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Limit distribution of distances in biased random tries

Part of: Limit theorems Theory of data Graph theory Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Rafik Aguech ,
Nabil Lasmar  and
Hosam Mahmoud
Rafik Aguech*
Affiliation:
Faculté des Sciences de Monastir, Tunisia
Nabil Lasmar*
Affiliation:
IPEIT, Tunisia
Hosam Mahmoud*
Affiliation:
The George Washington University
*
∗Postal address: Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia. Email address: [email protected]
∗∗Postal address: Département de Mathématiques, IPEIT, 2 rue Jawaher Lel Nehru 1008 Montfleury, Tunis, Tunisia. Email address: [email protected]
∗∗∗Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA. Email address: [email protected]
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Abstract

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The trie is a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

Keywords

Random treerecurrenceMellin transformpoissonizationfixed pointcontraction method

MSC classification

Primary: 05C05: Trees 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 68P05: Data structures 68P10: Searching and sorting 68P20: Information storage and retrieval
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 377 - 390
Copyright
© Applied Probability Trust 2006 

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