Abstract
Tries and PATRICIA tries are fundamental data structures in computer science with numerous applications. In a recent paper, a general framework for obtaining the mean and variance of additive shape parameters of tries and PATRICIA tries under the Bernoulli model was proposed. In this note, we show that a slight modification of this framework yields a central limit theorem for shape parameters, too. This central limit theorem contains many of the previous central limit theorems from the literature and it can be used to prove recent conjectures and derive new results. As an example, we will consider a refinement of the size of tries and PATRICIA tries, namely, the number of nodes of fixed outdegree and obtain (univariate and bivariate) central limit theorems. Moreover, trivariate central limit theorems for size, internal path length and internal Wiener index of tries and PATRICIA tries are derived as well.
Highlights
Introduction and ResultsTries (from the word data retrival) have, since their introduction by de la Briandais [2] in 1959, found many applications, e.g., in searching, sorting, dynamic hashing, coding, polynomial factorization, regular languages, contention tree algorithms, automatically correcting words in texts, retrieving IP addresses and satellite data, internet routing,
Introduction and ResultsTries have, since their introduction by de la Briandais [2] in 1959, found many applications, e.g., in searching, sorting, dynamic hashing, coding, polynomial factorization, regular languages, contention tree algorithms, automatically correcting words in texts, retrieving IP addresses and satellite data, internet routing, the electronic journal of combinatorics 21(1) (2014), #P1.68 and molecular biology; see Park et al [23] for details and many references
In Fuchs et al [8], the authors used Mellin transform together with the theory of JSadmissibility which was introduced in Hwang et al [15] and the idea of “corrected poissonized variance” which is from [15] to propose a general framework for deriving asymptotic expansions of mean and variance of additive shape parameters in random tries and random PATRICIA tries
Summary
Tries (from the word data retrival) have, since their introduction by de la Briandais [2] in 1959, found many applications, e.g., in searching, sorting, dynamic hashing, coding, polynomial factorization, regular languages, contention tree algorithms, automatically correcting words in texts, retrieving IP addresses and satellite data, internet routing,. In Fuchs et al [8], the authors used Mellin transform together with the theory of JSadmissibility which was introduced in Hwang et al [15] (and which is largely based on analytic Depoissonization) and the idea of “corrected poissonized variance” which is from [15] to propose a general framework for deriving asymptotic expansions of mean and variance of additive shape parameters in random tries and random PATRICIA tries. In this article, which is intended to be a supplement to [8], we will show that the same framework with only minor modifications gives a general central limit theorem for a large class of additive shape parameters, which in particular covers most of the previous central limit theorems for shape parameters in random tries and random Patricia tries.
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