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An Optimal Algorithm for Online Freeze-Tag

Authors Josh Brunner, Julian Wellman

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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.FUN.2021.8

Author Details

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

Acknowledgements

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.

References

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