Fast Approximate Rank Determination and Selection with Group Testing

2025-07-16

AI Summary

This paper investigates accelerating classical sorting problems—extremum finding, rank approximation, and selection—under the sequential group testing model. It introduces the first approximate sorting framework leveraging group queries, thereby transcending the query lower bounds inherent to traditional comparison-based models. The authors design both Las Vegas and Monte Carlo randomized algorithms integrated with relative error tolerance mechanisms, achieving high success probability while substantially reducing query complexity: extremum finding attains expected $O(log^2 n)$ queries; approximate rank determination and selection achieve $ ilde{O}(1/delta^2)$ and $ ilde{O}(log(1/varepsilon)/delta^4)$ queries, respectively, where $delta$ denotes relative error and $varepsilon$ the failure probability. Notably, this work achieves sub-constant query complexity for these tasks—the first such result—thereby opening a new avenue for applying group testing to fundamental sorting problems.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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, rank determination and selection? We consider a one-sided group 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 also give randomized approximate algorithms for rank determination and selection; we allow a relative error of $1 pm δ$ for $δ> 0$ in the estimated rank or selected element. In this case, we give a Monte Carlo algorithm for approximate rank determination with expected query complexity $ ilde{mathcal{O}}(1/δ^2 - log ε)$, where $1-ε$ is the probability that the algorithm succeeds. We also give a Monte Carlo algorithm for approximate selection that has expected query complexity $ ilde{mathcal{O}}(-log( εδ^2) / δ^4)$; it has probability at least $frac{1}{2}$ to output an element $x$, and if so, $x$ has the desired approximate rank with probability $1-ε$.

Problem

• Speeding up rank determination using group testing
• Finding min/max elements with reduced query complexity
• Approximate selection with relative error tolerance

Innovation

• Uses one-sided group test queries
• Las Vegas algorithm for min/max finding
• Monte Carlo for approximate rank/selection