Results 1  10
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18
Euclidean algorithms are Gaussian
, 2003
"... Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further an ..."
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Cited by 28 (12 self)
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Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further and prove a Local Limit Theorem (LLT), with speed of convergence O((log N) −1/4+ǫ). This extends and improves the LLT obtained by Hensley [27] in the case of the standard Euclidean algorithm. We use a “dynamical analysis ” methodology, viewing an algorithm as a dynamical system (restricted to rational inputs), and combining tools imported from dynamics, such as the crucial transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, the saddle point method. Dynamical analysis had previously been used to perform averagecase analysis of algorithms. For the present (dynamical) analysis in distribution, we require precise estimates on the transfer operators, when a parameter varies along vertical lines in the complex plane. Such estimates build on results obtained only recently by Dolgopyat in the context of continuoustime dynamics [20]. 1.
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 21 (6 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 18 (7 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.
Dynamical analysis of αEuclidean algorithms
 J. Algorithms
"... Abstract We study a class of Euclidean algorithms related to divisions where the remainder is constrained to belong to [α −1, α], for some α ∈ [0, 1]. The paper is devoted to the averagecase analysis of these algorithms, in terms of number of steps or bitcomplexity. This is a new instance of the s ..."
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Cited by 9 (3 self)
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Abstract We study a class of Euclidean algorithms related to divisions where the remainder is constrained to belong to [α −1, α], for some α ∈ [0, 1]. The paper is devoted to the averagecase analysis of these algorithms, in terms of number of steps or bitcomplexity. This is a new instance of the socalled "dynamical analysis" method, where dynamical systems are made a deep use of. Here, the dynamical systems of interest have an infinite number of branches and they are not Markovian, so that the general framework of dynamical analysis is more complex to adapt to this case than previously. 2002 Elsevier Science (USA). All rights reserved.
Lattice reduction in two dimensions: analyses under realistic probabilistic models
, 2003
"... The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm “inside ” the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All th ..."
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Cited by 7 (1 self)
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The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm “inside ” the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All the main parameters are studied: execution parameters which describe the behaviour of the algorithm itself as well as output parameters, which describe the geometry of reduced bases.
Sharp estimates for the main parameters of the Euclid Algorithm
 Proceedings of LATIN’06, LNCS 3887
"... Abstract. We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid algorithm, and we study in particular the distribution of the bitcomplexity which involves two main parameters: digit–costs and length of continuants. We perform a “dynamical analysis ” which h ..."
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Cited by 5 (5 self)
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Abstract. We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid algorithm, and we study in particular the distribution of the bitcomplexity which involves two main parameters: digit–costs and length of continuants. We perform a “dynamical analysis ” which heavily uses the dynamical system underlying the Euclidean algorithm. Baladi and Vallée [2] have recently designed a general framework for “distributional dynamical analysis”, where they have exhibited asymptotic gaussian laws for a large class of digit–costs. However, this family contains neither the bit–complexity cost nor the length of continuants. We first show here that an asymptotic gaussian law also holds for the length of continuants at a fraction of the execution. There exist two gcd algorithms, the standard one which only computes the gcd, and the extended one which also computes the Bezout pair, and is widely used for computing modular inverses. The extended algorithm is more regular than the standard one, and this explains that our results are more precise for the extended algorithm. We prove that the bit–complexity of the extended Euclid algorithm asymptotically follows a gaussian law, and we exhibit the speed of convergence towards the normal law. We describe also conjectures [quite plausible], under which we can obtain an asymptotic gaussian law for the plain bitcomplexity, or a sharper estimate of the speed of convergence towards the gaussian law. 1
Distributional analyses of Euclidean algorithms
 Proceedings ANALCO04, submitted
, 2004
"... We provide a complete analysis of the standard Euclidean algorithm and two of its “fast ” variants, the nearestinteger and the oddquotient algorithm. For a whole family of costs, including the number of iterations, we show that the distribution of the cost is asymptotically normal, and obtain the ..."
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Cited by 5 (2 self)
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We provide a complete analysis of the standard Euclidean algorithm and two of its “fast ” variants, the nearestinteger and the oddquotient algorithm. For a whole family of costs, including the number of iterations, we show that the distribution of the cost is asymptotically normal, and obtain the optimal speed of convergence. Precisely, we establish both local and central limit theorems, which characterize respectively distribution functions and their cumulative versions. Our results widely extend earlier results of Hensley (1994) regarding the number of steps of the standard algorithm and even in this particular case provide improved error estimates. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, and the saddlepoint method. Such dynamical analyses had previously been used to perform the averagecase analysis of algorithms. The present (dynamical) analysis in distribution relies on a novel approach based on bivariate transfer operators and builds upon recent results of Dolgopyat (1998) by providing polefree regions for certain associated Dirichlet series. 1
Gaussian laws for the main parameters of the Euclid algorithms
 Algorithmica
, 2008
"... Abstract. We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bitcomplexity which involves two main parameters: digit–costs and length of remainders. We ..."
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Cited by 4 (2 self)
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Abstract. We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bitcomplexity which involves two main parameters: digit–costs and length of remainders. We first show here that an asymptotic gaussian law holds for the length of remainders at a fraction of the execution, which exhibits a deep regularity phenomenon. Then, we study in each framework –polynomials (P) and integer numbers (I) – two gcd algorithms, the standard one (S) which only computes the gcd, and the extended one (E) which also computes the Bezout pair, and is widely used for computing modular inverses. The extended algorithm is more regular than the standard one, and this explains that our results are more precise for the Extended algorithm: we exhibit an asymptotic gaussian law for the bit–complexity of the extended algorithm, in both cases (P) and (I). We also prove that an asymptotic gaussian law for the bitcomplexity of the standard gcd in case (P), but we do not succeed obtaining a similar result in case (I). The integer study is more involved than the polynomial study, as it is usually the case. In the polynomial case, we deal with the central tools of the distributional analysis of algorithms, namely
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.
The Lyapunov tortoise and the dyadic hare
"... 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 ..."
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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 / log 2 ∼ 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 [1]...], and the algorithm terminates. 1