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

Abstract. We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures. Given a tree t and one of its leaves a, the w(t, a) parameter denotes the number of internal nodes in the subtree rooted at a’s father. The closely-related w(t, a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a’s father. We define the cumulative w parameter as W(t) = ∑ a w(t, a), i.e., as the sum of w(t, a) over all leaves a of t. The w parameter not only plays an important rôle in the analysis of the Lempel-Ziv ’77 data compression algorithm, but it is captivating from a combinatorial viewpoint too. In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries. The final section of this report briefly summarizes and improves the previously known results about the w parameter’s behavior on tries and suffix trees, originally published in one author’s thesis (see [War05], [WS05], [LSW07]). This survey of new results about the w parameter is very instructive since a variety of different combinatorial methods are used in tandem to carry out the analysis. 1.

Abstract. We study the asymptotic probability that a random composition of an integer n is gap-free, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise gap-free. 1. introduction A composition of a natural number n is said to be gap-free if the part sizes occuring in it form an interval. In addition if the interval starts at 1, the composition is said to be complete. Example Of the 32 compositions of n = 6, there are 21 gap-free compositions arising from permuting the order of the parts of the partitions

Abstract: We consider a stochastic process (Xt)t≥0 that grows linearly in time and experiences collapses at times governed by a Poisson process with rate λ. The collapses are modeled by multiplying the process level by a random variable supported on [0, 1). For the hitting time defined as τy = inf{t> 0|Xt = y} we derive power series for the Laplace transform and all moments. We further discuss the asymptotic behavior of the mean of τy as y tends to infinity. 1.

Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n(log n) 2-variance for certain notions of total path-length is also clarified.

This paper continues the study of gaps in sequences of geometrically distributed random variables, as started by Hitczenko and Knopfmacher [9], who concentrated on sequences which were gap-free. Now we allow gaps, and count some related parameters. Our notation of gaps just means empty “urns ” (within the range of occupied urns). This might be called weak gaps, as opposed to maximal gaps, as in [9]. If one considers only “gap-free ” sequences, both notions coincide. First, the probability that a sequence of length n has a fixed number r of gaps is studied; apart from small oscillations, this probability tends to a constant p ∗ (r). When p = q = 1/2, everything simplifies drastically; there are no oscillations. Then, the random variable ‘number of gaps ’ is studied; all moments are evaluated asymptotically. Furthermore, samples that have r gaps, in particular the random variable ‘largest non-empty urn ’ are studied. All moments of this distribution are evaluated asymptotically. The behaviour of the quantities obtained in our asymptotic formulæ is also studied for p → 0 resp. p → 1, through a variety of analytic techniques. The last section discusses the concept called ‘super-gap-free. ’ A sample is super-gapfree, if it is gap-free and each non-empty urn contains at least 2 items (and d-supergap-free, if they contain ≥ d items). For the instance p = q = 1/2, we sketch how the asymptotic probability (apart from small oscillations) that a sample is d-super-gap-free can be computed. 1

Krichevsky has shown that the average redundancy rate of an adaptive block code for memoryless sources is R = n+t t+1 + O , where m is a cardinality of source's alphabet, n is a block size, and t is a size of a sample used to construct this code. This led him to a conclusion that using samples with t = n is sucient to make R = O , which is the order of the redundancy rate of block codes for a known source.

We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; A shortcut to the limit is needed via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution (in the Wasserstein space) of a distributional equation. An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian.

Distances between nodes in random trees is a popular topic, and several classes of trees have recently been investigated. We look into this matter in digital search trees. By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. In addition to various asymptotics, we give an exact expression for the mean and for the variance in the unbiased case. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; it is prudent to seek a shortcut to the limit via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution of a distributional equation (distances being measured in the Wasserstein metric space). An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian.

Consider N balls that are distributed among V urns according to some distribution G. We do not see the outcome and now have to place one ball into one urn with the goal of maximizing the probability that it will be the left-most urn containing a single ball. How should we proceed? This is the urn-model translation of an interesting problem posed by an internetauction offered by a German real-estate company. In the real problem only V is known (upper-price limit), whereas neither G (the way in which participants choose their offer) nor N (number of offers) is known. We would like to make an offer in such a way to maximize the probability that it turns out to be the minimum of the random set of single offers. We face a two-sided problem. On the one side we would like to choose a model which is convincing in terms of the expected behaviour of participants. On the other side, we want to solve an optimization problem; that is, the model should also be tractable and allow for asymptotic expansions, leading to a computable algorithm. Our attack is based on arguing that G should be (essentially) geometric and that some information on E(N) (expectation of N) and V(N) (variance of N) can be obtained in practice. Under certain conditions on possible dependencies of G and N, we can give answers. Poissonization (namely, changing the number N ofballs from a fixed quantity into a random quantity with Poisson distribution and mean N) and dePoissonization (i.e. reconciling with the original model) play here an important role to make the answers explicit.