We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

Computational geometry

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.

http://algo.stat.sinica.edu.tw/hk/files/2005_07/pdf/On_convergence_rates_in_the_central_limit_theorems.pdf

On Convergence Rates in the Central Limit Theorems for Combinatorial Structures

We characterize all limit laws of the quicksort-type random variables defined recursively by ${\cal L}(X_n)= {\cal L}(X_{I_n}+X^*_{n-1-I_n}+T_n)$ when the "toll function" Tn varies and satisfies general conditions, where (Xn), (Xn*), (In, Tn) are independent, In is uniformly distributed over {0, . . .,n-1}, and ${\cal L}(X_n)={\cal L}(X_n^\ast)$. When the "toll function" Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly $\limsup_{n\rightarrow\infty}\log E(T_n)/\log n\le 1/2$), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on a broadcast communication model, from an in-situ permutation algorithm to tree traversal algorithms, etc.

Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions

We propose a uniform approach to describe the phase change of the limiting distribution of space measures in random m-ary search trees: the space requirement, when properly normalized, is asymptotically normally distributed for m≤26 and does not have a fixed limiting distribution for m>26. Our tools are based on the method of moments and asymptotic solutions of differential equations, and are applicable to secondary cost measures of quicksort with median-of-(2t+1) for which the same phase change occurs at t=58. Both problems are essentially special cases of the generalized quicksort of Hennequin in which a sample of m(t+1)−1 elements are used to select m−1 equi-spaced ranks that are used in turn to partition the input into m subfiles. A complete description of the numbers at which the phase change occurs is given. For example, when m is fixed and t varies, the phase change occurs at (m, t)={(2, 58), (3, 19), (4, 10), (5, 6), (6, 4), …}. We also indicate some applications of our approach to other problems including bucket recursive trees. A general framework on “asymptotic transfers” of the underlying recurrence is also given. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 316–358, 2001

Phase changes in random m -ary search trees and generalized quicksort

We prove a general central limit theorem for probabilities of large deviations for sequences of random variables satisfying certain natural analytic conditions. This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius’ version of Cram er’s classical method based on analytic moment generating functions. We thus generalize Cram er’s and Kubilius’ theorems on large deviations.

/pdf/large-deviations-for-combinatorial-distributions-i-central-elihlpha5t.pdf

Large deviations for combinatorial distributions. I. Central limit theorems

We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio α of the level and the logarithm of tree size lies in [0,e). Convergence of all moments is shown to hold only for α ∈ [0,1] (with only convergence of finite moments when α ∈ (1,e)). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for α = 0 and a "quicksort type" limit law for α = 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on the contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.

/pdf/profiles-of-random-trees-limit-theorems-for-random-recursive-5mbnx9l4q8.pdf

Hsien-Kuei Hwang

Papers

On Convergence Rates in the Central Limit Theorems for Combinatorial Structures

Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions

Phase changes in random m -ary search trees and generalized quicksort

Large deviations for combinatorial distributions. I. Central limit theorems

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees