2 01 2 Necklaces , Convolutions , and X +

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 even, and p = ∞. For p even, 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(n) time, whereas the obvious algorithms for these problems run in Θ(n) time. Faculty of Computer Science, University of New Brunswick, Fredericton, New Brunswick, Canada, [email protected]. Supported by NSERC. School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada, [email protected]. Supported by NSERC. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA, [email protected]. Supported in part by NSF grants CCF-0430849 and OISE-0334653 and by an Alfred P. Sloan Fellowship. Computer Science Department, University of Illinois, Urbana-Champaign, IL, USA, [email protected]. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain, [email protected]. Supported in part by projects MICINN MTM2009-07242, Gen. Cat. DGR 2009SGR1040, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain. Department of Computer and Information Science, Polytechnic University, Brooklyn, NY, USA, http://john.poly.edu. Supported in part by NSF grants CCF-0430849 and OISE-0334653 and by an Alfred P. Sloan Fellowship. Directeur de Recherches du FRS–FNRS, Départment d’Informatique, Université Libre de Bruxelles, Brussels, Belgium, [email protected]. Chercheur qualifié du FNRS, Départment d’Informatique, Université Libre de Bruxelles, Brussels, Belgium, [email protected]. College of Science and Engineering, American University of Armenia, Yerevan, Armenia, [email protected].