An Optimal Algorithm for Online Freeze-Tag

Authors Josh Brunner, Julian Wellman

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Josh Brunner
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Julian Wellman
  • Massachusetts Institute of Technology, Cambridge, MA, USA


The generalization of the metric in Section 4 to the construction of Section 5 was inspired by a discussion with Yuan Yao. We thank David Karger and Aleksander Madry for teaching the algorithms course where this project began, and for their useful feedback on a draft of this paper. We also thank three peer editors, Leo Castro, Roberto Ortiz, and Tianyi Zeng for their helpful comments on an early draft. Finally, we appreciate the helpful comments of the anonymous reviewers of this paper.

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Josh Brunner and Julian Wellman. An Optimal Algorithm for Online Freeze-Tag. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 8:1-8:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In the freeze-tag problem, one active robot must wake up many frozen robots. The robots are considered as points in a metric space, where active robots move at a constant rate and activate other robots by visiting them. In the (time-dependent) online variant of the problem, each frozen robot is not revealed until a specified time. Hammar, Nilsson, and Persson have shown that no online algorithm can achieve a competitive ratio better than 7/3 for online freeze-tag, and posed the question of whether an O(1)-competitive algorithm exists. We provide a (1+√2)-competitive algorithm for online time-dependent freeze-tag, and show that this is the best possible: there does not exist an algorithm which achieves a lower competitive ratio on every metric space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Online algorithm
  • competitive ratio
  • freeze-tag


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  1. Esther M Arkin, Michael A Bender, Sándor P Fekete, Joseph SB Mitchell, and Martin Skutella. The freeze-tag problem: how to wake up a swarm of robots. Algorithmica, 46(2):193-221, 2006. Google Scholar
  2. Esther M Arkin, Michael A Bender, and Dongdong Ge. Improved approximation algorithms for the freeze-tag problem. In Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures, pages 295-303. ACM, 2003. Google Scholar
  3. Mikael Hammar, Bengt J Nilsson, and Mia Persson. The online freeze-tag problem. In Latin American Symposium on Theoretical Informatics, pages 569-579. Springer, 2006. Google Scholar
  4. Jochen Könemann, Asaf Levin, and Amitabh Sinha. Approximating the degree-bounded minimum diameter spanning tree problem. Algorithmica, 41(2):117-129, 2005. Google Scholar
  5. Zahra Moezkarimi and Alireza Bagheri. A PTAS for geometric 2-FTP. Information Processing Letters, 114(12):670-675, 2014. Google Scholar
  6. Marcelo O Sztainberg, Esther M Arkin, Michael A Bender, and Joseph SB Mitchell. Analysis of heuristics for the freeze-tag problem. In Scandinavian Workshop on Algorithm Theory, pages 270-279. Springer, 2002. Google Scholar
  7. Ehsan Najafi Yazdi, Alireza Bagheri, Zahra Moezkarimi, and Hamidreza Keshavarz. An O(1)-approximation algorithm for the 2-dimensional geometric freeze-tag problem. Information Processing Letters, 115(6-8):618-622, 2015. Google Scholar
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