A Characterization of the Set of Fixed Points of the Quicksort Transformation

Abstract

The limiting distribution $\mu$ 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 $\mu$ with a Cauchy distribution of arbitrary center and scale. In particular, therefore, $\mu$ is the unique fixed point of $T$ having zero mean.