Lower Bounds on the Redundancy of Huffman Codes with Known and Unknown Probabilities

In this paper we provide a method to obtain tight lower bounds on the minimum redundancy achievable by a Huffman code when the probability distribution underlying an alphabet is only partially known. In particular, we address the case where the occurrence probabilities are unknown for some of the symbols in an alphabet. Bounds can be obtained for alphabets of a given size, for alphabets of up to a given size, and for alphabets of arbitrary size. The method operates on a Computer Algebra System, yielding closed-form numbers for all results. Finally, we show the potential of the proposed method to shed some light on the structure of the minimum redundancy achievable by the Huffman code

Minimax Trees in Linear Time with Applications

A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves\u27 depths, it minimizes the maximum of any leaf\u27s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, $O (n log n)$-time algorithm for building them. Drmota and Szpankowski (2002) gave another $O (n log n)$-time algorithm, which takes linear time when the weights are already sorted by their fractional parts. In this paper we give the first linear-time algorithm for building minimax trees for unsorted real weights

Codes robustes et codes joints source-canal pour transmission multimédia sur canaux mobiles

Some new error-resilient source coding and joint source/channel coding techniquesare proposed for the transmission of multimedia sources over error-prone channels.First, we introduce a class of entropy codes providing unequal error-resilience, i.e.providing some protection to the most sensitive information. These codes are thenextended to exploit the temporal dependencies. A new state model based on the aggregation of some states of the trellis is thenproposed and analyzed for soft source decoding of variable length codes with a lengthconstraint. It allows the weighting of the compromise between the estimation accuracyand the decoding complexity.Next, some paquetization methods are proposed to reduce the error propagationphenomenon of variable length codes.Finally, some re-writing rules are proposed to extend the binary codetree representationof entropy codes. The proposed representation allows in particular the designof codes with improved soft decoding performances.Cette thèse propose des codes robustes et des codes conjoints source/canal pourtransmettre des signaux multimédia sur des canaux bruités. Nous proposons des codesentropiques offrant une résistance intrinsèque aux données prioritaires. Ces codes sontétendus pour exploiter la dépendance temporelle du signal.Un nouveau modèle d’état est ensuite proposé et analysé pour le décodage souplede codes à longueur variable avec une contrainte de longueur. Il permet de réglerfinement le compromis performance de décodage/complexité.Nous proposons également de séparer, au niveau du codage entropique, les étapesde production des mots de codes et de paquétisation. Différentes stratégies de constructionde train binaire sont alors proposées.Enfin, la représentation en arbre binaire des codes entropiques est étendue enconsidérant des règles de ré-écriture. Cela permet en particulier d’obtenir des codesqui offrent des meilleures performances en décodage souple

The Minimum Description Length Principle for Pattern Mining: A Survey

This is about the Minimum Description Length (MDL) principle applied to pattern mining. The length of this description is kept to the minimum. Mining patterns is a core task in data analysis and, beyond issues of efficient enumeration, the selection of patterns constitutes a major challenge. The MDL principle, a model selection method grounded in information theory, has been applied to pattern mining with the aim to obtain compact high-quality sets of patterns. After giving an outline of relevant concepts from information theory and coding, as well as of work on the theory behind the MDL and similar principles, we review MDL-based methods for mining various types of data and patterns. Finally, we open a discussion on some issues regarding these methods, and highlight currently active related data analysis problems