We generalize the celebrated isoperimetric inequality of Khot, Minzer, andSafra~(SICOMP 2018) for Boolean functions to the case of real-valued functions . We show a matching lower bound for nonadaptive, 1-sided error testers for monotonicity . We apply our generalized isoperic inequality to improve algorithms fortesting monotonic and approximating the distance to monotone forreal-valued function $f$ . We also show that the distance of $O(\sqrt{d\logr)$ can be approximated nonadaptively with query complexity in $1/\alpha$ and the dimension $d$ This query complexity is known to be nearlyoptimal fornonadaptive algorithms even for the special case of Booleanfunctions . We conclude that the distances to monotonied functions that are $\alpha$-far from monotones are $O(d\Logr) and $O (D) is $O(‘D’) is $

Author(s) : Hadley Black, Iden Kalemaj, Sofya Raskhodnikova

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Keywords : functions - valued - algorithms - complexity - distance -

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