Refined Quicksort asymptotics

TL;DR

This paper establishes a central limit theorem for the normalized complexity of Quicksort, showing that the scaled difference between the complexity and its limit converges in distribution to a normal distribution.

Contribution

It introduces a central limit theorem for the almost sure convergence of Quicksort's normalized complexity, providing new probabilistic insights into its asymptotic behavior.

Findings

The scaled error term converges to a normal distribution.

Quantifies the fluctuations of Quicksort's complexity around its limit.

Enhances understanding of the probabilistic structure of Quicksort asymptotics.

Abstract

The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of $n$ data, permuted uniformly at random, the appropriately normalized complexity $Y_n$ is known to converge almost surely to a non-degenerate random limit $Y$. This assumes a natural embedding of all $Y_n$ on one probability space, e.g., via random binary search trees. In this note a central limit theorem for the error term in the latter almost sure convergence is shown: $$\sqrt{\frac{n}{2\log n}}(Y_n-Y) \stackrel{d}{\longrightarrow} {\cal N} \qquad (n\to\infty),$$ where ${\cal N}$ denotes a standard normal random variable.