Randomized Quicksort and the Entropy of the Random Number Generator

The worst-case complexity of an implementation of Quicksort depends on the random number generator that is used to select the pivot elements. In this paper we estimate the expected number of comparisons of Quicksort as a function in the entropy of the random source. We give upper and lower bounds and show that the expected number of comparisons increases from n log n to n, if the entropy of the random source is bounded. As examples we show explicit bounds for distributions with bounded min-entropy, the geometrical distribution and the δ-random source.