Showing papers by "Michael Drmota published in 2006"
TL;DR: The purpose of this article is to show that the distribution of the longest fragment in the random multisection problem after k steps and the height of m-ary search trees are not only closely related in a formal way but both can be asymptotically described with the same distribution function that has to be shifted in a proper way (travelling wave).
Abstract: The purpose of this article is to show that the distribution of the longest fragment in the random multisection problem after k steps and the height of m-ary search trees (and some extensions) are not only closely related in a formal way but both can be asymptotically described with the same distribution function that has to be shifted in a proper way (travelling wave). The crucial property for the proof is a so-called intersection property that transfers inequalities between two distribution functions (resp. of their Laplace transforms) from one level to the next. It is conjectured that such intersection properties hold in a much more general context. If this property is verified convergence to a travelling wave follows almost automatically.
24 citations
09 Jul 2006
TL;DR: An algebraic characterization of the Tunstall code is proposed which, together with tools like the Mellin transform and the Tauberian theorems, leads to new results on the variance and a central limit theorem for dictionary phrase lengths.
Abstract: We study the Tunstall code using the machinery from the analysis of algorithms literature. In particular, we propose an algebraic characterization of the Tunstall code which, together with tools like the Mellin transform and the Tauberian theorems, leads to new results on the variance and a central limit theorem for dictionary phrase lengths. This analysis also provides a new argument for obtaining asymptotic results about the mean dictionary phrase length and average redundancy rates.
23 citations
TL;DR: For the register function for t-ary trees, recently introduced by Auber et al., it is proved that the average is log4n + O(1), if all such trees with n internal nodes are considered to be equally likely.
Abstract: For the register function for t-ary trees, recently introduced by Auber et al., we prove that the average is log4n p O(1), if all such trees with n internal nodes are considered to be equally likely.This result remains true for rooted trees where the set of possible out-degrees is finite. Furthermore we obtain exponential tail estimates for the distribution of the register function. Thus, the distribution is highly concentrated around the mean value.
15 citations
Posted Content•
TL;DR: It is shown that the limiting distribution as n goes to infinity of the number of occurrences of ℳ is asymptotically normal with mean value and variance asymPTotically equivalent to μn and ρ2n, respectively, where the constants μ>0 and σ≥0 are computable.
Abstract: Let $T\_n$ denote the set of unrooted labeled trees of size $n$ and let $T\_n$ be a particular (finite, unlabeled) tree. Assuming that every tree of $T\_n$ is equally likely, it is shown that the limiting distribution as $n$ goes to infinity of the number of occurrences of $M$ as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to $\mu n$ and $\sigma^2n$, respectively, where the constants $\mu>0$ and $\sigma\ge 0$ are computable.
4 citations
01 Jan 2006
TL;DR: The present volume of UDT is devoted to Robert F. Tichy on the occasion of his 50th birthday as discussed by the authors, and we cordially congratulate him on this occasion and wish him the best for the future.
Abstract: The present volume of UDT is devoted to Robert F. Tichy on the occasion of his 50th birthday. We cordially congratulate him on this occasion and wish him the best for the future. In this short note we collect highlights of his scientific work. Together with more than 70 coauthors he has written over 200 papers so far, with topics that range from number theory to applications in actuarial mathematics and also mathematical chemistry. But, of course, the scientific work is only one facet of Robert Tichy’s personality. It is remarkable how many students graduated under his supervision since 1984. At least 25 PhD students are known to us, including the five authors of the
3 citations
TL;DR: In this article, the authors present recent results concerning concentration properties of extremal parameters of random discrete structures and provide exponential tail estimates for the height distribution of scale-free trees, as well as the maximum degree of several kinds of random trees.
Abstract: The purpose of this survey is to present recent results concerning concentration properties of extremal parameters of random discrete structures. A main emphasis is placed on the height and maximum degree of several kinds of random trees. We also provide exponential tail estimates for the height distribution of scale-free trees.
1 citations
TL;DR: In this article, it was shown that the recurrence does converge for all positive values of the parameter β, which leads to an extension of some limit theorems for the corresponding branching processes.
Abstract: Some classes of controlled branching processes (with nonhomo-geneous migration or with nonhomo-geneous state-dependent immigration) lead in the critical case to a recurrence for the extinction probabilities. Under some additional conditions it is known that this recurrence depends on some parameter β and converges for 0 < β < 1. Now we show that the recurrence does converge for all positive values of the parameter β, which leads to an extension of some limit theorems for the corresponding branching processes. We also give a generalization of the recurrence and an asymptotic analysis of its behavior.
1 citations
TL;DR: In this article, a functional limit theorem of the normalized profile of random search trees was proved for a certain range of α > 0, where α is a function in a complex domain.
Abstract: We study the profile $X_{n,k}$ of random search trees including binary search trees and $m$-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for $k=\lfloor\alpha\log n\rfloor$ in a certain range of $\alpha$. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.