Freeze-Tag in $L_1$ has Wake-up Time Five

TL;DR

This work resolves the wake-up (Freeze-Tag) problem in the plane under the $L_1$-norm by proving a tight makespan bound of $5$ for waking all sleeping robots from a single active robot, with an $O(n)$-time construction of the wake-up tree. The approach partitions the unit $L_1$-disk into squares and isosceles triangles, proving key lemmas (S5, S6, and T) that enable a four-case strategy to achieve the bound, and extending these ideas to obtain a scalable linear-time algorithm. The results imply a new upper bound of $5\sqrt{2}$ on the makespan in the $L_2$-norm, and the paper introduces a general framework using wake-up ratios $\gamma(\eta)$ and the parameter $\Lambda(\eta)$ with conjectures and corollaries across norms. Two constructive strategies, Heap-Strategy and Split-Cone-Strategy, underpin the linear-time algorithm, enabling practical wake-up scheduling in normed planes and offering directions for future work, including tighter bounds for $L_2$, fixed-$n$ wake-up ratios, and higher-dimensional extensions.

Abstract

The Freeze-Tag Problem, introduced in Arkin et al. (SODA'02) consists of waking up a swarm of $n$ robots, starting from a single active robot. In the basic geometric version, every robot is given coordinates in the plane. As soon as a robot is awakened, it can move towards inactive robots to wake them up. The goal is to minimize the wake-up time of the last robot, the makespan. Despite significant progress on the computational complexity of this problem and on approximation algorithms, the characterization of exact bounds on the makespan remains one of the main open questions. In this paper, we settle this question for the $\ell_1$-norm, showing that a makespan of at most $5r$ can always be achieved, where $r$ is the maximum distance between the initial active robot and any sleeping robot. Moreover, a schedule achieving a makespan of at most $5r$ can be computed in optimal time $O(n)$. Both bounds, the time and the makespan are optimal. This implies a new upper bound of $5\sqrt{2}r \approx 7.07r$ on the makespan in the $\ell_2$-norm, improving the best known bound so far $(5+2\sqrt{2}+\sqrt{5})r \approx 10.06r$.

Figures (10)

• Figure 1: Example of a (here, euclidean) instance of FTP (on the left). The robot at $p_0$ must wake up $n = 11$ sleeping robots at $p_1,\dots,p_n$. In this example, positions are normalized in the unit $\ell_2$-disk, $p_0$ being at the center. The optimal solution, depicted by arrows, can be represented as a binary weighted tree (right). The makespan is the length of the longest (weighted) branch in that tree, here $2.594$, corresponding to the path $(p_0,p_1,p_2,p_8,p_9)$. Observe that, even if the sleeping robots are in a convex configuration, the optimal solution may have multiple edge crossings.
• Figure 2: The unit $\ell_1$-disk, divided into squares and triangles of diameter $1$.
• Figure 3: The triangle $T$ with vertices $B = (0,0)$, $C = (1,0)$ and $A = (1/2,1/2)$. In red, the possible starting points covered by the lemma.
• Figure 4: In case A, the robot located at the top awakes the closest sleeping robot, then each of them applies recursively the algorithm in one of the two subtriangles.
• Figure 5: Canonical subdivision of the triangle $T$, with vertices $B = (0,0)$, $C = (1,0)$ and $A = (1/2,1/2)$.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Corollary 4
• Conjecture 4
• Lemma 4
• Lemma 4
• Lemma 4
• Lemma 5