A Characterization of the Set of Fixed Points of the Quicksort Transformation
James Allen Fill, The Johns Hopkins University
Svante Janson, Uppsala University
Abstract
The limiting distribution m of the
normalized number of key
comparisons required by the Quicksort sorting algorithm is known to be the
unique fixed point
of a certain distributional transformation T -- unique, that is,
subject to the
constraints of zero mean and finite variance. We show that a distribution
is a fixed point
of T if and only if it is the convolution of m with a
Cauchy distribution of arbitrary center and scale. In particular, therefore,
m is the unique fixed point of T
having zero mean.
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