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40
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 78 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
On Convergence Rates in the Central Limit Theorems for Combinatorial Structures
, 1998
"... Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is a ..."
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Cited by 67 (8 self)
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Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is also applicable to arithmetical functions. Moreover, asymptotic expressions are derived for moments of integral order. Many examples from different applications are discussed.
Structure Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction
, 1998
"... We introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum... ..."
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Cited by 66 (0 self)
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We introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum...
Varieties of Increasing Trees
, 1992
"... An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We ..."
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Cited by 55 (7 self)
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An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.
A recurrence related to trees
 Proceedings of the American Mathematical Society
, 1989
"... Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer c ..."
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Cited by 34 (4 self)
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Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer coefficients, which may be called the "tree polynomials." There are n" ~ (n 1)! sequences of edges between vertices (0.1) ux—vx.un_x—vn_x, \<uk<vk<n, that define a free tree on {1,...,«}, because there are n" ~ free trees on n labeled vertices and every such tree has n 1 edges. If we consider each of these n (n — 1)! sequences to be equally likely, the probability that unX and vn_x belong respectively to components of sizes k and n k based on the first « 2 edges is '^oer^r'■•<*< • ■ Knuth and Schönhage [9, §§912] considered treeconstruction algorithms whose analysis depended on the solution of the recurrence (°3) Xn=Cn+ E Pnk(xk+Xnk) 0<k<n for various sequences (cn). The purpose of the present note is to extend the results of [9] and to consider related sequences of functions whose exact and asymptotic values arise in a variety of algorithms. Much of the analysis below, as in [9], depends on properties of the formal power series tt\A \ TV \ \r^n"~[z " 2, 3 3, 8 4, 125 5, (0.4) T(z) = ^ — — = z + z +z +^z + —z +■■ ■, n>l Received by the editors March 18, 1988.
Large deviations of combinatorial distributions II: Local limit theorems
, 1997
"... This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a seq ..."
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Cited by 32 (5 self)
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This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables n#1 each of maximal span 1 (see below for definition), we are interested in the asymptotic behavior of the probabilities n = m} (m N, m = n x n # n , n := n , # n := n ), ##, where x n can tend to with n at a rate that is restricted to O(# n ). Our interest here is not to derive asymptotic expression for n = m} valid for the widest possible range of m, but to show that for m lying in the interval n O(# n ), very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17]. Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b + hk, k Z, for some constants b and h > 0; and there does not exist b # and h # > h such that X takes only values of the form b # + h # k
Analytic Variations on QuadTrees
, 1991
"... Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtreesfully specified and partial match queries. The data model assumes random points with independently ..."
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Cited by 28 (4 self)
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Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtreesfully specified and partial match queries. The data model assumes random points with independently drawn coordinate values. The analysis leads to a class of "fullhistory" divideandconquer recurrences. These recurrences are solved using generating functions, either exactly for dimension d = 2, or asymptotically for higher dimensions. The exact solutions involve hypergeometric functions. The general asymptotic solutions relie on the classification of singularities of linear differential equations with analytic coefficients, and on singularity analysis techniques. These methods are applicable to the asymptotic solution of a wide range of linear recurrences, as may occur in particular in the analysis of multidimensional searching problems.
General combinatorial schemas: Gaussian limit distributions and exponential tails
 Discrete Math
, 1993
"... Under general conditions, the number of components in combinatorial structures defined as sequences, cycles or sets of components admits a Gaussian limit distribution together with an exponential tail. The results are valid, assuming simple analytic conditions on the generating functions of the comp ..."
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Cited by 25 (6 self)
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Under general conditions, the number of components in combinatorial structures defined as sequences, cycles or sets of components admits a Gaussian limit distribution together with an exponential tail. The results are valid, assuming simple analytic conditions on the generating functions of the components. The proofs rely on the continuity theorem for characteristic functions and Laplace transforms as well as techniques of singularity analysis applied to algebraic and logarithmic singularities. Combinatorial applications are in the fields of graphs, permutations, random mappings, ordered partitions and polynomial factorizations. 1.