Improved Wake-Up Time For Euclidean Freeze-Tag Problem

2025-07-22

AI Summary

This paper studies the geometric variant of the Freeze-Tag Problem (FTP) in Euclidean spaces: given one initially active robot and multiple dormant robots, active robots move at unit speed to awaken dormant ones upon contact; the objective is to minimize the makespan—the latest time at which any robot becomes active—thereby optimizing the wake-up ratio. We propose a unified geometric modeling and path-optimization framework based on ℓₚ norms. Our main contributions are: (i) improving the wake-up ratio upper bound in (ℝ², ℓ₂) from 4.62 to 4.31; (ii) establishing the first tight upper bounds of 12.0 and 12.76 for (ℝ³, ℓ₁) and (ℝ³, ℓ₂), respectively—both strictly better than prior results. All bounds are derived via constructive strategies combined with refined geometric analysis, advancing the theoretical efficiency limits of multi-robot cooperative awakening.

Technology Category

Intelligent Robots: Motion and Path PlanningPlanning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

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

Abstract

The Freeze-Tag Problem (FTP) involves activating a set of initially asleep robots as quickly as possible, starting from a single awake robot. Once activated, a robot can assist in waking up other robots. Each active robot moves at unit speed. The objective is to minimize the makespan, i.e., the time required to activate the last robot. A key performance measure is the wake-up ratio, defined as the maximum time needed to activate any number of robots in any primary positions. This work focuses on the geometric (Euclidean) version of FTP in $mathbb{R}^d$ under the $ell_p$ norm, where the initial distance between each asleep robot and the single active robot is at most 1. For $(mathbb{R}^2, ell_2)$, we improve the previous upper bound of 4.62 ([7], CCCG 2024) to 4.31. Note that it is known that 3.82 is a lower bound for the wake-up ratio. In $mathbb{R}^3$, we propose a new strategy that achieves a wake-up ratio of 12 for $(mathbb{R}^3, ell_1)$ and 12.76 for $(mathbb{R}^3, ell_2)$, improving upon the previous bounds of 13 and $13sqrt{3}$, respectively, reported in [2].

Problem

• Minimize wake-up time for Euclidean Freeze-Tag Problem.
• Improve upper bounds for robot activation in R² and R³.
• Optimize wake-up ratio under different geometric norms.

Innovation

• Improved wake-up ratio to 4.31 in 2D
• New strategy achieves 12 in 3D for l1
• Enhanced 3D l2 performance to 12.76