On the Tails of the Limiting QuickSort Density

We give upper and lower asymptotic bounds for the left tail and for the right tail of the continuous limiting QuickSort density f that are nearly matching in each tail. The bounds strengthen results from a paper of Svante Janson (2015) concerning the corresponding distribution function F. Furthermore, we obtain similar upper bounds on absolute values of derivatives of f of each order

On the tails of the limiting Quicksort distribution

We give asymptotics for the left and right tails of the limiting Quicksort distribution. The results agree with, but are less precise than, earlier non-rigorous results by Knessl and Spankowski.Comment: 8 pages. v2: Typos corrected and some formulations improve

On the densities of the limiting distributions for QuickSort and QuickQuant

In this dissertation, we study in depth the limiting distribution of the costs of running the randomized sorting algorithm QuickSort and the randomized selection algorithm QuickQuant when the cost of sorting/selecting is measured by the number of key comparisons. It is well established in the literature that the limiting distribution F of the centered and scaled number of key comparisons required by QuickSort is infinitely differentiable and that the corresponding density function f enjoys superpolynomial decay in both tails. The first contribution of this dissertation is to establish upper and lower asymptotic bounds for the left and right tails of f that are nearly matching in each tail. The literature study of the scale-normalized number of key comparisons used by the algorithm QuickQuant(t) for 0 ≤ t ≤ 1, on the other hand, is somewhat limited and focuses on (non-limiting and limiting) moments and the limiting distribution function Ft. In particular, except knowing that t = 0 and t = 1 corresponds to the well-known Dickman distribution, from the literature we do not know much about smoothness or decay properties of Ft for 0 min{t, 1 − t} and infinite right differentiability at x = t. In particular, we prove that the survival function 1 − Ft(x) and the density function ft(x) both have the right-tail asymptotics exp[−x ln x − x ln ln x + O(x)]. The third contribution of this dissertation is to study large deviations of the number of key comparisons needed for both algorithms by using knowledge of the limiting distribution. In particular, we sharpen the large-deviation results of QuickSort established by McDiarmid and Hayward (1996) and produce similar new (as far as we know) results for QuickQuant

Approximating the limiting Quicksort distribution

The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique fixed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions obtained by iterating the transformation S, beginning with a (nearly) arbitrary starting distribution. We demonstrate geometrically fast convergence for various metrics and discuss some implications for numerical calculations of the limiting Quicksort distribution. Finally, we give companion lower bounds which show that the convergence is not faster than geometric.Comment: 30 pages. See also http://www.mts.jhu.edu/~fill/ and http://www.math.uu.se/~svante/ . Submitted for publication in January, 200

On martingale tail sums in affine two-color urn models with multiple drawings

In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.Comment: 17 page

ELECTRONIC COMMUNICATIONS in PROBABILITY On the tails of the limiting Quicksort distribution *

Abstract We give asymptotics for the left and right tails of the limiting Quicksort distribution. The results agree with, but are less precise than, earlier non-rigorous results by Knessl and Spankowski