Abstract not found

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.

We study the convergence rate to normal limit law for the space requirement of random m-ary search trees. While it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m ? 26, we show that the convergence rate is O(n ) for 3 m 19 and is O(n ), where 4=3 ! ff ! 3=2 is a parameter depending on m for 20 m 26. Our approach is based on a refinement to the method of moments and applicable to other recursive random variables; we briefly mention the applications to quicksort proper and the generalized quicksort of Hennequin, where more phase changes are given. These results provide natural, concrete examples for which the Berry-Esseen bounds are not necessarily proportional to the reciprocal of the standard deviation. Local limit theorems are also derived. Abbreviated title. Phase changes in search trees.

. Consider a tree partitioning process in which n elements are split into b at the root of a tree (b a design parameter), the rest going recursively into two subtrees with a binomial probability distribution. This extends some familiar tree data structures of computer science like the digital trie and the digital search tree. The exponential generating function for the expected size of the tree satisfies a difference-- differential equation of order b, d b dz b f(z) = e z + 2e z=2 f( z 2 ): The solution involves going to ordinary (rather than exponential) generating functions, analyzing singularities by means of Mellin transforms and contour integration. The method is of some general interest since a large number of related problems on digital structures can be treated in this way via singularity analysis of ordinary generating functions. Work of this author was supported in part by the Basic Research Action of the E.C. under contract No. 3075 (Project ALCOM). y The resea...

This booklet develops in nearly 200 pages the basics of combinatorial enumeration through an approach that revolves around generating functions. The major objects of interest here are words, trees, graphs, and permutations, which surface recurrently in all areas of discrete mathematics. The text presents the core of the theory with chapters on unlabelled enumeration and ordinary generating functions, labelled enumeration and exponential generating functions, and finally multivariate enumeration and generating functions. It is largely oriented towards applications of combinatorial enumeration to random discrete structures and discrete mathematics models, as they appear in various branches of science, like statistical physics, computational biology, probability theory, and, last not least, computer science and the analysis of algorithms.

It is shown explicitly how self-similar graphs can be obtained as ‘blow-up ’ constructions of finite cell graphs Ĉ. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals. For a class of symmetrically self-similar graphs we study the simple random walk on a cell graph Ĉ, starting in a vertex v of the boundary of Ĉ. It is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions we compute the asymptotic behaviour of the n-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically self-similar graphs and at the same time the error term of the asymptotic expression is improved. Finally we present a criterion for the occurrence of oscillating phenomena of the n-step transition probabilities.

Let ρn be the fraction of structures of “size ” n which are “connected”; e.g., (a) the fraction of labeled or unlabeled n-vertex graphs having one component, (b) the fraction of partitions of n or of an n-set having a single part or block, or (c) the fraction of n-vertex forests that contain only one tree. Various authors have considered limρn, provided it exists. It is convenient to distinguish three cases depending on the nature of the power series for the structures: purely formal, convergent on the circle of convergence, and other. We determine all possible values for the pair (liminf ρn, limsupρn) in these cases. Only in the convergent case can one have 0 < limρn < 1. We study the existence of limρn in this case.

This paper studies coefficients y h,n of sequences of polynomials y h (x) = n³0 S y h,n x n defined by non-linear recurrences. A typical example to which the results of this paper apply is that of the sequence B 0 (x) = 1 , B h + 1 (x) = 1 + xB h (x) 2 for h ³ 0 , which arises in the study of binary trees. For a wide class of similar sequences a general distribution law for the coefficients y h,n as functions of n with h fixed is established. It follows from this law that in many interesting cases the distribution is asymptotically Gaussian near the peak. The proof relies on the saddle point method applied in a region where the polynomials grow doubly exponentially as h ® . Applications of these results include enumerations of binary trees and 2-3 trees. Other structures of interest in computer science and combinatorics can also be studied by this method or its extensions. Limit Distributions for Coefficients of Iterates of Polynomials with Applications to Combinatorial Enum...

In this paper we give a survey of the symbolic operator methods to do statistics on random trees. We present some examples and apply the techniques to find their asymptotic behaviour. 1 Introduction Let us consider a class E of combinatorial objects, let A be an algorithm defined over the class E, and let denote the complexity measure we are interested in. Such a class E of combinatorial objects consists on a set, usually denoted by the same name as the class, and a size measure j \Delta j E : E \Gamma! IN. The subscript E in j \Delta j E will be dropped whenever it is clear from the context. We shall denote by E n the set of objects in E of size n. To analyze the average behaviour of A on an input e 2 E n with respect to measure means to compute A (n) = EfA (e) j e 2 E n g; (1:1) where EfXg denotes the expectation of the random variable X [Knu68, VF90]. By definition of expectation, Equation (1.1) can be written as A (n) = X k k PrfA (e) = k j e 2 E n g = X e2En Prfeg ...

Abstract not found