Quicksort asymptotics

• Published in: Journal of algorithms Vol. 44; no. 1; pp. 4 - 28
• Main Authors: Fill, James Allen, Janson, Svante
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-07-2002
• Subjects: Asymptotics; Coupling; Density; dp-metric; Kolmogorov–Smirnov distance; Limit distribution; Moment generating function; Numerical analysis; Quicksort; Rate of convergence; Sorting algorithm; Quicksort; Numerical analysis; Limit distribution; d p -metric; Coupling; Asymptotics; Rate of convergence; Kolmogorov–Smirnov distance; Density; Sorting algorithm; Moment generating function
• ISSN: 0196-6774; 1090-2678
• DOI: 10.1016/S0196-6774(02)00216-X

The number of comparisons 
X
n
used by
Quicksort to sort an array of 
n distinct numbers has mean
μ
n
of order
nlog
n and standard deviation of order 
n. Using different methods, Régnier and Rösler each showed that the normalized variate
Y
n
:=(
X
n
−
μ
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
ε−(1/2)
) for Kolmogorov–Smirnov distance, for any positive 
ε.