We give an $O(\log^2 n)$-query algorithm for finding a Tarski fixed point over the $4$-dimensional lattice $[n]^4$, matching the $Ω(\log^2 n)$ lower bound of [EPRY20]. Additionally, our algorithm yields an ${O(\log^{\lceil (k-1)/3\rceil+1} n)}$-query algorithm for any constant $k$, improving the previous best upper bound ${O(\log^{\lceil (k-1)/2\rceil+1} n)}$ of [CL22]. Our algorithm uses a new framework based on \emph{safe partial-information} functions. The latter were introduced in [CLY23] to give a reduction from the Tarski problem to its promised version with a unique fixed point. This is the first time they are directly used to design new algorithms for Tarski fixed points.

翻译：我们给出了一个在$4$维格$[n]^4$上寻找Tarski不动点的$O(\log^2 n)$查询算法，匹配了[EPRY20]中的$Ω(\log^2 n)$下界。此外，对于任意常数$k$，我们的算法产生了一个${O(\log^{\lceil (k-1)/3\rceil+1} n)}$查询算法，改进了[CL22]之前的最佳上界${O(\log^{\lceil (k-1)/2\rceil+1} n)}$。我们的算法采用了一个基于\emph{安全部分信息}函数的新框架。后者由[CLY23]引入，用于将Tarski问题归约到其承诺存在唯一不动点的版本。这是首次直接利用这些函数设计Tarski不动点的新算法。