Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(log^k n) queries [Chuangyin Dang et al., 2020]. Multiple authors have conjectured that this algorithm is optimal [Chuangyin Dang et al., 2020; Kousha Etessami et al., 2020], and indeed this has been proven for two-dimensional instances [Kousha Etessami et al., 2020]. We show that these conjectures are false in dimension three or higher by giving an O(log² n) query algorithm for the three-dimensional Tarski problem, which generalises to give an O(log^{k-1} n) query algorithm for the k-dimensional problem when k ≥ 3.
@InProceedings{fearnley_et_al:LIPIcs.STACS.2021.29, author = {Fearnley, John and Savani, Rahul}, title = {{A Faster Algorithm for Finding Tarski Fixed Points}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {29:1--29:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.29}, URN = {urn:nbn:de:0030-drops-136741}, doi = {10.4230/LIPIcs.STACS.2021.29}, annote = {Keywords: query complexity, Tarski fixed points, total function problem} }
Feedback for Dagstuhl Publishing