Typical Depth of a Digital Search Tree built on a general source

Abstract

The digital search tree (dst) plays a central role in compression algorithms, of Lempel-Ziv type. This important structure can be viewed as a mixing of a digital structure (the trie) with a binary search tree. Its probabilistic analysis is thus involved, even in the case when the text is produced by a simple source (a memoryless source, or a Markov chain). After the seminal paper of Flajolet and Sedgewick (1986) [11] which deals with the memoryless unbiased case, many papers, due to Drmota, Jacquet, Louchard, Prodinger, Szpankowski, Tang, published between 1990 and 2005, dealt with general memoryless sources or Markov chains, and performed the analysis of the main parameters of dst's--namely, internal path length, profile, typical depth-- (see for instance [7, 15, 14]). Here, we are interested in a more realistic analysis, when the words are emitted by a general source, where the emission of symbols may depend on the whole previous history. There exist previous analyses of text algorithms or digital structures that have been performed for general sources, for instance for tries ([3, 2]), or for basic sorting and searching algorithms ([22, 4]). However, the case of digital search trees has not yet been considered, and this is the main subject of the paper. The idea of this study is due to Philippe Flajolet and the first steps of the work were performed with him, during the end of 2010. This paper is dedicated to Philippe's memory.