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Size and Path length of Patricia Tries: Dynamical Sources Context.
, 2001
"... Digital trees, also known as tries, and Patricia tries are flexible data structures that occur in a variety of computer and communication algorithms including dynamic hashing, partial match retrieval, searching and sorting, conflict resolution algorithms for broadcast communication, data compression ..."
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Cited by 8 (1 self)
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Digital trees, also known as tries, and Patricia tries are flexible data structures that occur in a variety of computer and communication algorithms including dynamic hashing, partial match retrieval, searching and sorting, conflict resolution algorithms for broadcast communication, data compression, and so forth. We consider here tries and Patricia tries built from $n$ words emitted by a probabilistic dynamical source. Such sources encompass classical and many more models of sources as memoryless sources and finite Markov chains. The probabilistic behavior of the main parameters, namely the size and path length, appears to be determined by some intrinsic characteristics of the source, namely the entropy and two other constants, themselves related in a natural way to spectral properties of specific transfer operators of Ruelle type. Keywords: Averagecase Analysis of datastructures, Information Theory, Trie, Mellin analysis, Dynamical systems, Ruelle operator, Functional Analysis.
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.
On Differences of Zeta Values
 in "Journal of Computational and Applied Mathematics
, 2008
"... ABSTRACT. Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Ma´slanka, Coffey, BáezDuarte, Voros and others. We apply the theory of Nörlu ..."
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Cited by 4 (1 self)
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ABSTRACT. Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Ma´slanka, Coffey, BáezDuarte, Voros and others. We apply the theory of NörlundRice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li’s criterion for the Riemann hypothesis.
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 GAUSSKUZMINWIRSING OPERATOR
"... ABSTRACT. This paper presents a review of the GaussKuzminWirsing (GKW) operator. The GKW operator is the transfer operator of the Gauss map, and thus has connections to the theory of continued fractions – specifically, it is the shift operator for continued fractions. The operator appears to have ..."
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ABSTRACT. This paper presents a review of the GaussKuzminWirsing (GKW) operator. The GKW operator is the transfer operator of the Gauss map, and thus has connections to the theory of continued fractions – specifically, it is the shift operator for continued fractions. The operator appears to have a reasonably smooth, wellbehaved structure, however, no closedform analytic solutions are known, and these are not easy to obtain. Eigenvalues and eigenfunctions can be obtained numerically, but little else is known in the mathematical literature. While this paper does attempt to be a review, it is incomplete; it is more of a diary of research results. Connections to the Minkowski Question Mark Function are probed. In particular, the Question Mark is used to define a transfer operator which is conjugate to the GKW. This conjugate operator is solvable, and can be shown to have fractal eigenfunctions. However, the spectrum of this operator is not at all the same as that of the GKW. This is because the Jacobian of the transformation relating the two is given by (? ′ ◦? −1)(x) , which is wellknown as the prototypical “multifractal measure”. Nonetheless, conjugacy allows the eigenfunctions of the one to be used to construct eigenfunctions of the other; in this sense,