The randomized query complexity of finding a Tarski fixed point on the Boolean hypercube

The Knaster-Tarski theorem, also known as Tarski's theorem, guarantees that every monotone function defined on a complete lattice has a fixed point. We analyze the query complexity of finding such a fixed point on the k-dimensional grid of side length n under the ≤ relation. Specifically, there is an unknown monotone function $f:{\{0,1,\dots ,n-1\}}^k\to {\{0,1,\dots ,n-1\}}^k$ and an algorithm must query a vertex v to learn $f\left(v\right)$.

A key special case of interest is the Boolean hypercube ${\{0,1\}}^k$, which is isomorphic to the power set lattice—the original setting of the Knaster-Tarski theorem. We prove a lower bound that characterizes the randomized and deterministic query complexity of the Tarski search problem on the Boolean hypercube as $\Theta \left(k\right)$. More generally, we give a randomized lower bound of $\Omega (k+\frac{k\cdot \log n}{\log k})$ for the k-dimensional grid of side length n, which is asymptotically tight in high dimensions when k is large relative to n.