Fast Order Statistics with Group Inequality Testing
Adiesha Liyanage, Brendan Mumey, Braeden Sopp
TL;DR
The paper investigates speedups for order-statistics problems under a group inequality testing oracle, focusing on totally ordered sets. It introduces a Las Vegas min-finding algorithm with $O( ext{log}^2 n)$ expected group-test queries and shows that max-finding follows by reversing query direction. It also provides a Monte Carlo approach for approximate rank with complexity $Oig( rac{ ext{log} n}{ ext{delta}^2}( ext{log} ext{log} n+ ext{log}(1/ ext{epsilon})ig)$ and an approximate selection method with $Oig( rac{1}{ ext{delta}^2} ext{log}^2 N+ rac{1}{ ext{delta}^4} ext{log} rac{1}{ ext{epsilon} ext{delta}^2}ig)$ queries, both with probabilistic performance guarantees. These results extend group-testing techniques to ranking and selection, providing new query-efficient strategies and highlighting the potential and limitations of group tests for order-based problems. The work offers practical Monte Carlo schemes with provable bounds and lays groundwork for future exploration in posets and noisy testing settings.
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 $\tilde{\mathcal{O}}(\frac{1}{δ^2} \log \frac{1}ε)$, where $1-ε$ is the probability that the algorithm succeeds and we allow a relative error of $1 \pm δ$ for $δ> 0$ in the estimated rank. We then give a Monte Carlo algorithm for approximate selection that has expected query complexity $\tilde{\mathcal{O}}(\frac{1}{δ^4}\log \frac{1}{εδ^2} )$; it has probability at least $\frac{1}{2}$ to output an element $x$, and if so, $x$ has the desired approximate rank with probability $1-ε$. Keywords: Order statistics, Group inequality testing, Randomized algorithms