Interchange rearrangement: The element-cost model

Given an input string S and a target string T when S is a permutation of T, the interchange rearrangement problem is to apply on S a sequence of interchanges, such that S is transformed into T. The interchange operation exchanges the position of the two elements on which it is applied. The goal is to transform S into T at the minimum cost possible, referred to as the distance between S and T. The distance can be defined by several cost models that determine the cost of every operation. There are two known models: The Unit-cost model and the Length-cost model. In this paper, we suggest a natural cost model: The Element-cost model. In this model, the cost of an operation is determined by the elements that participate in it. Though this model has been studied in other fields, it has never been considered in the context of rearrangement problems. We consider both the special case where all elements in S and T are distinct, referred to as a permutation string, and the general case, referred to as a general string. An efficient optimal algorithm for the permutation string case and efficient approximation algorithms for the general string case, which is NP-hard, are presented. The study is broadened to include the transposition rearrangement problem under the Element-cost model and under the other known models, in order to provide additional perspective on the new model.