A faster algorithm for finding Tarski fixed points

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. We also give a new decomposition theorem for $k$-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an $O({\log }^{2\lceil k/3\rceil }n)$ query algorithm for the $k$-dimensional problem.