Dynamical sources in information theory: A general analysis of trie structures

Abstract

Digital trees, also known as tries, are a general purpose flexible 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 tries”). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of the 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 finite 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 specific transfer operators of the Ruelle type.