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

In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers $$U_k(n)$$Uk(n) of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$Uk(n)~ckn-5/2?k-n, where $$c_k> 0$$ck>0 and $$\rho _{k}>0$$?k>0 denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$U(z)=?n?0Uk(n)zn.

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An Asymptotic Analysis of Labeled and Unlabeled k-Trees

Extremal statistics on non-crossing configurations☆

In his book Topics in Analytic Number Theory (1973), Hans Rademacher considered the generating function of integer partitions into at most N parts and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19 (2013)), who gave strong numerical evidence against the conjecture.

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Disproof of a conjecture by Rademacher on partial fractions

The term analytic information theory has been coined to describe problems of information theory studied by analytic tools The approach of applying tools from analysis of algorithms to problems of source coding and, in general, to information theory lies at the crossroad of computer science and information theory Combining the tools from both areas often provides powerful results, such as computer scientist Abraham Lempel and information theorist Jacob Ziv working together in the late 1970s to develop compression algorithms that are now widely referred to as Lempel-Ziv algorithms and are the basis of the ZIP compression still used extensively in computing today This monograph surveys the use of these techniques for the rigorous analysis of code redundancy for known sources in lossless data compression A separate chapter is devoted to precise analyses of each of three types of lossless data compression schemes, namely fixed-to-variable length codes, variable-to-fixed length codes, and variable-to-variable length codes Each one of these schemes is described in detail, building upon work done in the latter part of the 20th century to present new and powerful techniques For the first time, this survey presents redundancy for universal variable-to-fixed and variable-to-variable length codes in a comprehensive and coherent manner The monograph will be of interest to computer scientists and information theorists working on modern coding techniques Written by two leading experts, it provides the reader with a unique, succinct starting point for their own research into the area

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Redundancy of Lossless Data Compression for Known Sources by Analytic Methods

This special issue is devoted to the Analysis of Algorithms (AofA). Most of the papers are from the Eighth Seminar on Analysis of Algorithms, held in Strobl, Austria, June 23–29, 2002.

Michael Drmota

Papers

An Asymptotic Analysis of Labeled and Unlabeled k-Trees

Extremal statistics on non-crossing configurations☆

Disproof of a conjecture by Rademacher on partial fractions

Redundancy of Lossless Data Compression for Known Sources by Analytic Methods

Special Issue on Analysis of Algorithms