Fast Order Statistics with Group Inequality Testing

TL;DR

This paper introduces new randomized algorithms leveraging group inequality testing to efficiently compute order statistics, including min, max, and approximate rank, with significantly reduced query complexity.

Contribution

It presents novel Las Vegas and Monte Carlo algorithms for order statistics that improve query efficiency using group inequality testing in total orders.

Findings

Las Vegas algorithm for min/max with O(log^2 n) expected queries

Monte Carlo approximation for rank with complexity depending on δ and ε

Monte Carlo approximate selection with probabilistic guarantees

Abstract

Suppose that a group test operation is available for checking order relations in a set, can this speed up problems like finding the minimum/maximum element, determining the rank of element, and computing order statistics? We consider a one-sided group inequality test to be available, where queries are of the form $u \le_Q V$ or $V \le_Q u$, and the answer is `yes' if and only if there is some $v \in V$ such that $u \le v$ or $v \le u$, respectively. We restrict attention to total orders and focus on query-complexity; for min or max finding, we give a Las Vegas algorithm that makes $\mathcal{O}(\log^2 n)$ expected queries. We observe that rank determination can be solved with existing ``defect-counting'' algorithms, but also give a simple Monte Carlo approximation algorithm with expected query complexity $\tilde{\mathcal{O}}(\frac{1}{\delta^2} \log \frac{1}{\epsilon})$, where $1-\epsilon$ is the probability that the algorithm succeeds and we allow a relative error of $1 \pm \delta$ for $\delta > 0$ in the estimated rank. We then give a Monte Carlo algorithm for approximate selection that has expected query complexity $\tilde{\mathcal{O}}(\frac{1}{\delta^4}\log \frac{1}{\epsilon \delta^2} )$; it has probability at least $\frac{1}{2}$ to output an element $x$, and if so, $x$ has the desired approximate rank with probability $1-\epsilon$. Keywords: Order statistics, Group inequality testing, Randomized algorithms