Results 1  10
of
19
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 ..."
Abstract

Cited by 54 (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.
On the Analysis of Stochastic Divide and Conquer Algorithms.
, 1999
"... This paper develops general tools for the analysis of stochastic divide and conquer algorithms. We concentrate on the average performance and the distribution of the duration of the algorithm. In particular we analyse the average performance and the running time distribution of the 2k + 1median ..."
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Cited by 48 (1 self)
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This paper develops general tools for the analysis of stochastic divide and conquer algorithms. We concentrate on the average performance and the distribution of the duration of the algorithm. In particular we analyse the average performance and the running time distribution of the 2k + 1median version of Quicksort.
Phase Change of Limit Laws in the Quicksort Recurrence Under Varying Toll Functions
, 2001
"... We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "to ..."
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Cited by 46 (19 self)
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We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "toll function" Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n## log E(Tn )/ log n 1/2), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an insitu permutation algorithm to tree traversal algorithms, etc.
On the internal path length of ddimensional quad trees
, 1999
"... It is proved that the internal path length of a d–dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limit ..."
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Cited by 16 (9 self)
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It is proved that the internal path length of a d–dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the mary search trees.
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 16 (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.
The Wiener index of random trees
, 2001
"... The Wiener index is analyzed for random recursive trees and random binary search trees in the uniform probabilistic models. We obtain the expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures ..."
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Cited by 16 (4 self)
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The Wiener index is analyzed for random recursive trees and random binary search trees in the uniform probabilistic models. We obtain the expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixedpoint equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.
Limit laws for partial match queries in quadtrees
 ANN. APPL. PROBAB
, 2001
"... It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for th ..."
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Cited by 10 (4 self)
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It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for the variance of the cost is derived and results on exponential moments are given. The analysis is based on the contraction method.
Density Approximation and Exact Simulation of Random Variables that are Solutions of FixedPoint Equations
 Adv. Appl. Probab
, 2002
"... An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic an ..."
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Cited by 10 (6 self)
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An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixedpoints with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method. AMS subject classifications. Primary: 65C10; secondary: 65C05, 68U20, 11K45.
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.