On the Stack-Size of General Tries

Bourdon, Jérémie; Nebel, Markus E.; Vallée, Brigitte

doi:10.1051/ita:2001114

Cited by 6 publications

(8 citation statements)

References 23 publications

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“…For our purposes, we need also a transfer theorem for entire functions satisfying the functional equation (5). Proof.…”

Section: Asymptotic Transfermentioning

confidence: 99%

“…Under different guises and different initial conditions, this is the most studied random variable defined on tries or related structures in the literature, most of them dealing with the expected value and very few of them with the variance. See, for example, [5,9,15,62,79,85,99] and the references therein for the mean, and [59,64,65,67,68,77,93] for the variance.…”

Section: Size Of Random Triesmentioning

confidence: 99%

“…While the dominant term involving the entropy for the expected value is well-known (see [5,9,62]), the corresponding term G(−1)/h for the variance is far from being intuitive. On the other hand, if we start with X 0 = a and X 1 = b, then…”

Section: Contention Resolution In Multi-access Channel Using Tree Algmentioning

confidence: 99%

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An analytic approach to the asymptotic variance of trie statistics and related structures

Fuchs

Hwang

Zacharovas

2014

Theoretical Computer Science

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We develop analytic tools for the asymptotics of general trie statistics, which are particularly advantageous for clarifying the asymptotic variance. Many concrete examples are discussed for which new Fourier expansions are given. The tools are also useful for other splitting processes with an underlying binomial distribution. We specially highlight Philippe Flajolet's contribution in the analysis of these random structures.

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“…For our purposes, we need also a transfer theorem for entire functions satisfying the functional equation (5). Proof.…”

Section: Asymptotic Transfermentioning

confidence: 99%

Section: Size Of Random Triesmentioning

confidence: 99%

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An analytic approach to the asymptotic variance of trie statistics and related structures

Fuchs

Hwang

Zacharovas

2014

Theoretical Computer Science

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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%

“…, 0.9 (the spans of the curves increase as p grows). The vertical lines represent the positions of α 2 (to the right of which the curves are straight lines); see(4).…”

mentioning

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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Number of Frequent Patterns in Random Databases

Lhote¹

2009

Advances in Data Analysis

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In a tabular database, patterns which occur over a frequency threshold are called frequent patterns. They are central in numerous data processes and various efficient algorithms were recently designed for mining them. Unfortunately, very few is known about the real difficulty of this mining, which is closely related to the number of frequent patterns. The worst case analysis always leads to an exponential number of frequent patterns, but experimentations show that algorithms become efficient for reasonable frequency thresholds. We perform here a probabilistic analysis of the number of frequent patterns. We first introduce a general model of random databases that encompasses all the previous classical models. In this model, the rows of the database are seen as independent words generated by the same probabilistic source [i.e. a random process that emits symbols]. Under natural conditions on the source, the average number of frequent patterns is studied for various frequency thresholds. Then, we exhibit a large class of sources, the class of dynamical sources, which is proven to satisfy our general conditions. This finally shows that our results hold in a quite general context of random databases.

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On the Stack-Size of General Tries

Cited by 6 publications

References 23 publications

An analytic approach to the asymptotic variance of trie statistics and related structures

An analytic approach to the asymptotic variance of trie statistics and related structures

Profiles of Tries

Number of Frequent Patterns in Random Databases

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