Noisy Computing of the OR and MAX Functions

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.