Analysis of Quickselect under Yaroslavskiy's Dual-Pivoting Algorithm

There is excitement within the algorithms community about a new partitioning method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly faster than the case when it runs under classic partitioning methods. We show that this improved performance in Quicksort is not sustained in Quickselect; a variant of Quicksort for finding order statistics. We investigate the number of comparisons made by Quickselect to find a key with a randomly selected rank under Yaroslavskiy's algorithm. This grand averaging is a smoothing operator over all individual distributions for specific fixed order statistics. We give the exact grand average. The grand distribution of the number of comparison (when suitably scaled) is given as the fixed-point solution of a distributional equation of a contraction in the Zolotarev metric space. Our investigation shows that Quickselect under older partitioning methods slightly outperforms Quickselect under Yaroslavskiy's algorithm, for an order statistic of a random rank. Similar results are obtained for extremal order statistics, where again we find the exact average, and the distribution for the number of comparisons (when suitably scaled). Both limiting distributions are of perpetuities (a sum of products of independent mixed continuous random variables).Comment: full version with appendices; otherwise identical to Algorithmica versio

The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance

For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenetic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Under the uniform model, the limiting distribution is the Airy distribution. The cornerstone of this study is the fact that the probabilistic models for phylogenetic trees are strongly related to the random permutation and the Catalan models for binary search trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000547 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Polya urns via the contraction method

We propose an approach to analyze the asymptotic behavior of P\'olya urns based on the contraction method. For this, a new combinatorial discrete time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each color. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete P\'olya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behavior.Comment: minor revision; accepted for publication in Combinatorics, Probability & Computing (Special issue dedicated to the memory of Philippe Flajolet

On weighted depths in random binary search trees

Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133-141], the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyze weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search process, the weighted path length and the weighted Wiener index in a random binary search tree. We establish three regimes of nodes depending on whether the second order behaviour of their weighted depths follows from fluctuations of the keys on the path, the depth of the nodes, or both. Finally, we investigate a random distribution function on the unit interval arising as scaling limit for weighted depths of nodes with at most one child