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22
Exponential decay of correlations for surface semiflows without finite Markov partitions
 Proc. Amer. Math. Soc
"... Abstract. We extend Dolgopyat’s bounds on iterated transfer operators to suspensions of interval maps with infinitely many intervals of monotonicity. 1. Statement of results Let 0 < c1 <... < cm < cm+1 <... < 1 be a finite or countable partition of I = [0, 1] into subintervals, and ..."
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Cited by 16 (6 self)
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Abstract. We extend Dolgopyat’s bounds on iterated transfer operators to suspensions of interval maps with infinitely many intervals of monotonicity. 1. Statement of results Let 0 < c1 <... < cm < cm+1 <... < 1 be a finite or countable partition of I = [0, 1] into subintervals, and let T: I → I be so that T  (cm,cm+1) is C 2 and extends to a homeomorphism from [cm, cm+1] to I. We assume that T is piecewise uniformly expanding: there are C ≥ 1 and ˆρ < 1 so that h(x) −h(y)  ≤ Cˆρ n x −y for every inverse branch h of T n and all n. Let H be the set of inverse branches h: I → [cm, cm+1] of T. We suppose (Renyi’s condition) that there is K> 0 so that every h ∈ H satisfies h ′ ′  ≤ Kh ′ . Let r: I → R+ be so that r  (cm,cm+1) is C 1, and inf r> 0. Assume that there is σ0 < 0 so that ∑ h∈H supexp(−σ(r ◦ h))h ′  < ∞ for all σ> σ0, and that r ′ ◦ h  · h ′  ≤ K for all h ∈ H. For n ≥ 1, write r (n)(x) = ∑n−1 k=0 r(T k)(x). We study the transfer operators, indexed by s = σ + it, Lsf(x) = ∑
The Number of Symbol Comparisons in QuickSort and QuickSelect
"... We revisit the classical QuickSort and QuickSelect algorithms, under a complexity model that fully takes into account the elementary comparisons between symbols composing the records to be processed. Our probabilistic models belong to a broad category of information sources that encompasses memory ..."
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Cited by 9 (3 self)
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We revisit the classical QuickSort and QuickSelect algorithms, under a complexity model that fully takes into account the elementary comparisons between symbols composing the records to be processed. Our probabilistic models belong to a broad category of information sources that encompasses memoryless (i.e., independentsymbols) and Markov sources, as well as many unboundedcorrelation sources. We establish that, under our conditions, the averagecase complexity of QuickSort is O(n log² n) [rather than O(n log n), classically], whereas that of QuickSelect remains O(n). Explicit expressions for the implied constants are provided by our combinatorial–analytic methods.
Digital Trees and Memoryless Sources: from Arithmetics to Analysis
 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’10), Discrete Math. Theor. Comput. Sci. Proc
, 2010
"... Digital trees, also known as “tries”, are fundamental to a number of algorithmic schemes, including radixbased searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstr ..."
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Cited by 7 (1 self)
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Digital trees, also known as “tries”, are fundamental to a number of algorithmic schemes, including radixbased searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the memoryless source, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters’ expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous Dirichlet series. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific arithmetic properties, especially irrationality measures, of the sources under consideration.
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 4 (4 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
Euclidean dynamics
 Discrete and Continuous Dynamical Systems
"... Abstract. 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 tran ..."
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Cited by 2 (1 self)
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Abstract. 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. 1. Introduction. Computing the Greatest Common Divisor [Gcd
The dynamics of Pythagorean triples
, 2008
"... We construct a piecewise onto 3to1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and is equal to the address of the PPT on Barning’s [9] ternary ..."
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Cited by 1 (0 self)
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We construct a piecewise onto 3to1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and is equal to the address of the PPT on Barning’s [9] ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugate to a modified Euclidean algorithm. The invariant measure is identified, and the system is shown to be conservative and ergodic. We also show, based on a result of Aaronson and Denker [2], that the dynamical system can be obtained as a factor map of a crosssection of the geodesic flow on a quotient space of the hyperbolic plane by the group Γ(2), a free subgroup of the modular group with two generators. 1
Analysis of fast versions of the Euclid Algorithm
 Proceedings of ANALCO’07, Janvier 2007
"... There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schönhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications and stop the recursion at a depth slightly smaller than lg n. A rough estimate of the wors ..."
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Cited by 1 (1 self)
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There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schönhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications and stop the recursion at a depth slightly smaller than lg n. A rough estimate of the worst–case complexity of these fast versions provides the bound O(n(log n) 2 log log n). However, this estimate is based on some heuristics and is not actually proven. Here, we provide a precise probabilistic analysis of some of these fast variants, and we prove that their average bit–complexity on random inputs of size n is Θ(n(log n) 2 log log n), with a precise remainder term. We view such a fast algorithm as a sequence of what we call interrupted algorithms, and we obtain three results about the (plain) Euclid Algorithm which may be of independent interest. We precisely describe the evolution of the distribution during the execution of the (plain) Euclid Algorithm; we obtain a sharp estimate for the probability that all the quotients produced by the (plain) Euclid Algorithm are small enough; we also exhibit a strong regularity phenomenon, which proves that these interrupted algorithms are locally “similar ” to the total algorithm. This finally leads to the precise evaluation of the average bit–complexity of these fast algorithms. This work uses various tools, and is based on a precise study of generalised transfer operators related to the dynamical system underlying the Euclid Algorithm. 1
DOI: 10.1214/07AIHP140 c ○ Association des Publications de l’Institut Henri Poincaré, 2008
, 2007
"... www.imstat.org/aihp ..."
The Euclid algorithm is “totally ” gaussian
"... We consider Euclid’s gcd algorithm for two integers (p, q) with 1 ≤ p ≤ q ≤ N, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function d on the set of possible digits, asymptotically for N → ∞. For any addi ..."
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We consider Euclid’s gcd algorithm for two integers (p, q) with 1 ≤ p ≤ q ≤ N, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function d on the set of possible digits, asymptotically for N → ∞. For any additive cost of moderate growth d, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost d is lattice – a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost d together with the “canonical ” cost c of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost d. This “replacement ” is made possible by using the additivity of cost d, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost c is both “strongly ” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just ” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories.