On the Horton-Strahler Number for Combinatorial Tries

Nebel, Markus E.

doi:10.1051/ita:2000117

Cited by 8 publications

(6 citation statements)

References 19 publications

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“…• fill-up (or saturation) level : the largest full level, or max{j : I n,j = 2 j }, where the levels of a tree denote the sets of nodes with the same distance from the root; see [49]; • Horton-Strahler number and stack-size: certain notions of heights related to the traversal of tries; see [4,17,54,55,56]; • distance of two randomly chosen nodes; see [1,7];…”

Section: Introductionmentioning

confidence: 99%

Profiles of Tries

Park¹,

Hwang²,

Nicodème³

et al. 2009

SIAM J. Comput.

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Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to the (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) with the same distance from the root. It is a function of the number of strings stored in a trie and the distance from the root. Several, if not all, trie parameters such as height, size, depth, shortest path, and fill-up level can be uniformly analyzed through the (external and internal) profiles. Although profiles represent one of the most fundamental parameters of tries, they have hardly been studied in the past. The analysis of profiles is surprisingly arduous, but once it is carried out it reveals unusually intriguing and interesting behavior. We present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source. We first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root. It appears that profiles of tries exhibit several fascinating phenomena. When moving from the root to the leaves of a trie, the growth of the expected profiles varies. Near the root, the external profiles tend to zero at an exponential rate, and then the rate gradually rises to being logarithmic; the external profiles then abruptly tend to infinity, first logarithmically and then polynomially; they then tend polynomially to zero again. Furthermore, the expected profiles of asymmetric tries are oscillating in a range where profiles grow polynomially, while symmetric tries are nonoscillating, in contrast to most shape parameters of random tries studied previously. Such a periodic behavior for asymmetric tries implies that the depth satisfies a central limit theorem but not a local limit theorem of the usual form. Also the widest levels in symmetric tries contain a linear number of nodes, differing from the order n/ √ log n for asymmetric tries, n being the size of the trees. Finally, it is observed that profiles satisfy central limit theorems when the variance goes unbounded, while near the height they are distributed according to Poisson laws. As a consequence of these results we find typical behaviors of the height, shortest path, fill-up level, and depth. These results are derived here by methods of analytic algorithmics such as generating functions, Mellin transform, Poissonization and de-Poissonization, the saddle-point method, singularity analysis, and uniform asymptotic analysis.

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Section: Introductionmentioning

confidence: 99%

Profiles of Tries

Park¹,

Hwang²,

Nicodème³

et al. 2009

SIAM J. Comput.

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“…Other applications of this parameter are related to geology, molecular biology, synthetic images of trees and channel networks (see [29] for details). For tries in the Bernoulli model, the same parameter has been studied by Devroye and Kruszewski [4] and Nebel [20] provides results in a combinatorial model. The present paper considers the natural model for characterizing the performance of the trie data structure.…”

Section: Introductionmentioning

confidence: 87%

On the Stack-Size of General Tries

Bourdon¹,

Nebel²,

Vallée³

2001

RAIRO-Theor. Inf. Appl.

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Digital trees or tries are a general purpose flexible data structure that implements dictionaries built on words. The present paper is focussed on the average-case analysis of an important parameter of this tree-structure, i.e., the stack-size. The stack-size of a tree is the memory needed by a storage-optimal preorder traversal. The analysis is carried out under a general model in which words are produced by a source (in the information-theoretic sense) that emits symbols. Under some natural assumptions that encompass all commonly used data models (and more), we obtain a precise average-case and probabilistic analysis of stack-size. Furthermore, we study the dependency between the stack-size and the ordering on symbols in the alphabet: we establish that, when the source emits independent symbols, the optimal ordering arises when the most probable symbol is the last one in this order.

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“…As in the specific case of extended binary trees [8] or combinatorial tries [26], it is possible to solve this recursion by a trigonometric change of variables. In this way it is possible to prove that…”

Section: A Unified Analysis Of Horton-strahler Parametersmentioning

confidence: 99%

“…(a) The average Horton-Strahler number of a tree of size (n, ᐉ) is asymptotically given by [26] log 4 ͑2 2 n͒ Ϫ 2 ϩ ␥ 2 ln͑2͒ ϩ ⌬͑n͒, :ϭ ᐉ n Ͻ 1 fix, n 3 ϱ.…”

Section: Nebelmentioning

confidence: 99%

“…Here, a node is called critical if its two subtrees possess the same value of hs, i.e., if it is responsible for a growth of the Horton-Strahler number. Recently, the author determined the average Horton-Strahler number of a class of trees (called Ꮿ-tries), which model the trie data structure in a combinatorial way [26]; the Horton-Strahler number of tries in the Bernoulli model has been investigated in [6]. The combinatorics of the Horton-Strahler analysis has been used in computer graphics for the creation of faithful synthetic images of natural trees (see [34]) and for information visualization [16].…”

Section: Introductionmentioning

confidence: 99%

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A unified approach to the analysis of Horton‐Strahler parameters of binary tree structures

Nebel

2002

Random Struct Algorithms

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ABSTRACT:The Horton-Strahler number naturally arose from problems in various fields, e.g., geology, molecular biology and computer science. Consequently, detailed investigations of related parameters for different classes of binary tree structures are of interest. This paper shows one possibility of how to perform a mathematical analysis for parameters related to the Horton-Strahler number in a unified way such that only a single analysis is needed to obtain results for many different classes of trees. The method is explained by the examples of the expected Horton-Strahler number and the related rth moments, the average number of critical nodes, and the expected distance between critical nodes.

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On the Horton-Strahler Number for Combinatorial Tries

Cited by 8 publications

References 19 publications

Profiles of Tries

Profiles of Tries

On the Stack-Size of General Tries

A unified approach to the analysis of Horton‐Strahler parameters of binary tree structures

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