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Necklaces, Convolutions, and X + Y

  • David Bremner
  • Timothy M. Chan
  • Erik D. Demaine
  • Jeff Erickson
  • Ferran Hurtado
  • John Iacono
  • Stefan Langerman
  • Perouz Taslakian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)

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.

Keywords

Time Algorithm Decision Tree Model Median Element Asymptotic Number Circular Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

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
  • Perouz Taslakian
    • 8
  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.Chercheur qualifié du FNRS, Départment d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  8. 8.School of Computer ScienceMcGill UniversityMontréalCanada

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