Noisy Computing of the 𝖮𝖱 and 𝖬𝖠𝖷 Functions
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We consider the problem of computing a function of n variables using noisy queries, where each query is incorrect with some fixed and known probability p ∈ (0,1/2). Specifically, we consider the computation of the 𝖮𝖱 function of n bits (where queries correspond to noisy readings of the bits) and the 𝖬𝖠𝖷 function of n real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of (1 ± o(1)) nlog1/δ/D_𝖪𝖫(p 1-p) is both sufficient and necessary to compute both functions with a vanishing error probability δ = o(1), where D_𝖪𝖫(p 1-p) denotes the Kullback-Leibler divergence between 𝖡𝖾𝗋𝗇(p) and 𝖡𝖾𝗋𝗇(1-p) distributions. Compared to previous work, our results tighten the dependence on p in both the upper and lower bounds for the two functions.
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