par Bremner, David;Chan, Timothy M.;Demaine, Erik D. 
;Erickson, Jeff J.;Hurtado, Ferran ;Iacono, John ;Langerman, Stefan ;Pǎtraşcu, Mihai;Taslakian, Perouz
Référence arXiv.org
Publication Publié, 2012
;Erickson, Jeff J.;Hurtado, Ferran ;Iacono, John ;Langerman, Stefan ;Pǎtraşcu, Mihai;Taslakian, Perouz Référence arXiv.org
Publication Publié, 2012
Article sans comité de lecture
| Résumé : | 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 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 = \infty. For p even, we reduce the problem to standard convolution, while for p = \infty 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^2) time, whereas the obvious algorithms for these problems run in \Theta(n^2) time. |
