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12
A general limit theorem for recursive algorithms and combinatorial structures
 ANN. APPL. PROB
, 2004
"... Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer ..."
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Cited by 53 (25 self)
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Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer theorem is derived which allows us to establish a limit law on the basis of the recursive structure and to use the asymptotics of the first and second moments of the sequence. In particular, a general asymptotic normality result is obtained by this theorem which typically cannot be handled by the more common ℓ2 metrics. As applications we derive quite automatically many asymptotic limit results ranging from the size of tries or mary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proofs of these results we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric, which is the main tool in this article.
An asymptotic theory for CauchyEuler differential equations with applications to the analysis of algorithms
, 2002
"... CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We ..."
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Cited by 22 (10 self)
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CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for CauchyEuler equations and propose an asymptotic theory that covers almost all applications where CauchyEuler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.
A general contraction theorem and asymptotic normality in combinatorial structures
 Annals of Applied Probability
, 2001
"... Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method a general transfer t ..."
Abstract

Cited by 7 (2 self)
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Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method a general transfer theorem is derived, which allows to establish a limit law on the basis of the recursive structure and using the asymptotics of the first and second moments of the sequence. In particular a general asymptotic normality result is obtained by this theorem, which typically cannot be handled by the more common ` 2metrics. As applications we derive quite automatically many asymptotic normality results ranging from the size of tries or mary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proof of these we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric which is the main tool in this paper.
Partial match queries in random quadtrees
 SIAM J. Comput
, 2003
"... We propose a simple, direct approach for computing the expected cost of random partial match queries in random quadtrees. The approach gives not only an explicit expression for the leading constant in the asymptotic approximation of the expected cost but also more terms in the asymptotic expansion i ..."
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Cited by 7 (1 self)
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We propose a simple, direct approach for computing the expected cost of random partial match queries in random quadtrees. The approach gives not only an explicit expression for the leading constant in the asymptotic approximation of the expected cost but also more terms in the asymptotic expansion if desired. Key words. Quadtrees, partial match queries, binomial transform, Mellin transform, Euler transform, Rice’s integral.
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 ..."
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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.
Analysis in Distribution of Two Randomized Algorithms for Finding the Maximum in a Broadcast Communication Model
, 2002
"... The limit laws of three cost measures are derived of two algorithms for finding the maximum in a singlechannel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method o ..."
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Cited by 5 (4 self)
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The limit laws of three cost measures are derived of two algorithms for finding the maximum in a singlechannel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method of proof proceeds along the line via the method of moments and the "asymptotic transfers", which roughly bridges the asymptotics of the "conquering cost of the subproblems" and that of the total cost. Such a general approach has proved very fruitful for a number of problems in the analysis of recursive algorithms. 1
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 ..."
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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.
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. 1
Geometric Bucket Trees: Analysis of Linear Bucket Tree
"... We analyse the average number of buckets in a Linear Bucket tree created by n points uniformly dispatched on an interval of length y. A new bucket is created when a point does not fall in an existing bucket. The bucket is the interval of length 2 centered on the point. We illustrate this concept by ..."
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We analyse the average number of buckets in a Linear Bucket tree created by n points uniformly dispatched on an interval of length y. A new bucket is created when a point does not fall in an existing bucket. The bucket is the interval of length 2 centered on the point. We illustrate this concept by an interesting tale of how the moon’s surface took on its present form. Thanks to an explicit Laplace transform of the Poissonized sequence, and the use of dePoissonization tools, we obtain the explicit asymptotic expansions of the average number of buckets in most of the asymptotic regimes relative to n and y.