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/Exposé /Talk/

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 makespan 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 time /O(n)/. Both bounds, the time and the makespan are optimal. Our results also imply for the $\ell_2$-norm a new upper bound of /5√2r ≈ 7.07r/ on the makespan, improving the best known bound of /(5+2√2+√5)r ≈ 10.06r/.

Along the way, we introduce new linear time wake-up strategies, that apply to any norm and show that an optimal bound on the makespan can always be achieved by a schedule computable in linear time.

[Nicolas Bonichon ] (Université de Bordeaux, LaBRI )
https://www.labri.fr/perso/bonichon/wiki/pmwiki.php

Remarks / Remarques

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