Partial quicksort and weighted branching processes

In this dissertation we look at different two models of sorting algorithms based on divideand-conquer algorithms. Quicksort algorithm, sort an unsorted array of n distinct elements. Partial Quicksort sorts the l smallest elements in a list of length n. Both stochastic divide-and-conquer algorithms are widely studied. Our algorithm Quicksort on the fly provides online the first smallest, then second smallest and so on. If we stop at the l-th smallest, we obtain Partial Quicksort. We analyze the running time performance Yn of Quicksort on the fly using the parameter l as time index. We show that, the process Yn converges not only in distribution, but also uniformly as n → ∞ almost everywhere in D to a random variable Y . The distribution of Y is characterized as a solution of a stochastic fixed point equation