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We characterize all limit laws of the quicksort type random variables defined recursively by X n d = X In + X * n−1−In + T n when the " toll function " T n varies and satisfies general conditions, When the " toll function " T n (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n→∞ log E(T n)/ log n ≤ 1/2), X(More)
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(More)
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 OE0; e/. Convergence of all moments is shown to hold only for ˛ 2 OE0; 1 (with only convergence of finite moments when ˛ 2 .1; e/). When the(More)
We consider the problem of recovering items matching a partially specified pattern in multidi-mensional trees (quad trees and k-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on n points, it is known that the number of nodes C n (ξ) to visit in order to(More)
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 xed point of a random aane 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(More)
The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of n data, permuted uniformly at random, the appropriately normalized complexity Yn is known to converge almost surely to a non-degenerate random limit Y. This assumes a natural embedding of all Yn on one(More)
For the random binary search tree with n nodes inserted the number of ancestors of the elements with ranks k and , 1 ≤ k < ≤ n, as well as the path distance between these elements in the tree are considered. For both quantities central limit theorems for appropriately rescaled versions are derived. For the path distance the condition − k → ∞ as n → ∞ is(More)
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(More)