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Asymptotic variance of random symmetric digital search trees
, 2009
"... Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more caref ..."
Abstract

Cited by 6 (5 self)
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Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic dePoissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n(log n) 2variance for certain notions of total pathlength is also clarified.
Asymptotic variance of random digital search trees
"... Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more caref ..."
Abstract

Cited by 1 (1 self)
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Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic dePoissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n.log n / 2variance for certain notions of total pathlength is also clarified.
A Multivariate Study on the Size of mary Tries and
"... 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. 1
Smoothing equations for large Pólya urns 1
, 2013
"... Abstract. Consider a balanced non triangular twocolor PólyaEggenberger urn process, assumed to be large which means that the ratio σ of the replacement matrix eigenvalues satisfies 1/2 < σ < 1. The composition vector of both discrete time and continuous time models admits a drift which is carried ..."
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Abstract. Consider a balanced non triangular twocolor PólyaEggenberger urn process, assumed to be large which means that the ratio σ of the replacement matrix eigenvalues satisfies 1/2 < σ < 1. The composition vector of both discrete time and continuous time models admits a drift which is carried by the principal direction of the replacement matrix. In the second principal direction, this random vector admits also an almost sure asymptotics and a realvalued limit random variable arises, named W DT in discrete time and W CT in continous time. The paper deals with the distributions of both W. Appearing as martingale limits, known to be nonnormal, these laws remain up to now rather mysterious. Exploiting the underlying tree structure of the urn process, we show that W DT and W CT are the unique solutions of two distributional systems in some suitable spaces of integrable probability measures. These systems are natural extensions of distributional equations that already appeared in famous algorithmical problems like Quicksort analysis. Existence and unicity of the solutions of the systems are obtained by means of contracting smoothing transforms. Via the equation systems, we find upperbounds for the moments of W DT and W CT and we show that the laws of W DT and W CT are momentdetermined. We also prove that their densities are not bounded at the origin.