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31
NonUniform Random Variate Generation
, 1986
"... Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various ..."
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Cited by 620 (21 self)
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Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.
A Note on the Height of Binary Search Trees
, 1986
"... Let H. be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of I,..., n. It is shown that HJog n + c = 4.3 I 107... in probability as n + 00, where c is the unique solution of c log((2e)lc) = 1, c 2 2. Also, for all p> 0, lim,,E(H$)/ log ..."
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Cited by 79 (23 self)
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Let H. be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of I,..., n. It is shown that HJog n + c = 4.3 I 107... in probability as n + 00, where c is the unique solution of c log((2e)lc) = 1, c 2 2. Also, for all p> 0, lim,,E(H$)/ log % = cp. Finally, it is proved that &/log n, c * = 0.3733..., in probability, where c * is defined by c log((2e)lc) = 1, c 5 1, and.S, is the saturation level of the same tree, that is, the number of full levels in the tree.
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.
Branching processes in the analysis of the heights of trees
 Acta Informatica
, 1987
"... Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random kd trees, quadtrees and unionend trees under various models of randomization. For example, ..."
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Cited by 58 (19 self)
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Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random kd trees, quadtrees and unionend trees under various models of randomization. For example, for the random binary search tree constructed from a random permutation of 1,..., n, it is shown that H„/(c log (n)) tends to 1 in probability and in the mean as n oo, where H „ is the height of the tree, and c =4.31107... is a solution of the equation c log (2e / = 1. In addition, we ~c ~ show that H „clog (n) = O (/log (n) loglog (n)) in probability.
Probability laws related to the Jacobi theta and Riemann zeta functions, and the Brownian excursions
 Bulletin (New series) of the American Mathematical Society
"... Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional ..."
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Cited by 57 (11 self)
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Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents
Longest increasing subsequences in patternrestricted permutations
 Electron. J. Combin., 9(2):Research paper
, 2002
"... Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions for patternrestricted permutations in which the pattern is any one of the six patterns of length 3. We ..."
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Cited by 21 (0 self)
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Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions for patternrestricted permutations in which the pattern is any one of the six patterns of length 3. We show that the (132)avoiding case is identical to the distribution of heights of ordered trees, and that the (321)avoiding case has interesting connections with a well known theorem of ErdősSzekeres. 1
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 20 (5 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
Distributional limits for critical random graphs
 In preparation
, 2009
"... We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metri ..."
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Cited by 12 (5 self)
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We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean. Keywords: Random graphs, GromovHausdorff distance, scaling limits, continuum random tree, diameter. 2000 Mathematics subject classification: 05C80, 60C05.