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An Averagecase Analysis of the Gaussian Algorithm for Lattice Reduction
, 1996
"... .The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a ..."
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Cited by 47 (9 self)
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.The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, the probability distribution decays geometrically, and the dynamics is characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented. Une analyse en moyenne de l'algorithme de Gauss de r'eduction des r'eseaux R'esum'e. L'algorithme de r'eduction des r'eseaux en dimension 2 qui est du `a Gauss est analys'e sous sa forme dite standard. Il est 'etabli ici que, sous un mod`ele continu, sa complexit'e est constante en moyenne et que la distribution de probabilit'es associ'ee decroit g'eom'etriquement tandis que la dynamique est caract'eris'ee par une densit'e conditionnelle invariante. Les preuves f...
Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators
 ALGORITHMICA
, 1998
"... We provide here a complete averagecase analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N. We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of ..."
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Cited by 24 (3 self)
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We provide here a complete averagecase analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N. We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of the binary Euclidean algorithm, and finally the sum of the exponents of powers of 2 contained in the numerators of the binary continued fraction. The average values of these parameters are shown to be asymptotic to Ai log N, and the three constants Ai are related to the invariant measure of the Perron–Frobenius operator linked to this dynamical system. The binary Euclidean algorithm was previously studied in 1976 by Brent who provided a partial analysis of the number of steps, based on a heuristic model and some unproven conjecture. Our methods are quite different, not relying on unproven assumptions, and more general, since they allow us to study all the parameters of the binary continued fraction expansion.
Dynamical Analysis of a Class of Euclidean Algorithms
"... We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properti ..."
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Cited by 19 (5 self)
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We develop a general framework for the analysis of algorithms of a broad Euclidean type. The averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcdlike algorithms together with new results regarding the probable behaviour of their cost functions. 1
Digits and Continuants in Euclidean Algorithms. Ergodic versus Tauberian Theorems
, 2000
"... We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviou ..."
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Cited by 16 (6 self)
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We obtain new results regarding the precise average case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the averagecase complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters digits and continuants that intervene in an entire class of gcdlike algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.
Euclidean dynamics
 DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
, 2006
"... We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer ope ..."
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Cited by 4 (2 self)
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We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various tools of analytic combinatorics: generating functions, Dirichlet series, Tauberian theorems, Perron’s formula and quasipowers theorems. Such dynamical analyses can be used to perform the averagecase analysis of algorithms, but also (dynamical) analysis in distribution.