Digital Trees and Memoryless Sources: from Arithmetics to Analysis

Philippe Flajolet 1 Mathieu Roux 2, 3 Brigitte Vallée 3
1 ALGORITHMS - Algorithms
Inria Paris-Rocquencourt
3 Equipe AMACC - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : Digital trees, also known as "tries", are fundamental to a number of algorithmic schemes, including radix-based searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the memoryless source, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters' expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous Dirichlet series. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific arithmetic properties, especially irrationality measures, of the sources under consideration.
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Philippe Flajolet, Mathieu Roux, Brigitte Vallée. Digital Trees and Memoryless Sources: from Arithmetics to Analysis. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10) , Jun 2010, Vienne, Austria. pp.231-258. ⟨hal-01083405v1⟩



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