Exact L^2-distance from the limit for QuickSort key comparisons (extended abstract)

TL;DR

This paper derives an exact expression for the L^2-distance from the limit in QuickSort key comparisons, showing it is asymptotically equivalent to the square root of (2/n log n), refining previous bounds.

Contribution

It provides a simple recursive method to obtain an exact formula for the L^2-distance, improving understanding of the convergence rate in QuickSort analysis.

Findings

L^2-distance is asymptotically equivalent to (2 n^{-1} log n)^{1/2}

Previous bounds on the distance are refined by this exact expression

The recursive approach simplifies the analysis of convergence in QuickSort

Abstract

Using a recursive approach, we obtain a simple exact expression for the L^2-distance from the limit in R\'egnier's (1989) classical limit theorem for the number of key comparisons required by QuickSort. A previous study by Fill and Janson (2002) using a similar approach found that the d_2-distance is of order between n^{-1} log n and n^{-1/2}, and another by Neininger and Ruschendorf (2002) found that the Zolotarev zeta_3-distance is of exact order n^{-1} log n. Our expression reveals that the L^2-distance is asymptotically equivalent to (2 n^{-1} ln n)^{1/2}.