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\lceil \rceil } n) query algorithm for the k-dimensional problem when k%DIFDELCMD < \ge %%% 3.
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 %DIFDELCMD < \ge %%% .
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, which generalises to give
. We also give a new decomposition theorem for k-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives
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k-1
2
A
k/3
D
when k
R
3.
.