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38
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...
Dynamical Sources in Information Theory: Fundamental intervals and Word Prefixes.
, 1998
"... A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewe ..."
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Cited by 37 (7 self)
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A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some important problems about prefixes that intervene in algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewed as a "generating" operator. Its dominant spectral objects are linked with important parameters of the source such as the entropy, and play a central role in all the results. 1 Introduction. In information theory contexts, data items are (infinite) words that are produced by a common mechanism, called a source. Realistic sources are often complex objects. We work here inside a quite general framework of sources related to dynamical systems theory which goes beyond the cases of memoryless and Markov sources. This model can describe nonmarkovian processes, where the dependency on past history is unbounded, and as such, they attain a high level of generality. A probabilistic dynamical source ...
Continued Fraction Algorithms, Functional Operators, and Structure Constants
, 1996
"... Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer o ..."
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Cited by 31 (6 self)
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Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2dimensional generalization. This paper surveys the main properties of functional operators,  transfer operators  due to Ruelle and Mayer (also following Lévy, Kuzmin, Wirsing, Hensley, and others) that describe precisely the dynamics of the continued fraction transformation. Spectral characteristics of transfer operators are shown to have many consequences, like the normal law for logarithms of continuants associated to the basic continued fraction algorithm and a purely analytic estimation of the average number of steps of the Euclidean algorithm. Transfer operators also lead to a complete analysis of the "Hakmem" algorithm for comparing two rational numbers via partial continued fraction expansions and of the "digital tree" algorithm for completely sorting n real numbers by means of ...
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 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
Average BitComplexity of Euclidean Algorithms
 Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387
, 2000
"... We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set ..."
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Cited by 18 (7 self)
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We obtain new results regarding the precise average bitcomplexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where 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 algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of an entire class of gcdlike algorithms. Keywords: Averagecase Analysis of algorithms, BitComplexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...
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.
A oneparameter family of analytic markov maps with an intermittency transition. Discrete and Continuous Dyn
 Syst
, 2005
"... Abstract. In this paper we introduce and study a oneparameter family of piecewise analytic interval maps having the tent map and the Farey map as extrema. Among other things, we construct a Hilbert space of analytic functions left invariant by the PerronFrobenius operator of all these maps and stu ..."
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Cited by 7 (1 self)
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Abstract. In this paper we introduce and study a oneparameter family of piecewise analytic interval maps having the tent map and the Farey map as extrema. Among other things, we construct a Hilbert space of analytic functions left invariant by the PerronFrobenius operator of all these maps and study the transition between discrete and continuous spectrum when approaching the intermittent situation. 1. Introduction. Expanding
A Unifying Framework for the Analysis of a Class of Euclidean Algorithms
 the proceedings of LATIN'2000, LNCS
, 2000
"... . 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 algorithms. The methods rely on p ..."
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Cited by 6 (2 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 algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise averagecase analyses of four algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods 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 Introduction Euclid's algorithm, discovered as early as 300BC, was analysed first in the worst case in 1733 by de Lagny, then in the averagecase around 1969 independently by Heilbronn [8] and Dixon [5], and finally in distribut...
Remarks on modular symbols for Maass wave forms
"... To Professor F. Hirzebruch, with friendship and admiration, for his anniversary. Abstract. In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non–positive rational ends, with ..."
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Cited by 4 (0 self)
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To Professor F. Hirzebruch, with friendship and admiration, for his anniversary. Abstract. In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non–positive rational ends, with values in analytic functions (pseudo–measures in the sense of [MaMar2]). After explaining the basic issues and analogies in the extended Introduction, I construct modular symbols in the sec. 1 and the related Lévy–Mellin transforms in the sec. 2. The whole paper is an extended footnote to the Lewis–Zagier fundamental study [LZ2]. §0.