A faster algorithm for finding Tarski fixed points

10/06/2020

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by   John Fearnley, et al.

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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. Multiple authors have conjectured that this algorithm is optimal [Dang et al., Etessami et al.], and indeed this has been proven for two-dimensional instances [Etessami et al.]. We show that these conjectures are false in dimension three or higher by giving an O(log^2 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.