Showing papers by "Michael Drmota published in 2004"

Precise minimax redundancy and regret

Abstract: Recent years have seen a resurgence of interest in redundancy of lossless coding. The redundancy (regret) of universal fixed-to-variable length coding for a class of sources determines by how much the actual code length exceeds the optimal (ideal over the class) code length. In a minimax scenario one finds the best code for the worst source either in the worst case (called also maximal minimax) or on average. We first study the worst case minimax redundancy over a class of stationary ergodic sources and replace Shtarkov's bound by an exact formula. Among others, we prove that a generalized Shannon code minimizes the worst case redundancy, derive asymptotically its redundancy, and establish some general properties. This allows us to obtain precise redundancy for memoryless, Markov, and renewal sources. For example, we present the exact constant of the redundancy for memoryless and Markov sources by showing that the integer nature of coding contributes log(logm/(m-1))/logm+o(1) where m is the size of the alphabet. Then we deal with the average minimax redundancy and regret. Our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources. After formulating some general bounds relating these two redundancies, we prove our assertion for memoryless and Markov sources. Nevertheless, we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy (however, our general results show that maximal and average regrets are asymptotically equivalent).

Mathematics and Computer Science Iii: "Algorithms, Trees, Combinatorics And Probabilities"

Abstract: Mathematics and Computer Science III contains invited and contributed papers on combinatorics, random graphs and networks, algorithms analysis and trees, branching processes, constituting the Proceedings of the Third International Colloquium on Mathematics and Computer Science, held in Vienna in September 2004. It addresses a large public in applied mathematics, discrete mathematics and computer science, including researchers, teachers, graduate students and engineers.

Stochastic analysis of tree–like data structures

Abstract: The purpose of this article is to present two types of data structures, binary search trees and usual (combinatorial) binary trees. Although they constitute the same set of (rooted) trees, they are constructed via completely different rules and thus the underlying probabilistic models are different too. Both kinds of data structure can be analysed by probabilistic and stochastic tools, binary search trees (more or less) with martingales and binary trees (which can be considered as a special case of Galton–Watson trees) with stochastic processes. It is also an aim of this article to demonstrate the strength of analytical methods in specific parts of probability theory related to combinatorial problems, and we especially make use of the concept of generating functions. One reason for the use of generating functions is that recursive combinatorial descriptions can be translated to relations for generating functions, and another is that analytical properties of these generating functions can be used to derive asymptotic (probabilistic) relations.

The width of Galton-Watson trees conditioned by the size †

Abstract: It is proved that the moments of the width of Galton-Watson trees of size n and with offspring variance σ ^2 are asymptotically given by (σ √n)^pm_p where m_p are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications.

Special Issue on Analysis of Algorithms

Abstract: This special issue is devoted to the Analysis of Algorithms (AofA). Most of the papers are from the Eighth Seminar on Analysis of Algorithms, held in Strobl, Austria, June 23–29, 2002.

Profile and Height of Random Binary Search Trees

Abstract: The purpose of this article is to survey recent results on distributional properties of random binary search trees. In particular we consider the profile and the height.

General urn models with several types of balls and Gaussian limiting fields

Abstract: We study a system of m urns, where several types of balls are thrown, and an additive valuation is assigned to each urn depending on its state. Examples are the join models studied in a database context, and some models with two types of balls. The object of our investigation is the evolution of the valuation with time, when a ball is thrown at each time unit. By means of a generating function approach we show the weak convergence of the valuation process to a Gaussian field.

Variable-to-variable codes with small redundancy rates

Abstract: There are three major classes of lossless compression: fixed-to-variable (FV) length codes, variable-to-fixed (VF) length codes, and finally variable-to-variable (VV) length codes. This paper presents the construction and analysis of a VV-code with small average and maximal redundancy that decays to zero as the average code length increases. A variable-to-variable (VV) code is a concatenation of variable-to-fixed and fixed-to-variable codes.