A Faster Algorithm for Finding Tarski Fixed Points
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.
query complexity
Tarski fixed points
total function problem
Theory of computation~Design and analysis of algorithms
29:1-29:16
Regular Paper
https://arxiv.org/abs/2010.02618
We would like to thank Kousha Etessami, Thomas Webster, and an anonymous reviewer for pointing out that the proof of Lemma 12 could be drastically simplified from its original version.
John
Fearnley
John Fearnley
Department of Computer Science, University of Liverpool, UK
Rahul
Savani
Rahul Savani
Department of Computer Science, University of Liverpool, UK
https://orcid.org/0000-0003-1262-7831
10.4230/LIPIcs.STACS.2021.29
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John Fearnley and Rahul Savani
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