On the height of digital trees and related problems

Wojciech Szpankowski

doi:10.1007/bf01759045

Digital Trees and Memoryless Sources: from Arithmetics to Analysis

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2799 ◽

2010 ◽

Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽

Cited By ~ 1

Author(s):

Philippe Flajolet ◽

Mathieu Roux ◽

Brigitte Vallée

Keyword(s):

Probabilistic Models ◽

Asymptotic Form ◽

Leader Election ◽

Text Compression ◽

Digital Trees ◽

Arithmetic Properties ◽

International Audience ◽

Error Terms ◽

Dynamic Hashing

International audience Digital trees, also known as $\textit{"tries''}$, 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 $\textit{memoryless source}$, 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 $\textit{Dirichlet series}$. 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 $\textit{arithmetic properties}$, especially irrationality measures, of the sources under consideration.

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Dependence between path-length and size in random digital trees

Journal of Applied Probability ◽

10.1017/jpr.2017.56 ◽

2017 ◽

Vol 54 (4) ◽

pp. 1125-1143

Author(s):

Michael Fuchs ◽

Hsien-Kuei Hwang

Keyword(s):

Path Length ◽

Normal Limit ◽

Limit Laws ◽

Normal Distributions ◽

Digital Trees ◽

Bivariate Normal ◽

Asymmetric Case ◽

Periodic Fluctuations ◽

Type Of Behavior ◽

Small Periodic

AbstractWe 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.

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Asymptotic behavior of the Lempel-Ziv parsing scheme and digital trees

Proceedings of 1995 IEEE International Symposium on Information Theory ◽

10.1109/isit.1995.531116 ◽

2002 ◽

Author(s):

P. Jacquet ◽

W. Szpankowski

Keyword(s):

Asymptotic Behavior ◽

Digital Trees

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On the variance of the internal path length of generalized digital trees – The Mellin convolution approach

Theoretical Computer Science ◽

10.1016/s0304-3975(98)00213-8 ◽

2000 ◽

Vol 242 (1-2) ◽

pp. 143-168 ◽

Cited By ~ 4

Author(s):

Friedrich Hubalek

Keyword(s):

Path Length ◽

Digital Trees ◽

Convolution Approach ◽

Mellin Convolution

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The variance for partial match retrievals in $k$-dimensional bucket digital trees

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2796 ◽

2010 ◽

Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽

Author(s):

Michael Fuchs

Keyword(s):

Search Trees ◽

Partial Match ◽

Digital Trees ◽

Digital Search Trees ◽

International Audience ◽

Digital Search

International audience The variance of partial match queries in $k$-dimensional tries was investigated in a couple of papers in the mid-nineties, the resulting analysis being long and complicated. In this paper, we are going to re-derive these results with a much easier approach. Moreover, our approach works for $k$-dimensional PATRICIA tries, $k$-dimensional digital search trees and bucket versions as well.

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