Necklaces, convolutions, and X + Y

Authors:
David Bremner;Timothy M. Chan;Erik D. Demaine;Jeff Erickson;Ferran Hurtado;John Iacono;Stefan Langerman;Perouz Taslakian
Affiliations:
Faculty of Computer Science, University of New Brunswick, Fredericton, New Brunswick, Canada;School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada;Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA;Computer Science Department, University of Illinois, Urbana-Champaign, IL;Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain;Department of Computer and Information Science, Polytechnic University, Brooklyn, NY;Chercheur qualifié du FNRS, Départment d'Informatique, Université Libre de Bruxelles, Brussels, Belgium;School of Computer Science, McGill University, Montréal, Québec, Canada
Venue:
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Year:
2006

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Abstract

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the lp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p=2, and p=∞. For p=2, we reduce the problem to standard convolution, while for p=∞ and p=1, we reduce the problem to (min,+) convolution and (median,+) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n2) time, whereas the obvious algorithms for these problems run in Θ (n2) time.