PapersViola considered the complexity of (approximately) sampling from a given distribution. We consider the uniform distribution over vectors in $\{0,1\}^n$ of weight $k$. We show that when $k$ is constant, any approximate sampler must have locality $\tilde\Omega(\log n)$, almost matching the upper bound $O(\log n)$.
Beyersdorff et al. considered the complexity of generating from a given set. One natural example is the set of vectors in $\{0,1\}^n$ with a majority of ones. They gave a “proof system” with locality $O(\log^2 n)$, and proved a lower bound of $\Omega(\log^* n)$. We improve the lower bound to $\Omega(\sqrt{\log n})$.
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