Noisy Computing of the OR\mathsf{OR} and MAX\mathsf{MAX} Functions

Abstract

We consider the problem of computing a function of nn variables using noisy queries, where each query is incorrect with some fixed and known probability p(0,1/2)p \in (0,1/2). Specifically, we consider the computation of the OR\mathsf{OR} function of nn bits (where queries correspond to noisy readings of the bits) and the MAX\mathsf{MAX} function of nn real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of (1±o(1))nlog1δDKL(p1p) (1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)} is both sufficient and necessary to compute both functions with a vanishing error probability δ=o(1)\delta = o(1), where DKL(p1p)D_{\mathsf{KL}}(p \| 1-p) denotes the Kullback-Leibler divergence between Bern(p)\mathsf{Bern}(p) and Bern(1p)\mathsf{Bern}(1-p) distributions. Compared to previous work, our results tighten the dependence on pp in both the upper and lower bounds for the two functions

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