A Detailed Analysis of Quicksort Running Time
by Shalosh B. Ekhad and Doron Zeilberger
.pdf
.ps
.tex
First Written: March 8, 2019
One of the greatest algorithms of all time is Quicksort,
(it is ranked 7th in this top 10 list).
Its average running time is famously O(n log(n)), and its variance, less famously, is O(n^{2}) (hence its
standard deviation is O(n)). But what about higher moments? Here we find explicit expressions for the first eight moments,
their scaled limits, and we describe how to compute, approximately (but very accurately), percentiles of running time for any
listlength.
Added March 12, 2019:
Helmut Prodinger pointed our attention to this humangenerated gem.
Added March 22, 2019:
Sumit Kumar Jha pointed our attention to yet another humangenerated gem.
Maple package
Sample Input and Output for QuickSortAnalsis.txt

If you want to see the first 130 terms in the sequence of probablity generating functions of the
random variable "Number of comparisons in Quicksort with lists of length n"
the input file generates the
output file.

If you want to see the the probablity generating function of the
random variable "Number of comparisons in Quicksort with lists of length 150, in floating point
the input file generates the
output file.

If you want to see explicit expressionf or the first eight moments of the random variable "Number of comparisons in Quicksort with lists of length n"
as a function of n, as well as their asympotic expressions and scaled limits
the input file generates the
output file.

If you want to see a computergenerated article about the above
the input file generates the
output file.

If you want to see a the approximate percentiles for the running time of Quicksort with a list with 10000 items
the input file generates the
output file.

If you want to see a the results of simulating quicksort on 10000 randomly chosen sequences of length 200, and
also the theoretical values of the expectation, variance, skewness, and kurtosis,
the input file generates the
output file.

If you want to see a the results of simulating quicksort on 50000 randomly chosen sequences of length 200, and
also the theoretical values of the expectation, variance, skewness, and kurtosis,
the input file generates the
output file.
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