Document Open Access Logo

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

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

PDF
Thumbnail PDF

File

LIPIcs.DISC.2024.9.pdf
  • Filesize: 0.9 MB
  • 16 pages

Document Identifiers

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 As Get 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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Zachary Abel, Hugo A. Akitaya, and Yu Jingjin. Freeze tag awakening in 2D is NP-hard. In 27th Annual Fall Workshop on Computational Geometry (FWCG), November 2017. URL: https://www.ams.stonybrook.edu/~jsbm/fwcg17/proceedings.html.
  2. Esther M. Arkin, Michael A. Bender, Sándor P. Fekete, Joseph S.B. Mitchell, and Martin Skutella. The freeze-tag problem: How to wake up a swarm of robots. In 13th Symposium on Discrete Algorithms (SODA), pages 568-577. ACM-SIAM, 2002. URL: http://dl.acm.org/citation.cfm?id=545381.545457.
  3. Esther M. Arkin, Michael A. Bender, Sándor P. Fekete, Joseph S.B. Mitchell, and Martin Skutella. The freeze-tag problem: How to wake up a swarm of robots. Algorithmica, 46:193-221, 2006. URL: https://doi.org/10.1007/s00453-006-1206-1.
  4. Esther M. Arkin, Michael A. Bender, Dongdong Ge, Simai He, and Joseph S.B. Mitchell. Improved approximation algorithms for the freeze-tag problem. In 15th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 295-303. ACM Press, June 2003. URL: https://doi.org/10.1145/777412.777465.
  5. Amitai Armona, Adi Avidora, and Oded Schwartz. Cooperative tsp. Theoretical Computer Science, 411(31-33):2847-2863, June 2010. URL: https://doi.org/10.1016/j.tcs.2010.04.016.
  6. Sanjeev Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753-782, September 1998. URL: https://doi.org/10.1145/290179.290180.
  7. Nicolas Bonichon, Arnaud Casteigts, Cyril Gavoille, and Nicolas Hanusse. Freeze-tag in L1 has wake-up time five. Technical Report 2402.03258v1 [cs.DS], arXiv, February 2024. URL: https://doi.org/2402.03258v1.
  8. Josh Brunner and Julian Wellman. An optimal algorithm for online freeze-tag. In 10th International Conference Fun with Algorithms (FUN), volume 157 of LIPIcs, pages 8:1-11, September 2020. URL: https://doi.org/10.4230/LIPIcs.FUN.2021.8.
  9. Dan George Bucatanschi. The ant colony system for the freeze-tag problem. In Midstates Conference on Undergraduate Research in Mathematics and Computer Science (MCURCSM), pages 61-69, 2004. Google Scholar
  10. Dan Georges Bucatanschi, Blaine Hoffmann, Kevin R. Hutson, and R. Matthew Kretchmar. A neighborhood search technique for the freeze tag problem. In Extending the Horizons: Advances in Computing, Optimization, and Decision Technologies, volume 37 of Operations Research/Computer Science Interfaces Series, pages 97-113, 2007. URL: https://doi.org/10.1007/978-0-387-48793-9_7.
  11. Shantanu Das. Graph explorations with mobile agents. In Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors, Distributed Computing by Mobile Entities, volume 11340 of Lecture Notes in Computer Science, chapter 16, pages 403-422. Springer, Cham, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_16.
  12. L. Few. The shortest path and the shortest road through n points. Mathematika, 2(2):141-144, 1955. URL: https://doi.org/10.1112/S0025579300000784.
  13. Jil Gao. Normal structure and the arc length in banach spaces. Taiwanese Journal of Mathematics, 5(2):353-366, June 2001. URL: http://www.jstor.org/stable/43828249.
  14. Mikael Hammar, Bengt J. Nilsson, and Mia Persson. The online freeze-tag problem. In 7th Latin American Symposium on Theoretical Informatics (LATIN), volume 3887 of Lecture Notes in Computer Science, pages 569-579. Springer, March 2006. URL: https://doi.org/10.1007/11682462_53.
  15. Matthew Johnson. Easier hardness for 3D freeze-tag. In 27th Annual Fall Workshop on Computational Geometry (FWCG), November 2017. URL: https://www.ams.stonybrook.edu/~jsbm/fwcg17/proceedings.html.
  16. Marek Karpinski. Towards better inapproximability bounds for TSP: A challenge of global dependencies. In Electronic Colloquium on Computational Complexity (ECCC), TR15-097, June 2015. URL: https://eccc.weizmann.ac.il/report/2015/097/.
  17. Hamidreza Keshavarz. Applying tabu search to the freeze-tag. In 1st Conference on Swarm Intelligence and Evolutionary Computation (CSIEC), pages 37-41. IEEE Computer Society Press, March 2016. URL: https://doi.org/10.1109/CSIEC.2016.7482136.
  18. Jochen Könemann, Asaf Levin, and Amitabh Sinha. Approximating the degree-bounded minimum diameter spanning tree problem. Algorithmica, 41(2):117-129, 2005. URL: https://doi.org/10.1007/s00453-004-1121-2.
  19. Joseph S.B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing, 28(4):1298-1309, 1999. URL: https://doi.org/10.1137/S0097539796309764.
  20. Zahra Moezkarimi and Alireza Bagheri. A PTAS for geometric 2-FTP. Information Processing Letters, 114(12):670-675, 2014. URL: https://doi.org/10.1016/j.ipl.2014.06.017.
  21. Lehilton Lelis Chaves Pedrosa and Lucas de Oliveira Silva. Freeze-tag is NP-hard in 3D with L₁ distance. In 12th Latin-American Algorithms, Graphs and Optimization Symposium (LAGOS), volume 223:C, pages 360-366. Procedia Computer Science, September 2023. URL: https://doi.org/10.1016/j.procs.2023.08.248.
  22. Carsten Rössner and Jean-Pierre Seifert. Hardness of approximating shortest integer relations among rational numbers. Theoretical Computer Science, 209(1-2):287-297, December 1998. URL: https://doi.org/10.1016/S0304-3975(97)00118-7.
  23. Juan Jorge Schäffer. Geometry of Spheres in Normed Spaces, volume 20 of Lecture Notes in Pure and Applied Mathematics. Dekker, Marcel, 1976. Google Scholar
  24. Trevor Standley. Finding optimal solutions to cooperative pathfinding problems. In 24th AAAI Conference on Artificial Intelligence (AAAI), volume 24, pages 173-178. AAAI Press, July 2010. URL: https://doi.org/10.1609/aaai.v24i1.7564.
  25. Marcelo O. Sztainberg, Esther M. Arkin, Michael A. Bender, and Joseph S.B. Mitchell. Theoretical and experimental analysis of heuristics for the "freeze-tag" robot awakening problem. IEEE Transactions on Robotics, 20(4):691-701, August 2004. URL: https://doi.org/10.1109/TRO.2004.829439.
  26. Vera Traub, Jens Vygen, and Rico Zenklusen. Reducing path TSP to TSP. In 52nd Annual ACM Symposium on Theory of Computing (STOC), pages 14-27. ACM Press, June 2020. URL: https://doi.org/10.1145/3357713.3384256.
  27. Ehsan Najafi Yazdia, Alireza Bagheri, Zahra Moezkarimia, and Hamidreza Keshavarz. An O(1)-approximation algorithm for the 2-dimensional geometric freeze-tag problem. Information Processing Letters, 115(6-8):618-622, June 2015. URL: https://doi.org/10.1016/j.ipl.2015.02.011.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact