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2003b). On the contraction method with degenerate limit equation
"... A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X L = X. For nondegenerate limit equations the contraction method is a main tool to establis ..."
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Cited by 22 (12 self)
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A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X L = X. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the “unique ” solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.
A functional limit theorem for the profile of search trees
 Annals of Applied Probability
, 2008
"... We study the profile Xn,k of random search trees including binary search trees and mary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ in a certain range of α. A central feature of the proof is the use of the contraction method to ..."
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Cited by 21 (11 self)
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We study the profile Xn,k of random search trees including binary search trees and mary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ in a certain range of α. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinitedimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space. 1. Introduction. Search
Profile of Tries
, 2006
"... Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) wi ..."
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Cited by 18 (8 self)
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Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) with the same distance from the root. It is a function of the number of strings stored in a trie and the distance from the root. Several, if not all, trie parameters such as height, size, depth, shortest path, and fillup level can be uniformly analyzed through the (external and internal) profiles. Although profiles represent one of the most fundamental parameters of tries, they have been hardly studied in the past. The analysis of profiles is surprisingly arduous but once it is carried out it reveals unusually intriguing and interesting behavior. We present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source. We first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root. It appears that profiles of tries exhibit several fascinating phenomena. When moving from the root to the leaves of a trie, the growth of the expected profiles vary. Near the root, the external profiles tend to zero in an exponentially rate, then the rate gradually rises to being logarithmic; the external profiles then abruptly tend to infinity, first logarithmically
Profiles of random trees: Limit theorems for random recursive trees and binary search trees
, 2005
"... We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only con ..."
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Cited by 17 (11 self)
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We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only convergence of finite moments when ˛ 2.1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for ˛ D 0 and a “quicksort type ” limit law for ˛ D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.
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.
Probabilistic analysis for randomized game tree evaluation
 In Mathematics and computer science. III, Trends Math
, 2004
"... ABSTRACT: We give a probabilistic analysis for the randomized game tree evaluation algorithm of Snir. We first show that there exists an input such that the running time, measured as the number of external nodes read by the algorithm, on that input is maximal in stochastic order among all possible i ..."
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Cited by 5 (4 self)
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ABSTRACT: We give a probabilistic analysis for the randomized game tree evaluation algorithm of Snir. We first show that there exists an input such that the running time, measured as the number of external nodes read by the algorithm, on that input is maximal in stochastic order among all possible inputs. For this worst case input we identify the exact expectation of the number of external nodes read by the algorithm, give the asymptotic order of the variance including the leading constant, provide a limit law for an appropriate normalization as well as a tail bound estimating large deviations. Our tail bound improves upon the exponent of an earlier bound due to Karp and Zhang, where subgaussian tails were shown based on an approach using multitype branching processes and Azuma’s inequality. Our approach rests on a direct, inductive estimate of the moment generating function. 1
Multivariate Aspects of the Contraction Method
, 2003
"... We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open proble ..."
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Cited by 4 (0 self)
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We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open problems and give applications to the analysis of algorithms and branching recurrences.
On a functional contraction method
, 1202
"... Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the socalled contraction method to the space C[0, 1] of continuous functions endowed with unifo ..."
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Cited by 4 (2 self)
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Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the socalled contraction method to the space C[0, 1] of continuous functions endowed with uniform topology and the space D[0, 1] of càdlàg functions with the Skorokhod topology. The contraction method originated form the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences. It is based on stochastic fixedpoint equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach’s fixedpoint theorem. We develop the use of the Zolotarev metrics on the spaces C[0, 1] and D[0, 1] in this context. As an application a short proof of Donsker’s functional limit theorem is given. AMS 2010 subject classifications. Primary 60F17, 68Q25; secondary 60G18, 60C05. Key words. Functional limit theorem, contraction method, recursive distributional equation, Zolotarev
The size of random fragmentation trees
 In preparation
"... Abstract. We consider the random fragmentation process introduced by Kolmogorov, where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given ..."
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Cited by 3 (1 self)
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Abstract. We consider the random fragmentation process introduced by Kolmogorov, where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution (the dislocation law); this is repeated recursively for all pieces. More precisely, we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold, and we study the associated random tree. Dean and Majumdar found a phase transition for this process: the number of fragmentations is asymptotically normal for some dislocation laws but not for others, depending on the position of roots of a certain characteristic equation. This parallels the behaviour of discrete analogues with various random trees that have been studied in computer science. We give rigorous proofs of this phase transition, and add further details. The proof uses the contraction method. We extend some previous results for recursive sequences of random variables to families of random variables with a