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The Wiener Index of Random Digital Trees
, 2012
"... The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An impo ..."
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The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An important class of random trees for which the Wiener index was not studied so far are random digital trees. In this work, we close this gap. More precisely, we derive asymptotic expansions of moments of the Wiener index and show that a central limit law for the Wiener index holds. These results are obtained for digital search trees and bucket versions as well as tries and PATRICIA tries. Our findings answer in affirmative two questions posed by Neininger.
A binomial splitting process in connection with corner parking
, 2013
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A Limit Theorem for Radix Sort and Tries with Markovian Input
, 2015
"... Tries are among the most versatile and widely used data structures on words. In particular, they are used in fundamental sorting algorithms such as radix sort which we study in this paper. While the performance of radix sort and tries under a realistic probabilistic model for the generation of words ..."
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Tries are among the most versatile and widely used data structures on words. In particular, they are used in fundamental sorting algorithms such as radix sort which we study in this paper. While the performance of radix sort and tries under a realistic probabilistic model for the generation of words is of significant importance, its analysis, even for simplest memoryless sources, has proved difficult. In this paper we consider a more realistic model where words are generated by a Markov source. By a novel use of the contraction method combined with moment transfer techniques we prove a central limit theorem for the complexity of radix sort and for the external path length in a trie. This is the first application of the contraction method to the analysis of algorithms and data structures with Markovian inputs; it relies on the use of systems of stochastic recurrences combined with a product version of the Zolotarev metric. 1
On 2Protected Nodes in Random Digital Trees
, 2015
"... In this paper, we consider the number of 2protected nodes in random digital trees. Results for the mean and variance of this number for tries have been obtained by Gaither, Homma, Sellke and Ward (2012) and Gaither and Ward (2013) and for the mean in digital search trees by Du and Prodinger (2012). ..."
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In this paper, we consider the number of 2protected nodes in random digital trees. Results for the mean and variance of this number for tries have been obtained by Gaither, Homma, Sellke and Ward (2012) and Gaither and Ward (2013) and for the mean in digital search trees by Du and Prodinger (2012). In this short note, we show that previous results and extensions such as the variance in digital search trees and limit laws in both cases can be derived in a systematic way by recent approaches of Fuchs, Hwang and Zacharovas (2010; 2014) and Fuchs and Lee (2014). Interestingly, the results for the moments we obtain by our approach are quite different from the previous ones and contain divergent series which have values by appealing to the theory of Abel summability. We also show that our tools apply to PATRICIA tries, for which the number of 2protected nodes has not been investigated so far.
A General Central Limit Theorem for Shape Parameters of
, 2014
"... 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 ..."
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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. 1
A Multivariate Study on the Size of mary Tries and PATRICIA Tries
"... 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 t ..."
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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 limit laws as well. As an application, we consider a refinement of the size of tries and PATRICIA tries, namely, the number of nodes of fixed outdegree and derive multivariate limit laws. Moreover, multivariate limit laws for size, internal path length and internal Wiener index of tries and PATRICIA tries are derived as well.
Analysis of radix selection on Markov sources
, 2014
"... The complexity of the algorithm Radix Selection is considered for independent data generated from a Markov source. The complexity is measured by the number of bucket operations required and studied as a stochastic process indexed by the ranks; also the case of a uniformly chosen rank is considered. ..."
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The complexity of the algorithm Radix Selection is considered for independent data generated from a Markov source. The complexity is measured by the number of bucket operations required and studied as a stochastic process indexed by the ranks; also the case of a uniformly chosen rank is considered. The orders of mean and variance of the complexity and limit theorems are derived. We find weak convergence of the appropriately normalized complexity towards a Gaussian process with explicit mean and covariance functions (in the space D[0,1] of càdlàg functions on [0,1] with the Skorokhod metric) for uniform data and the asymmetric Bernoulli model. For uniformly chosen ranks and uniformly distributed data the normalized complexity was known to be asymptotically normal. For ageneral Markov source (excludingthe uniform case) we findthat this complexity is less concentrated and admits a limit law with nonnormal limit distribution.