Noisy Computing of the OR and MAX Functions

Abstract

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