Similarity distances between permutations

We address the problem of computing distances between rankings that take into account similarities between elements. The need for evaluating such distances arises in applications such as machine learning, social sciences and data storage. The problem may be summarized as follows: Given two rankings and a positive cost function on transpositions that depends on the similarity of the elements involved, find a smallest cost sequence of transpositions that converts one ranking into another. Our focus is on costs that may be described via special tree structures and on rankings modeled as permutations. The presented results include a quadratic-time algorithm for finding a minimum cost transform for a single cycle; and a linear time, 5/3-approximation algorithm for permutations that contain multiple cycles.