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

International audienceMost 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 Λ(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 Λ(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. Plan of the paper. We first recall in Section 1 general facts on sources and trees, and define the prob-abilistic model chosen for the analysis. Then, we provide the statements of the main two theorems (Theorem 1 and 2) which establish the possible probabilistic behaviour of trees, provided that the source be tame. The tameness notions are defined in a general framework and then studied in the case of simple sources (memoryless sources and Markov chains). In Section 2, we focus on a general model of sources, the dynamical sources, that contains as a subclass the simple sources. We present sufficient conditions on these sources under which it is possible to prove tameness. We compare these tameness properties to those of simple sources, and exhibit both resemblances and differences between the two classes