The complexity probabilistic quasi-metric space

We introduce and study a probabilistic quasi-metric on the set of complexity functions, which provides an efficient framework to measure the distance from a complexity function f to another one g in the case that f is asymptotically more efficient than g. In this context we also obtain a version of the Banach fixed point theorem which allows us to show that the functionals associated both to Divide and Conquer algorithms and Quicksort algorithms have a unique fixed point.