. The average case analysis of algorithms can avail itself of the development of synthetic methods in combinatorial enumerations and in asymptotic analysis. Symbolic methods in combinatorial analysis permit to express directly the counting generating functions of wide classes of combinatorial structures. Asymptotic methods based on complex analysis permit to extract directly coefficients of structurally complicated generating functions without a need for explicit coefficient expansions. Three major groups of problems relative to algebraic equations, differential equations, and iteration are presented. The range of applications includes formal languages, tree enumerations, comparison--based searching and sorting, digital structures, hashing and occupancy problems. These analytic approaches allow an abstract discussion of asymptotic properties of combinatorial structures and schemas while opening the way for automatic analysis of whole classes of combinatorial algorithms. I...

Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. "Singularity analysis" reviewed here provides constructive estimates that are applicable in several areas of combinatorics. It constitutes a complex-analytic Tauberian procedure by which combinatorial constructions and asymptotic-probabilistic laws can be systematically related.

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Consider a given pattern H and a random text T generated by a Markovian source. We study the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately. We present exact and asymptotic formulae for all moments (including the variance), and probability of r pattern occurrences for three different regions of r, namely: (i) r = O(1), (ii) central limit regime, and (iii) large deviations regime. In order to derive these results, we first construct some language expressions that characterize pattern occurrences which are later translated into generating functions. Finally, we use analytical methods to extract asymptotic behaviors of the pattern frequency. Applications of these results include molecular biology, source coding, synchronization, wireless communications, approximate pattern matching, game theory, and stock market analysis. These findings are of particular interest to information theory (e.g., second-order properties of the re...

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.

An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc.

We present a complete analysis of the statistics of number of occurrences of a regular expression pattern in a random text. This covers "motifs" widely used in computational biology. Our approach is based on: (i) a constructive approach to classical results in theoretical computer science (automata and formal language theory), in particular, the rationality of generating functions of regular languages; (ii) analytic combinatorics that is used for deriving asymptotic properties from generating functions; (iii) computer algebra for determining generating functions explicitly, analysing generating functions and extracting coefficients efficiently. We provide constructions for overlapping or non-overlapping matches of a regular expression. A companion implementation produces multivariate generating functions for the statistics under study. A fast computation of Taylor coefficients of the generating functions then yields exact values of the moments with typical application to random t...

We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many copies of any fixed planar graph, and this implies that almost all such graphs have large automorphism groups.