Information theory: Sources, Dirichlet series, and realistic analyses of data structures

TL;DR

This paper analyzes data structures built on words emitted by general sources, using an analytic combinatorics approach with Dirichlet series, to provide a realistic and detailed understanding of their probabilistic and algorithmic behavior.

Contribution

It introduces a framework combining dynamical sources and Dirichlet series to analyze data structures with realistic cost models, extending beyond simple source assumptions.

Findings

The Dirichlet series Lambda(s) encodes source properties analytically.

Tameness of sources is characterized by Diophantine conditions on Lambda(s).

The approach links probabilistic source behavior with analytic properties of generating functions.

Abstract

Most of the text algorithms build data structures on words, mainly trees, as digital trees (tries) or binary search trees (bst). The mechanism which produces symbols of the words (one symbol at each unit time) is called a source, in information theory contexts. The probabilistic behaviour of the trees built on words emitted by the same source depends on two factors: the algorithmic properties of the tree, together with the information-theoretic properties of the source. Very often, these two factors are considered in a too simplified way: from the algorithmic point of view, the cost of the Bst is only measured in terms of the number of comparisons between words --from the information theoretic point of view, only simple sources (memoryless sources or Markov chains) are studied. We wish to perform here a realistic analysis, and we choose to deal together with a general source and a realistic cost for data structures: we take into account comparisons between symbols, and we consider a general model of source, related to a dynamical system, which is called a dynamical source. Our methods are close to analytic combinatorics, and our main object of interest is the generating function of the source Lambda(s), which is here of Dirichlet type. Such an object transforms probabilistic properties of the source into analytic properties. The tameness of the source, which is defined through analytic properties of Lambda(s), appears to be central in the analysis, and is precisely studied for the class of dynamical sources. We focus here on arithmetical conditions, of diophantine type, which are sufficient to imply tameness on a domain with hyperbolic shape.