We design a nonadaptive algorithm that, given oracle access to a function which is -far from monotone, makes poly queries and returns an estimate that, with high probability, is an -approximation to the distance of to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly-query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant every nonadaptive -approximation algorithm for this problem requires queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure-resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a -junta.

中文翻译：

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逼近布尔函数单调性的距离

我们设计了一个非自适应算法，在给定预言机访问一个远离单调的函数的情况下，进行多查询并返回一个估计值，该估计值很有可能是单调性距离的近似值。我们算法的分析依赖于对 Khot、Minzer 和 Safra 的有向等周不等式的改进（SIAM J. Comput.，2018）。此外，我们排除了一种多查询非自适应算法，该算法通过表明对于所有常量，该问题的每个非自适应逼近算法都需要查询，从而显着更好地逼近单调性的距离。这回答了 Seshadhri 的问题（属性测试评论, 2014) 对于非自适应算法的情况。我们通过证明擦除弹性（和容忍）测试人员的类似界限来获得我们的下限。我们的方法也为 unateness 和成为-junta 产生了相同的下限。