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Algorithmica

, Volume 69, Issue 2, pp 294–314 | Cite as

Necklaces, Convolutions, and X+Y

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

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 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 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time.

Keywords

Necklace alignment Cyclic swap distance Convolution Sorting X+Y All pairs shortest paths 

Notes

Acknowledgements

This work was initiated at the 20th Bellairs Winter Workshop on Computational Geometry held January 28–February 4, 2005. We thank the other participants of that workshop—Greg Aloupis, Justin Colannino, Mirela Damian-Iordache, Vida Dujmović, Francisco Gomez-Martin, Danny Krizanc, Erin McLeish, Henk Meijer, Patrick Morin, Mark Overmars, Suneeta Ramaswami, David Rappaport, Diane Souvaine, Ileana Streinu, David Wood, Godfried Toussaint, Remco Veltkamp, and Sue Whitesides—for helpful discussions and contributing to a fun and creative atmosphere. We particularly thank the organizer, Godfried Toussaint, for posing the problem to us. The last author would also like to thank Luc Devroye for pointing out the easy generalization of the 2 necklace alignment problem to p for any fixed even integer p.

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • David Bremner
    • 1
  • Timothy M. Chan
    • 2
  • Erik D. Demaine
    • 3
  • Jeff Erickson
    • 4
  • Ferran Hurtado
    • 5
  • John Iacono
    • 6
  • Stefan Langerman
    • 7
  • Mihai Pǎtraşcu
    • 8
  • Perouz Taslakian
    • 9
  1. 1.Faculty of Computer ScienceUniversity of New BrunswickFrederictonCanada
  2. 2.School of Computer ScienceUniversity of WaterlooWaterlooCanada
  3. 3.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.Computer Science DepartmentUniversity of IllinoisUrbana-ChampaignUSA
  5. 5.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain
  6. 6.Department of Computer and Information SciencePolytechnic UniversityBrooklynUSA
  7. 7.Directeur de Recherches du FRS-FNRS, Départment d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  8. 8.AT&T Labs—ResearchFlorham ParkUSA
  9. 9.College of Science and EngineeringAmerican University of ArmeniaYerevanArmenia

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