Probability metrics and recursive algorithms

Rachev, Svetlozar T.; Rüschendorf, Ludger

doi:10.2307/1428133

Cited by 85 publications

(88 citation statements)

References 34 publications

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“…The contraction method is an effective tool for proving limit theorems and existence and uniqueness results for recursive algorithms and, in particular, recursive equations. The method was introduced for the analysis of the Quicksort algorithm in [25] and then independently developed further in [26] and [24] (which was submitted prior to [26]). It was then used and extended to the analysis of a large variety of algorithms in a series of papers; see, in particular, [19], [20], [21], and [27], which give general and easy-to-apply conditions for convergence results.…”

Section: Introductionmentioning

confidence: 99%

“…The contraction method has also been successfully applied to some nonlinear stochastic equations, e.g. for the analysis of iterated function systems, random fractal measures, and fractal stochastic processes (see [13], [14], [15], and [24]). …”

Section: Introductionmentioning

confidence: 99%

“…Contraction arguments based on suitable probability metrics for this problem were given in [6], [24], and [26].…”

Section: Introductionmentioning

confidence: 99%

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On stochastic recursive equations of sum and max type

Rüschendorf

2006

J. Appl. Probab.

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In this paper we consider stochastic recursive equations of sum type,, and b are random, (X i ) are independent, identically distributed copies of X, and ' d =' denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal L s -metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum-and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On stochastic recursive equations of sum and max type

Rüschendorf

2006

J. Appl. Probab.

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“…This method was introduced by Rösler [43] for the derivation of the limit law of the number of key comparisons needed by Hoare's Quicksort algorithm to sort a list of randomly permuted items. The contraction method was further developed in Rösler [44] and independently in Rachev and Rüschendorf [40]. A guide for the use of this technique and an overview over the applications up to 1998 is given in the survey article of Rösler and Rüschendorf [46].…”

Section: Introductionmentioning

confidence: 99%

“…where γ q+s,d and γ q+t,d are the constants in (40). By (39) the L 2 convergence in (11) is satisfied for…”

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confidence: 99%

On a multivariate contraction method for random recursive structures with applications to Quicksort

Neininger

2001

Random Struct Algorithms

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The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an application the asymptotic correlations and a bivariate limit law for the number of key comparisons and exchanges of median-of-(2t + 1) Quicksort is given. Moreover, for the Quicksort programs analyzed by Sedgewick the exact order of the standard deviation and a limit law follow, considering all the parameters counted by Sedgewick.

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On the internal path length ofd-dimensional quad trees

Neininger

Rüschendorf

1999

Random Struct. Alg.

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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 m-ary search trees.

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Probability metrics and recursive algorithms

Cited by 85 publications

References 34 publications

On stochastic recursive equations of sum and max type

On stochastic recursive equations of sum and max type

On a multivariate contraction method for random recursive structures with applications to Quicksort

On the internal path length ofd-dimensional quad trees

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