Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions

Abstract

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, where (X n), (X * n), (I n , T n) are independent, X n d = X * n , and I n is uniformly distributed over {0,. .. , n − 1}. 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 n is asymptotically normally distributed; non-normal limit laws emerge when T n 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.

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

“…The same limit law X 0 .1/ also appeared in the total path length (which is P k kX n;k ) of recursive trees (see Dobrow and Fill, 1999), or essentially the total left path length of random binary search trees, and the cost of an in-situ permutation algorithm; see Hwang and Neininger (2002).…”

“…By considering N m .˛/ WD m .˛/.m˛C1/=m!, we easily obtain by induction that N m .˛/ D O.K m / for˛2 OE0; 1 (see Hwang and Neininger, 2002), and thus convergence in distribution of X n;k = n;k follows from (6) when˛2 OE0; 1.…”

“…From the recurrence (32) and the estimate (34), we deduce, by an induction similar to that used for (27), that…”

Limit laws for the Randić index of random binary tree models

“…This is a "Quicksort"-like recurrence, which has been treated in various works (see Hwang and Neininger, 2002, Kirschenhofer and Prodinger, 1998, Kirschenhofer, Prodinger and Martínez , 1997, and Panholzer, 2003, Panholzer and Prodinger, 1998, and Prodinger, 1995. We only outline here the strategy: One begins by differencing (n − 1) E(Y n−1 ) from n E(Y n ).…”

On a multivariate contraction method for random recursive structures with applications to Quicksort

“…These include in particular asymptotic expressions for the means and variances, as well as limit laws for the scaled quantities, and large deviation inequalities, see Hennequin [22,23], Régnier [42], Rösler [43,45], McDiarmid and Hayward [11], Bruhn [3], and for a detailed survey the book of Mahmoud [28]. For the number of exchanges B n the mean and variance were for general t ∈ N 0 studied in Hennequin [23], Chern and Hwang [5] refined the analysis of the mean, and Hwang and Neininger [25] gave a limit law for the standard case t = 0.…”