Dynamical Sources in Information Theory : A General Analysis of Trie StructuresJulien Clément

Digital trees, also known as tries, are a general purpose exible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of array-tries, list tries, and bst-tries ((ternary search triess). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of height of tries is given. The unifying data model used is that of dynamical sources and it encompasses classical models like those of memoryless sources with independent symbols, of nite Markov chains, and of nonuniform densities. The probabilistic behaviour of the main parameters, namely size, path length, or height, appears to be determined by two intrinsic characteristics of the source: the entropy and the probability of letter coincidence. These characteristics are themselves related in a natural way to spectral properties of speciic transfer operators of the Ruelle type. Sources dynamiques en thhorie de l'information: une analyse ggnnrale des arbres digitaux RRsumm : Les arbres digitaux, galement connus sous le nom de triess sont une structure de donne ggnnrique et exible qui permet d'implanter des dictionnaires construits sur des ensembles de mots. Nous donnons une analyse de troies reprrsentations principales de ces arbres, les arbres-tableaux, les arbres-listes, et les arbres ternaires de recherche. La taille et les coots de recherche de ces reprrsentations sont analysss prrcissment en moyenne, tandis qu'une analyse en distribution de la hauteur est obtenue. Le moddle uniicateur d'analyse est celui des sources dynamiquess, lesquelles recouvrent les moddles classiques comme les sources sans mmmoire ((symboles inddpendants), les chaines de Markov nies, et les densitts initiales non uniformes. Les propriitts probabilistes des principaux parammtres de taille, longueur de cheminement et hauteur apparaissent liies deux caracttristiques fondamentales de la source: l'entropie et la probabilitt de coincidence. Ces caracttristiques se trouvent elle-mmmems reliies aux propriitts spectrales d'oprateurs de transfert du type introduit par Ruelle. Abstract. Digital trees, also known as tries, are a general purpose exible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of array-tries, list tries, and bst-tries ((ternary search triess). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of height of tries is given. The unifying data model used is that of dynamical …