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

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

Quicksort asymptotics

The number of comparisons X_n used by Quicksort to sort an array of n distinct numbers has mean mu_n of order n log n and standard deviation of order n. Using different methods, Regnier and Roesler each showed that the normalized variate Y_n := (X_n - mu_n) / n converges in distribution, say to Y; the distribution of Y can be characterized as the unique fixed point with zero mean of a certain distributional transformation. We provide the first rates of convergence for the distribution of Y_n to that of Y, using various metrics. In particular, we establish the bound 2 n^{- 1 / 2} in the d_2-metric, and the rate O(n^{epsilon - (1 / 2)}) for Kolmogorov-Smirnov distance, for any positive epsilon.Comment: 23 pages. See also http://www.mts.jhu.edu/~fill/ and http://www.math.uu.se/~svante/ . To be submitted for publication in May, 200

A characterization of the set of fixed points of the Quicksort transformation

The limiting distribution \mu of the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation T -- unique, that is, subject to the constraints of zero mean and finite variance. We show that a distribution is a fixed point of T if and only if it is the convolution of \mu with a Cauchy distribution of arbitrary center and scale. In particular, therefore, \mu is the unique fixed point of T having zero mean.Comment: 9 pages. See also http://www.mts.jhu.edu/~fill/ and http://www.math.uu.se/~svante/papers . Submitted for publication in May,200

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 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

Distributional convergence for the number of symbol comparisons used by QuickSort

Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (i.i.d.) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness" condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by n. Additionally, under a condition that grows more restrictive as p increases, we have convergence of moments of orders p and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, that is, whenever each key is generated as an infinite string of i.i.d. symbols. This is somewhat surprising; even for the classical model that each key is an i.i.d. string of unbiased ("fair") bits, the mean exhibits periodic fluctuations of order n.Comment: Published in at http://dx.doi.org/10.1214/12-AAP866 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org