Freeze-Tag in L_1 has Wake-up Time Five

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 ℓ_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√(2)r ≈ 7.07r on the makespan in the ℓ_2-norm, improving the best known bound so far (5+2√(2)+√(5))r ≈ 10.06r.