Abstract. This chapter provides a survey of the main methods in non-uniform 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.

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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 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.

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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 k-d trees, quadtrees and union-end 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.

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 one-dimensional 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

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 one-to-one 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).

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 pattern-restricted 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 Erdős-Szekeres. 1

A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The DiameterConstrained MST (DCMST) problem can be stated as follows: given an undirected, edge-weighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NP-complete, for all values of k; 4 k #n - 2). In this paper, we investigate the behavior of the diameter of MST in randomly-weighted complete graphs (in Erds-Rnyi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is small---independent of n, we present a one-time-tree-construction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of t...