Abstract not found

The goal of this research is threefold: (i) to analyze generalized digital search trees, (ii) to derive the average profile (i.e., phrase length) of a generalization of the well known parsing algorithm due to Lempel and Ziv, and (iii) to provide analytical tools to analyze asymptotically certain partial differential functional equations often arising in the analysis of digital trees. In the generalized Lempel-Ziv parsing scheme, one partitions a sequence of symbols from a finite alphabet into phrases such that the new phrase is the shortest substring seen in the past by at most b \Gamma 1 phrases (b = 1 corresponds to the original Lempel--Ziv scheme). Such a scheme can be analyzed through a generalized digital search tree in which every node is capable of storing up to b strings. In this paper, we investigate the depth of a randomly selected node in such a tree and the length of a randomly selected phrase in the generalized Lempel-Ziv scheme. These findings and some recent results al...

Ramanujan's Q-function and the so called "tree function" T (z) defined implicitly by the equation T (z) = ze T (z) found applications in hashing, the birthday paradox problem, random mappings, caching, memory conflicts, and so forth. Recently, several novel applications of these functions to information theory problems such as linear coding and universal portfolios were brought to light. In this paper, we study them in the context of another information theory problem, namely: universal coding which was recently investigated by Shtarkov et al. [Prob. Inf. Trans., 31, 1995]. We provide asymptotic expansions of certain recurrences studied there which describe the optimal redundancy of universal codes. Our methodology falls under the so called analytical information theory that was recently applied successfully to a variety of information theory problems. Key Words: Source coding, multi-alphabet universal coding, redundancy, minimum description length, analytical information theory, si...

. In combinatorics and analysis of algorithms often a Poisson version of a problem (called Poisson model or poissonization) is easier to solve than the original one, which is known as the Bernoulli model. Poissonization is a technique that replaces the original input by a Poisson process. More precisely, an analytical Poisson transform maps a sequence (e.g., characterizing the Bernoulli model) into a generating function of a complex variable. However, after poissonization one must depoissonize in order to translate the results of the Poisson model into the original (i.e., Bernoulli) model. We present here some analytical depoissonization results that fall into the following general scheme: if the Poisson transform has an appropriate growth in the complex plane, then an asymptotic expansion of the sequence can be expressed in terms of the Poisson transform and its derivatives evaluated on the real line. We illustrate our results on a few examples from combinatorics and the analysis of...