Analysis of the Continued Logarithm Algorithm

TL;DR

This paper provides a detailed average-case analysis of the Continued Logarithm Algorithm, revealing precise asymptotics for its key parameters by studying an underlying dynamical system with a dyadic component.

Contribution

It introduces a novel dyadic component to analyze the algorithm's average behavior, extending previous studies and providing explicit asymptotic constants.

Findings

Precise asymptotics for the number of iterations

Asymptotic behavior of total shifts analyzed

Explicit constants derived for average-case parameters

Abstract

The Continued Logarithm Algorithm - CL for short- introduced by Gosper in 1978 computes the gcd of two integers; it seems very efficient, as it only performs shifts and subtractions. Shallit has studied its worst-case complexity in 2016 and showed it to be linear. We here perform the average-case analysis of the algorithm: we study its main parameters (number of iterations, total number of shifts) and obtain precise asymptotics for their mean values. Our 'dynamical' analysis involves the dynamical system underlying the algorithm, that produces continued fraction expansions whose quotients are powers of 2. Even though this CL system has already been studied by Chan (around 2005), the presence of powers of 2 in the quotients ingrains into the central parameters a dyadic flavour that cannot be grasped solely by studying the CL system. We thus introduce a dyadic component and deal with a two-component system. With this new mixed system at hand, we then provide a complete average-case analysis of the CL algorithm, with explicit constants.