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.
@InProceedings{brunner_et_al:LIPIcs.FUN.2021.8, author = {Brunner, Josh and Wellman, Julian}, title = {{An Optimal Algorithm for Online Freeze-Tag}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {8:1--8:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.8}, URN = {urn:nbn:de:0030-drops-127693}, doi = {10.4230/LIPIcs.FUN.2021.8}, annote = {Keywords: Online algorithm, competitive ratio, freeze-tag} }
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