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

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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

This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.

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Prediction, learning, and games

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

We consider an extension of Hayman's notion of admissibility to bivariate generating functions f(z,u) that have the property that the coefficients a nk satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a large class of functions for which it is possible to check this kind of admissibility automatically. This is realized with help of a MAPLE program that is also presented. We apply this concept to various combinatorial examples.

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Extended admissible functions and Gaussian limiting distributions

Let ℱn be the set of random mappings ϕ : {1,…,n} → {1,…,n} (such that every mapping is equally likely). For x e {l,…,n} the elements are called the predecessors of x. Let Nr denote the random variable which counts the number of points x e {l,…,n} with exactly r predecessors. In this paper we identify the limiting distribution of Nr as n → ∞. If r = r(n) = o(n⅔) then the limiting distribution is Gaussian, if r ˜ Cn⅔ then it is Poisson, and in the remaining case rn−⅔ → ∞ it is degenerate. Furthermore, it is shown that Nr is a Poisson approximation if r → ∞.

Predecessors in Random Mappings

We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures.



Given a tree t and one of its leaves a, the w(t, a) parameter denotes the number of internal nodes in the subtree rooted at a's father. The closely related w(t, a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a's father. We define the cumulative w parameter as W(t) = Σaw(t, a), i.e. as the sum of w(t, a) over all leaves a of t. The w parameter not only plays an important role in the analysis of the Lempel–Ziv '77 data compression algorithm, but it is captivating from a combinatorial viewpoint too.



In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries.



The final section of this report briefly summarizes and improves the previously known results about the w parameter's behavior on tries and suffix trees, originally published in one author's thesis (see Analysis of the multiplicity matching parameter in suffix trees. Ph.D. Thesis, Purdue University, West Lafayette, IN, U.S.A., May 2005; Discrete Math. Theoret. Comput. Sci. 2005; AD:307–322; IEEE Trans. Inform. Theory 2007; 53:1799–1813).



This survey of new results about the w parameter is very instructive since a variety of different combinatorial methods are used in tandem to carry out the analysis. Copyright © 2008 John Wiley & Sons, Ltd.

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On the shape of the fringe of various types of random trees

Catalytic equations appear in several combinatorial applications, most notably in the numeration of lattice path and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size $n$ in various planar map classes grows asymptotically like $c\cdot n^{-5/2} \gamma^n$, for suitable positive constants $c$ and $\gamma$. Essentially we have to distinguish between linear catalytic equations (where the subexponential growth is $n^{-3/2}$) and non-linear catalytic equations (where we have $n^{-5/2}$ as in planar maps). Furthermore we provide a quite general central limit theorem for parameters that can be encoded by catalytic functional equations, even when they are not positive.

Universal singular exponents in catalytic variable equations

This paper contains some general theorems on the limiting distribution of a certain random variable $S_T $ arising in the context of recurrent events. $\mathcal{S}_T $ is especially meaningful to the investigation of discrete-time queueing systems subjected to service time deadlines. It may be viewed as a sum of mutually independent conditional random variables $\mathcal{B}_T $ having the same distribution. It is assumed that $\mathcal{B}_T $ depends on a parameter T and tends to an unconditional random variable $\mathcal{B}$ for $T \to \infty $. Under some weak conditions concerning the conditional probability generating function $B_T ( z )$, it follows that $\mathcal{S}_T $ is approximately exponentially distributed with parameter $\lambda _T = 1/\mu _T ,\, \mu _T = B'_T ( 1 )/( 1 - B_T ( 1 ) )$, which tends to infinity for $T \to \infty $. Provided here are uniform asymptotic expansions for the appropriate probabilities and for all moments, also.

Michael Drmota

Papers

Extended admissible functions and Gaussian limiting distributions

Predecessors in Random Mappings

On the shape of the fringe of various types of random trees

Universal singular exponents in catalytic variable equations

Exponential limiting distributions in queueing systems with deadlines