Improved upper bound for sorting by short swaps

We consider the problem of sorting an arbitrary permutation of length n using substring reversals of length 2 or 3. This has been called "short swaps ". We give an upper bound of (5/24) n² + O(nlogn), improving the previous ( 1/4 ) n² upper bound. We also show that there is a short swap sorting network with ( 1/4 ) n² +O(nlogn) comparators and depth n.