Can a permutation be sorted by best short swaps

A short swap switches two elements with at most one element caught between them. Sorting permutation by short swaps asks to find a shortest short swap sequence to transform a permutation into another. A short swap can eliminate at most three inversions. It is still open for whether a permutation can be sorted by short swaps each of which can eliminate three inversions. In this paper, we present a polynomial time algorithm to solve the problem, which can decide whether a permutation can be sorted by short swaps each of which can eliminate 3 inversions in O(n) time, and if so, sort the permutation by such short swaps in O(n^2) time, where n is the number of elements in the permutation. A short swap can cause the total length of two element vectors to decrease by at most 4. We further propose an algorithm to recognize a permutation which can be sorted by short swaps each of which can cause the element vector length sum to decrease by 4 in O(n) time, and if so, sort the permutation by such short swaps in O(n^2) time. This improves upon the O(n^2) algorithm proposed by Heath and Vergara to decide whether a permutation is so called lucky.