Algorithms for Computing Geometric Measures of Melodic Similarity

Consider two orthogonal closed chains on a cylinder. These chains are monotone with respect to the tangential Θ direction. We wish to rigidly move one chain so that the total area between the two is minimized. This minimization is a geometric measure of similarity between two melodies proposed by Ó Maid́ın. The Θ direction represents time and the axial direction, z, represents pitch. Let the two chains have n and m vertices respectively, where n ≥ m, We present an O(n + m) time algorithm if Θ is fixed, and an O(nm log(n + m)) time algorithm for general rigid motions. These bounds also apply for planar orthogonal monotone open chains, where area is measured only within the common domain of the two chains in the direction This paper extends the results presented by the authors at CCCG’03 [1] School of Computer Science, McGill University. {athens,godfried}@uni.cs.mcgill.ca Department of Computer Science and Software Engineering, Concordia University. [email protected] Chercheur qualifié du FNRS, Département d’Informatique, Université de Bruxelles. [email protected] Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo. [email protected] Facultad de Matematica y Computacion, Universidad de la Habana. [email protected] , [email protected] School of Computing, Queen’s University. [email protected]