Quicksort asymptotics

TL;DR

This paper investigates the asymptotic distribution of the number of comparisons in Quicksort, providing the first explicit convergence rates to the limiting distribution using various metrics.

Contribution

It establishes the first quantitative bounds on the convergence rates of the normalized Quicksort comparison distribution to its limit.

Findings

Bound of 2 n^{-1/2} in the d_2-metric for convergence rate

O(n^{epsilon - 1/2}) rate in Kolmogorov-Smirnov distance

First explicit rates of convergence for the Quicksort comparison distribution

Abstract

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.