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

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 tries. 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 c hains, 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 speci c transfer operators of the Ruelle type.Key-words: Information theory, dynamical sources, analysis of algorithms, digital trees, tries, ternary search tries, transfer operator, continued fractions. 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 tries. 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 i n trinsic characteristics of the source: the entropy and the probability o f letter coincidence. These characteristics are themselves related in a natural way to spectral properties of speci c transfer operators of the Ruelle type.

The Number of Symbol Comparisons in QuickSort and QuickSelect

“…Our approach otherwise relies on methods from analytic combinatorics [10], as well as on information theory and the theory of dynamical systems. It relates to earlier analyses of digital trees ("tries") by Clément, Flajolet, and Vallée [3,19].…”

“…They are principally determined by a shift transformation T . As shown by Vallée [19] (see also [3]), the Dirichlet series of fundamental intervals can be analysed by means of transfer operators, specifically the ones of secant type. We discuss here the much researched case where T is a non-linear expanding map of the interval.…”

“…also known as "tries", in [3].) The surprise is that there are cases where this upper bound is tight, as in QuickSort; others where both costs are of the same order, as in QuickSelect.…”

The Ubiquitous Digital Tree

“…It's as simple as that! The TST was analysed by Clément, Flajolet, and Vallée [7,8]. Quoting from [7]: Ternary search tries are an efficient data structure from the information theoretic point of view since a search costs typically about log n comparisons.…”

“…Vallée's theory has been applied to tries, in particular in a joint work with Clément [8]. What it brings to the field is the unifying notion of fundamental intervals which are the subintervals of [0, 1] associated to places corresponding to potential nodes of tries.…”