Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution)

Algorithmic counting of nonequivalent compact Huffman codes

It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees (level-greedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively

Egyptian Fractions

Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.Comment: 10 pages, survey article written for Nieuw Archief voor Wiskunde. Comments welcome

Compositions into Powers of bb: Asymptotic Enumeration and Parameters

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated

Design and application of variable-to-variable length codes

This work addresses the design of minimum redundancy variable-to-variable length (V2V) codes and studies their suitability for using them in the probability interval partitioning entropy (PIPE) coding concept as an alternative to binary arithmetic coding. Several properties and new concepts for V2V codes are discussed and a polynomial-based principle for designing V2V codes is proposed. Various minimum redundancy V2V codes are derived and combined with the PIPE coding concept. Their redundancy is compared to the binary arithmetic coder of the video compression standard H.265/HEVC

Algorithms for Imaging Atmospheric Cherenkov Telescopes

Imaging Atmospheric Cherenkov Telescopes (IACTs) are complex instruments for ground-based -ray astronomy and require sophisticated software for the handling of the measured data. In part one of this work, a modular and efficient software framework is presented that allows to run the complete chain from reading the raw data from the telescopes, over calibration, background reduction and reconstruction, to the sky maps. Several new methods and fast algorithms have been developed and are presented. Furthermore, it was found that the currently used file formats in IACT experiments are not optimal in terms of flexibility and I/O speed. Therefore, in part two a new file format was developed, which allows to store the camera and subsystem data in all its complexity. It offers fast lossy and lossless compression optimized for the high data rates of IACT experiments. Since many other scientific experiments also struggle with enormous data rates, the compression algorithm was further optimized and generalized, and is now able to efficiently compress the data of other experiments as well. Finally, for those who prefer to store their data as ASCII text, a fast I/O scheme is presented, including the necessary compression and conversion routines. Although the second part of this thesis is very technical, it might still be interesting for scientists designing an experiment with high data rates