A faster algorithm for finding Tarski fixed points

16^th December 2020, 13:00
John Fearnley
University of Liverpool

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. 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.