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A Faster Algorithm for Finding Tarski Fixed Points

Authors John Fearnley, Rahul Savani

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John Fearnley
  • Department of Computer Science, University of Liverpool, UK
Rahul Savani
  • Department of Computer Science, University of Liverpool, UK

Acknowledgements

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.

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John Fearnley and Rahul Savani. A Faster Algorithm for Finding Tarski Fixed Points. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.STACS.2021.29

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • query complexity
  • Tarski fixed points
  • total function problem

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References

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