Towards constructing optimal strip move sequences

Abstract

The Sorting by Strip Moves problem, SBSM, was introduced in [6] as a variant of the well-known Sorting by Transpositions problem. A restriction called Block Sorting was shown in [2] to be NP-hard. In this article, we improve upon the ideas used in [6] to obtain a combinatorial characterization of the optimal solutions of SBSM. Using this, we show that a strip move which results in a permutation of two or three fewer strips or which exchanges a pair of adjacent strips to merge them into a single strip necessarily reduces the strip move distance. We also establish that the strip move diameter for permutations of size n is n - 1. Further, we exhibit an optimum-preserving equivalence between SBSM and the Common Substring Removals problem (CSR) - a natural combinatorial puzzle. As a consequence, we show that sorting a permutation via strip moves is as hard (or as easy) as sorting its inverse. © Springer-Verlag Berlin Heidelberg 2004.