Approximate selection with unreliable comparisons in sublinear time

Abstract

Given n elements, an integer $k\le \frac{n}{2}$ and a parameter $\epsilon \ge \frac{1}{n}$, we study the problem of selecting an element with rank in $(k-n\epsilon ,k+n\epsilon ]$ using unreliable comparisons where the outcome of each comparison is incorrect independently with a constant error probability, and multiple comparisons between the same pair of elements are independent. We develop a randomized algorithm that performs expected $O(\frac{k}{n}{\epsilon }^{-2}\log \frac{1}{Q})$ comparisons to achieve success probability at least $1-Q$. We also prove that even in the absence of comparison faults, any randomized algorithm with success probability at least $1-Q$ performs expected $\Omega (\min \{n,\frac{k}{n}{\epsilon }^{-2}\log \frac{1}{Q}\left\}\right)$ comparisons. In particular, our algorithm is optimal as long as n is large enough, i.e., when $n=\Omega (\frac{k}{n}{\epsilon }^{-2}\log \frac{1}{Q})$; outside this parameter range, no algorithm performs a sublinear number of comparisons. Surprisingly, for constant Q, our algorithm performs expected $O\left(\frac{k}{n}{\epsilon }^{-2}\right)$ comparisons with and without comparison faults, while for the exact selection problem, the expected number of comparisons is $\Theta (n\log k)$ with faults versus $\Theta \left(n\right)$ without faults.