Directed trees: A powerful representation for sorting and ordering problems

We present a simple framework for dealing with search spaces consisting of permutations. To demonstrate its usefulness, we build upon it a simple (1+1)-evolutionary algorithm for one of the most fundamental problems in computer science, namely the problem of sorting n pairwise comparable items. We give a rigorous proof that the optimization time is at most O(n²) with high probability. Our experimental evaluation shows that it is much better, namely around O(n log n). This compares favorably with the currently best (1 + 1)-EAs for sorting, for which an optimization time of O(n² log n) was proven (Scharnow, Tinnefeld and Wegener (2004)) and one of similar order is observed experimentally in this work. Our approach has the particular advantage that it does distinguish between wrong and unexplored information. This allows to retrieve partial, correct information even before the optimal solution has been found.