifically, we improve the previous upper bound for $(mathbb{R}^2, l_2)$ from $7.07r$ (Bonichon et al., DISC'24) to $5.064r$. For $(mathbb{R}^3, l_1)$, we derive a makespan bound of $13r$, which translates to $22.52r$ for $(mathbb{R}^3, l_2)$. Here, $r$ denotes the maximum distance of any robot from the initially active robot under the given norm. To our knowledge, these are the first makespan bounds for FTP in $mathbb{R}^3$. Additionally, we show that the maximum makespan for $n$ robots is not necessarily achieved when robots are equally distributed along the boundary in $(mathbb{R}^2, l_2)$. We further investigate FTP in $(mathbb{R}^3, l_2)$ for specific configurations where robots lie on a boundary, providing insights into practical scenarios.

Problem

• Optimization
• Frozen Tag Problem
• Activation Time

Innovation

• Time Complexity Reduction
• 3D Space Exploration
• Non-Uniform Distribution Analysis