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4 results on '"Necklace alignment"'

1. Necklaces, Convolutions, and X+ Y.

Author

Bremner, David, Chan, Timothy, Demaine, Erik, Erickson, Jeff, Hurtado, Ferran, Iacono, John, Langerman, Stefan, Pǎtraşcu, Mihai, and Taslakian, Perouz

Subjects
*MATHEMATICAL convolutions, *QUADRATIC equations, *ALGORITHMS, *ARBITRARY constants, *MEDIAN (Mathematics)
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 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) time, whereas the obvious algorithms for these problems run in Θ( n) time. [ABSTRACT FROM AUTHOR]

Published
2014
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2. Necklaces, convolutions, and X plus Y

Author

Bremner, David, Chan, Timothy M., Demaine, Erik D., Erickson, Jeff, Hurtado Díaz, Fernando Alfredo, Iacono, John, Langerman, Stefan, Patrascu, Mihai, Taslakian, Perouz, and Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta

Subjects
Necklace alignment, DONT CARES, PAIRS SHORTEST PATHS, ALGORITHMS, SIMILARITY, Cyclic swap distance, Sorting X plus Y, Algorismes, TRANSFORM, Convolution, All pairs shortest paths, Informàtica::Informàtica teòrica::Algorísmica i teoria de la complexitat [Àrees temàtiques de la UPC]
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 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=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(n 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time

Published
2014

3. Necklaces, convolutions, and X+Y

Author

Bremner, David, Chan, Timothy M., Demaine, Erik D., Erickson, Jeff J., Hurtado, Ferran, Iacono, John, Langerman, Stefan, Pǎtraşcu, Mihai, Taslakian, Perouz, Bremner, David, Chan, Timothy M., Demaine, Erik D., Erickson, Jeff J., Hurtado, Ferran, Iacono, John, Langerman, Stefan, Pǎtraşcu, Mihai, and Taslakian, Perouz

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 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 (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 Θ(n 2) time. © 2012 Springer Science+Business Media New York., SCOPUS: ar.j, info:eu-repo/semantics/published

Published
2014

4. Necklaces, convolutions, and X plus Y

Author

Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta, Bremner, David, Chan, Timothy M., Demaine, Erik D., Erickson, Jeff, Hurtado Díaz, Fernando Alfredo, Iacono, John, Langerman, Stefan, Patrascu, Mihai, Taslakian, Perouz, Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta, Bremner, David, Chan, Timothy M., Demaine, Erik D., Erickson, Jeff, Hurtado Díaz, Fernando Alfredo, Iacono, John, Langerman, Stefan, Patrascu, Mihai, and Taslakian, Perouz

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 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=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(n 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time, Peer Reviewed, Postprint (published version)

Published
2014

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