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Freeze-Tag in L₁ Has Wake-Up Time Five with Linear Complexity

Authors Nicolas Bonichon , Arnaud Casteigts , Cyril Gavoille , Nicolas Hanusse

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Author Details

Nicolas Bonichon
  • LaBRI, University of Bordeaux, CNRS, Bordeaux INP, France
Arnaud Casteigts
  • LaBRI, University of Bordeaux, CNRS, Bordeaux INP, France
  • CS Department, University of Geneva, Switzerland
Cyril Gavoille
  • LaBRI, University of Bordeaux, CNRS, Bordeaux INP, France
Nicolas Hanusse
  • LaBRI, University of Bordeaux, CNRS, Bordeaux INP, France

Funding

The last three authors are supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL).

Acknowledgements

We thank J.S. Mitchell and the anonymous referees for their advice on a previous version of this article.

Cite AsGet BibTex

Nicolas Bonichon, Arnaud Casteigts, Cyril Gavoille, and Nicolas Hanusse. Freeze-Tag in L₁ Has Wake-Up Time Five with Linear Complexity. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.DISC.2024.9

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 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 𝓁₁-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 𝓁₂-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • freeze-tag problem
  • metric
  • algorithm

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