The Freeze-Tag Problem (FTP) involves activating a set of initially inactive robots as quickly as possible, starting from a single active robot. Once activated, a robot can assist in activating 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 all of the robots in any initial configuration. This work focuses on the geometric (Euclidean) version of FTP in \(\varvec{\mathbb {R}}^{\varvec{d}}\) under the \(\varvec{\ell }_{\varvec{p}}\) norm, where the initial distance between each inactive robot and the single active robot is at most \(\varvec{1}\) . For \(\varvec{(\mathbb {R}}^{\varvec{2}}\varvec{, \ell }_{\varvec{2}}\varvec{)}\) , we improve the previous upper bound of \(\varvec{4.62}\) (Bonichon et al. [1], CCCG 2024) to \(\varvec{4.31}\) . The known lower bound for the wake-up ratio is \(\varvec{3.82}\) . In \(\varvec{\mathbb {R}}^{\varvec{3}}\) , we propose a new strategy that achieves a wake-up ratio of \(\varvec{12}\) for \(\varvec{(\mathbb {R}}^{\varvec{3}}\varvec{, \ell }_{\varvec{1}}\varvec{)}\) and \(\varvec{12.76}\) for \(\varvec{(\mathbb {R}}^{\varvec{3}}\varvec{, \ell }_{\varvec{2}}\varvec{)}\) . We also explore the FTP in \(\varvec{(\mathbb {R}}^{\varvec{3}}\varvec{, \ell }_{\varvec{2}}\varvec{)}\) for specific instances where robots are positioned on the boundary of a sphere, providing further insights into practical scenarios. Finally, we demonstrate the practical efficiency of our \(\varvec{(\mathbb {R}}^{\varvec{2}}\varvec{, \ell }_{\varvec{2}}\varvec{)}\) algorithm through simulations on real-world spatial data.
