Fast Rational Search via Stern-Brocot Tree

TL;DR

This paper introduces a novel rational search algorithm using Stern-Brocot tree traversal, addressing bounded, unbounded, and approximation problems with comparison queries, improving efficiency and extending previous methods.

Contribution

A new rational search algorithm based on Stern-Brocot tree traversal, extending to unbounded search and rational approximation problems.

Findings

Efficient rational search algorithm using Stern-Brocot tree.

Extension to unbounded rational search where bounds are unknown.

Application to computing best rational approximations of real numbers.

Abstract

We revisit the problem of rational search: given an unknown rational number $\alpha = \frac{a}{b} \in (0,1)$ with $b \leq n$, the goal is to identify $\alpha$ using comparison queries of the form ``$\beta \leq \alpha$?''. The problem has been studied several decades ago and optimal query algorithms are known. We present a new algorithm for rational search based on a compressed traversal of the Stern--Brocot tree, which appeared to have been overlooked in the literature. This approach also naturally extends to two related problems that, to the best of our knowledge, have not been previously addressed: (i) unbounded rational search, where the bound $n$ is unknown, and (ii) computing the best (in a precise sense) rational approximation of an unknown real number using only comparison queries.