Abstract not found

We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp sign-changes when one level is fixed and the other one is varying. An asymptotic estimate for the expected width is also derived.

this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the random variables in question are available, complex-analytic methods can be used to derive precise asymptotic results for the five distances above. Actually, we shall consider the following generalized distances: let ff ? 0 be a fixed positive number, (X; Y ) = FM (X; Y ) = (X; Y ) = sup K (X; Y ) = sup M (X; Y ) = jP(X = j) \Gamma P(Y = j) Note that d TV = d M . Besides the case ff = 1 (and ff = 1=2 for d M ), only the case d TV was previously studied by Franken [39] for Poisson approximation to the sum of independent but not identically distributed Bernoulli random variables. We take these quantities as our measures of degree of nearness of Poisson approximation, some of which may be interpreted as certain norms in suitable space as many authors did (cf. [12, 22, 23, 74, 96]). For a large class of discrete distributions, we shall derive an asymptotic main term together with an error estimate for each of these distances. Our results are thus "approximation theorems" rather than "limit theorems". The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]). Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall di...

Balancing of structured peer-to-peer graphs, including their zone sizes, has recently become an important topic of distributed hash table (DHT) research. To bring analytical understanding into the various peer-join mechanisms based on consistent hashing, we study how zone-balancing decisions made during the initial sampling of the peer space affect the resulting zone sizes and derive several asymptotic bounds for the maximum and minimum zone sizes that hold with high probability. Several of our results contradict those of prior work and shed new light on the theoretical performance limitations of consistent hashing. We use simulations to verify our models and compare the performance of the various methods using the example of recently proposed de

A brief survey, based mainly on my recent work with coauthors, is given of the different types of phase changes (or transitions) appearing in random discrete structures and in analysis of algorithms with a recursive character.

The limit laws of three cost measures are derived of two algorithms for finding the maximum in a single-channel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method of proof proceeds along the line via the method of moments and the "asymptotic transfers", which roughly bridges the asymptotics of the "conquering cost of the subproblems" and that of the total cost. Such a general approach has proved very fruitful for a number of problems in the analysis of recursive algorithms. 1

New uniform asymptotic approximations with error bounds are derived for a generalized total variation distance of Poisson approximations to the Poisson-binomial distribution. The method of proof is also applicable to other Poisson approximation problems. MSC 2000 Subject Classifications: Primary 62E17; secondary 60C05. 1

random samples

We compute the asymptotic expansion of the index of regularity for overdetermined quadratic semi-regular sequences of algebraic equations. This implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryptographic community. 1 Motivations and Results The worst-case complexity of Gröbner bases has been the object of extensive studies. In the most general case, it is well known after work by Mayr and Meyer that the complexity is doubly exponential in the number of variables. For subclasses of polynomial systems, the complexity may be much smaller. Of particular importance is the class of regular sequences of polynomials. There, it is known that after a generic linear change of variables the complexity of the computation for the degree-reverse-lexicographic order is simply exponential in the number of variables. Moreover, in characteristic 0, these systems are generic. Our goal is to give similar complexity bounds for overdetermined systems, for a class of systems that we