Necklaces, Convolutions, and X+Y
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Demaine_Necklaces, convolutions.pdf
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Author(s)
Bremner, David
Chan, Timothy M.
Demaine, Erik D.
Erickson, Jeff
Hurtado, Ferran
Iacono, John
Langerman, Stefan
Patrascu, Mihai
Taslakian, Perouz
Date Issued
December 2012
Journal
Algorithmica
Publisher
Springer-Verlag
Citation
Bremner, David, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Patrascu, and Perouz Taslakian. “Necklaces, Convolutions, and X+Y.” Algorithmica 69, no. 2 (December 28, 2012): 294–314.
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Author's final manuscript
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 ℓ [subscript p] 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 \((\operatorname {median},+)\) 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 [superscript 2]) time, whereas the obvious algorithms for these problems run in Θ(n [superscript 2]) time.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1007/s00453-012-9734-3
