Necklaces, Convolutions, and X+Y
Name
1212.4771.pdf
Description
Accepted version
Size
377.91 KB
Format
Adobe PDF
Checksum (MD5)
82a5b3b01fa4bd17f2962f68c2ef47bd
Author(s)
Bremner, David
Chan, Timothy M.
Demaine, Erik D
Erickson, Jeff
Hurtado, Ferran
Iacono, John
Langerman, Stefan
Taslakian, Perouz
Date Issued
December 2012
Journal
Algorithmica
Publisher
Springer Nature America, Inc
Citation
Bremner, David, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman and Perouz Taslakian. "Necklaces, Convolutions, and X + Y." Algorithmica, Vol. 69 (2) June 2014: 294-314.
Version
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 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(n²) time, whereas the obvious algorithms for these problems run in Θ(n 2 ) time.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/S00453-012-9734-3
