The Ubiquitous Digital Tree

Flajolet, Philippe

doi:10.1007/11672142_1

Cited by 13 publications

(12 citation statements)

References 48 publications

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“…The analyses in [11] and [9] feature trie recurrences involving incomplete binomial sums with extra terms with similarly complicated Mellin transforms; our recurrence, in contrast with these, is bivariate and features different additional terms, which complicates the analysis. For other parameters of interest, see, e.g., [22,6]. See also [8] for background on complex asymptotics.…”

Section: Introductionmentioning

confidence: 99%

Profiles of PATRICIA Tries

Magner

Szpankowski

2016

Algorithmica

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A PATRICIA trie is a trie in which non-branching paths are compressed. The external profile B n,k , defined to be the number of leaves at level k of a PATRICIA trie on n nodes, is an important "summarizing" parameter, in terms of which several other parameters of interest can be formulated. Here we derive precise asymptotics for the expected value and variance of B n,k , as well as a central limit theorem with error bound on the characteristic function, for PATRICIA tries on n infinite binary strings generated by a memoryless source with bias p > 1/2 for k ∼ α log n with α ∈ (1/ log(1/q) + , 1/ log(1/p) − ) for any fixed > 0. In this range, E[B n,k ] = Θ(Var[B n,k ]), and both are of the form Θ(n β(α) / √ log n), where the Θ hides bounded, periodic functions in log n whose Fourier series we explicitly determine. The compression property leads to extra terms in the Poisson functional equations for the profile which are not seen in tries or digital search trees, resulting in Mellin transforms which are only implicitly given in terms of the moments of B m,j for various m and j. Thus, the proofs require information about the profile outside the main range of interest. Our derivations rely on analytic techniques, including Mellin transforms, analytic de-Poissonization, the saddle point method, and careful bounding of complex functions.

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Section: Introductionmentioning

confidence: 99%

Profiles of PATRICIA Tries

Magner

Szpankowski

2016

Algorithmica

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“…where M [φ(z); s] := ∞ 0 φ(z)z s−1 dz denotes the Mellin transform of φ; see [7]. Next, again from (6) we see that M [h 1 (z); s] can be analytically continued to the vertical line (s) = −1 and has no singularities there. Thus, by shifting the line of integration in (10) and computing residues, we obtainC (z) ∼ zF [g (2) ](z), uniformly for z in a sector.…”

Section: Covariance and Correlation Coefficientmentioning

confidence: 89%

“…The dependence phenomena as those discovered in this paper are not limited to random tries and have indeed a wider range of connections. They also appear in different forms in other structures and algorithms with an underlying binomial splitting process; see Flajolet [6] and [10,13] for references on data structures, algorithms, conflict resolution protocols and stochastic models. A typical example is the dependence between the number of coin-tossings p = 0.4 p = 0.5 p = 0.6 p = 0.1 p = 0.2 p = 0.3 p = 0.7 p = 0.8 p = 0.9 Figure 3: Joint distributions of (S n , K n ) by Monte-Carlo simulations for n = 10 7 and varying p: the case p = 0.5 is seen to have stronger dependence than the others.…”

Section: Shape Parametersmentioning

confidence: 99%

Dependence between path-length and size in random digital trees

Fuchs¹,

Hwang

2017

J. Appl. Probab.

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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 correlation is 0, 1 and periodically oscillating. Moreover, the same type of behaviors is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

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“…There are natural instances of sources that belong to the Good-UNI Class, for instance the Euclidean dynamical system defined in (9), together with two other dynamical systems, of the Euclidean type.…”

Section: Tameness Of Dynamical Sourcesmentioning

confidence: 99%

“…Here, in the paper, we focus on three main parameters, two for Trie(X ) and one for Bst(X ). When restricted to simple sources, there exist many works that study the trie parameters (see [9,14,15,24]) or the symbol path length for Bst (see [8]). The same studies, in the case of a general source, are done in [4] for the Trie and in [26] for the Bst, and are summarized as follows:…”

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confidence: 99%

Information theory: Sources, Dirichlet series, and realistic analyses of data structures

Roux

Vallée

2011

Electron. Proc. Theor. Comput. Sci.

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Most of the text algorithms build data structures on words, mainly trees, as digital trees (tries) or binary search trees (bst). The mechanism which produces symbols of the words (one symbol at each unit time) is called a source, in information theory contexts. The probabilistic behaviour of the trees built on words emitted by the same source depends on two factors: the algorithmic properties of the tree, together with the information-theoretic properties of the source. Very often, these two factors are considered in a too simplified way: from the algorithmic point of view, the cost of the Bst is only measured in terms of the number of comparisons between words --from the information theoretic point of view, only simple sources (memoryless sources or Markov chains) are studied. We wish to perform here a realistic analysis, and we choose to deal together with a general source and a realistic cost for data structures: we take into account comparisons between symbols, and we consider a general model of source, related to a dynamical system, which is called a dynamical source. Our methods are close to analytic combinatorics, and our main object of interest is the generating function of the source Lambda(s), which is here of Dirichlet type. Such an object transforms probabilistic properties of the source into analytic properties. The tameness of the source, which is defined through analytic properties of Lambda(s), appears to be central in the analysis, and is precisely studied for the class of dynamical sources. We focus here on arithmetical conditions, of diophantine type, which are sufficient to imply tameness on a domain with hyperbolic shape

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The Ubiquitous Digital Tree

Cited by 13 publications

References 48 publications

Profiles of PATRICIA Tries

Profiles of PATRICIA Tries

Dependence between path-length and size in random digital trees

Information theory: Sources, Dirichlet series, and realistic analyses of data structures

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