Algorithmica manuscript No. (will be inserted by the editor) Computing Similarity Distances Between Rankings

Abstract We address the problem of computing distances between rankings that take into account similar-ities between candidates. The need for evaluating such distances is governed by applications as diverse as rank aggregation, bioinformatics, social sciences and data storage. The problem may be summarized as fol-lows. Given two rankings and a positive cost function on transpositions that depends on the similarity of the candidates 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 metric-tree structures and on full rankings mod-eled as permutations. The presented results include a quadratic-time algorithm for finding a minimum cost transform for simple cycles; and a linear time, 5/3-approximation algorithm for permutations that contain multiple cycles. In addition, for permutations with digraphs represented by non-intersecting cycles embed-ded in trees, we present a polynomial-time transform algorithm. The proposed methods rely on investigating a newly introduced balancing property of cycles embedded in trees, cycle-merging methods, and shortest-path optimization techniques