The Lyapunov tortoise and the dyadic hare

Benoît Daireaux 1 Véronique Maume-Deschamps 2 Brigitte Vallée 1
1 Equipe AMACC - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : We study a gcd algorithm directed by Least Significant Bits, the so―called LSB algorithm, and provide a precise average―case analysis of its main parameters [number of iterations, number of shifts, etc...]. This analysis is based on a precise study of the dynamical systems which provide a continuous extension of the algorithm, and, here, it is proved convenient to use both a 2―adic extension and a real one. This leads to the framework of products of random matrices, and our results thus involve a constant $γ$ which is the Lyapunov exponent of the set of matrices relative to the algorithm. The algorithm can be viewed as a race between a dyadic hare with a speed of 2 bits by step and a "real'' tortoise with a speed equal to $γ /\textit{log} \ 2 \sim 0.05$ bits by step. Even if the tortoise starts before the hare, the hare easily catches up with the tortoise [unlike in Aesop's fable [Ae]\ldots], and the algorithm terminates.
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Benoît Daireaux, Véronique Maume-Deschamps, Brigitte Vallée. The Lyapunov tortoise and the dyadic hare. 2005 International Conference on Analysis of Algorithms, 2005, Barcelona, Spain. pp.71-94. ⟨hal-01184044⟩

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