DMTCS Proceedings, 2005 International Conference on Analysis of Algorithms

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DMTCS Conference vol AD (2005), pp. 71-94


2005 International Conference on Analysis of Algorithms

Conrado Martínez (ed.)

DMTCS Conference Volume AD (2005), pp. 71-94

author: Benoît Daireaux, Véronique Maume-Deschamps and Brigitte Vallée
title: The Lyapunov tortoise and the dyadic hare
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
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/log2 ∼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]…], and the algorithm terminates.
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reference: Benoît Daireaux and Véronique Maume-Deschamps and Brigitte Vallée (2005), The Lyapunov tortoise and the dyadic hare, in 2005 International Conference on Analysis of Algorithms, Conrado Martínez (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AD, pp. 71-94
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