Improved Wake-Up Time For Euclidean Freeze-Tag Problem

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 $13\sqrt{3}$, respectively, reported in [2].