Abstract not found

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 + 1-median version of Quicksort.

We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n-1-In + 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; non-normal 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 in-situ permutation algorithm to tree traversal algorithms, etc.

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 m-ary 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.

this paper, we establish an integral equation for the probability density of the number of comparisons L n . Then, we investigate the large deviations of L n . We shall show that the left tail of the limiting distribution is much "thinner" (i.e., double exponential) than the right tail (which is only exponential). Our results contain some constants that must be determined numerically. We use formal asymptotic methods of applied mathematics such as the WKB method and matched asymptotics. Keywords: Algorithms, Analysis of algorithms, Asymptotic analysis, Binary search tree, Quicksort, Sorting. 1 Introduction

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.

Introduction Some thirty years ago Hoare (1962) introduced the algorithm QUICKSORT, now a widely applied and well-studied sorting method. A terse description of the algorithm is in Hoare (1961), which also contains a closely related selection algorithm. These algorithms are formulated in a recursive manner and contain a random element. In the case of the selection algorithm FIND, which is the primary object of the present paper, the following is a simple version: F IND(S; l) has as its input a set S IR and an integer l, 1 l n, where n = #S denotes the number of elements of S. If n = 1 the algorithm returns the element of S. If n > 1 it proceeds as follows: (i) Choose x uniformly at random from S. (ii) Determine the two sets S = fy 2 S : y xg and S> = fy 2 S : y > xg. (iii) Let m = #S . If m l then continue with FIND(S ; l), else continue with FIND(S> ; l m). Obviously, this will give the unique x 2 S with the property #fy 2 S : y xg = l, i.e. the algorithm nds the

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.

An algorithm is developed for the exact simulation from distributions that are defined as fixed-points of maps between spaces of probability measures. The fixed-points 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.

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 first-order asymptoticexpansion for the variance of the cost is derived and results on exponential moments are given. The analysis is based on the contraction method.