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Dependence between path-length and size in random digital trees

Part of: Combinatorial probability Combinatorics Theory of data Algorithms - Computer Science Limit theorems

Published online by Cambridge University Press:  30 November 2017

Michael Fuchs  and
Hsien-Kuei Hwang
Michael Fuchs*
Affiliation:
National Chiao Tung University
Hsien-Kuei Hwang*
Affiliation:
Academia Sinica
*
* Postal address: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan. Email address: mfuchs@math.nctu.edu.tw
** Postal address: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan.
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Abstract

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

Keywords

Random triescovariancetotal path lengthPearson's correlation coefficientasymptotic normalityPoissonizationde-Poissonizationintegral transformcontraction method

MSC classification

Primary: 05A16: Asymptotic enumeration 60F05: Central limit and other weak theorems 68W40: Analysis of algorithms
Secondary: 60C05: Combinatorial probability 68P05: Data structures
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1125 - 1143
Copyright
Copyright © Applied Probability Trust 2017 

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