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 2-dimensional 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 ...

We develop a general framework for the analysis of algorithms of a broad Euclidean type. The average-case 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 average-case 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 gcd-like algorithms together with new results regarding the probable behaviour of their cost functions. 1

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 average-case 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 gcd-like 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.

ABSTRACT. We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this process normalized linearly we determine a large deviation asymptotic.

Hartmann and Olmstead recently described an O(n log n) algorithm for solving sequential knapsack problems, and applied it to a relaxation of the 0--1 knapsack problem to compute a better bound than z LP in linear time after sorting the ratios p j =w j . This note extends these results in several directions. First we show how a similar bound can be obtained for the bounded knapsack problem. These bounds can be strengthened using Lagrangian relaxation, although the time required to solve the relaxation may increase by a factor of O(log n). In the 0--1 case, the optimal value of the Lagrangian dual can be interpreted as the result of adding a class of cutting planes to the linear programming relaxation of the 0--1 knapsack problem. We show that the separation problem for this class of cutting planes can be reduced to the separation problem over the bounded sequential knapsack polytope. We give a complete description of the bounded sequential knapsack problem in two special cases, and show...

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 quasi-powers theorems. Such dynamical analyses can be used to perform the average-case analysis of algorithms, but also (dynamical) analysis in distribution. 1. Introduction. Computing the Greatest Common Divisor [Gcd

Abstract. Motivated by the formula for the sum of the geometric series, we consider various classes of sets S ⊂ Z d of integer points for which an a priori “long ” Laurent series or polynomial � m∈S xm can be written as a “short ” rational function f(S; x). Examples include the sets of integer points in rational polyhedra, integer semigroups, and Hilbert bases of rational cones, among others. We discuss applications to efficient counting and optimization and open questions.

ABSTRACT. We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and lower fluctuation process. Also a large deviation asymptotic for single digits is given. 1.

We prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by F. Schweiger [14] [15] and studied also by C. Kraaikamp and A. Lopes [11]. Our main result is proven following the strategy used by Ya. Sinai and C. Ulcigrai [18] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

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