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Necklaces, Convolutions, and X + Y

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Algorithms – ESA 2006 (ESA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4168))

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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.

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Author information

Authors and Affiliations

  1. Faculty of Computer Science, University of New Brunswick, Fredericton, New Brunswick, Canada

    David Bremner

  2. School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada

    Timothy M. Chan

  3. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA

    Erik D. Demaine

  4. Computer Science Department, University of Illinois, Urbana-Champaign, IL, USA

    Jeff Erickson

  5. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

    Ferran Hurtado

  6. Department of Computer and Information Science, Polytechnic University, Brooklyn, NY, USA

    John Iacono

  7. Chercheur qualifié du FNRS, Départment d’Informatique, Université Libre de Bruxelles, Brussels, Belgium

    Stefan Langerman

  8. School of Computer Science, McGill University, Montréal, Québec, Canada

    Perouz Taslakian

Authors
  1. David Bremner
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  2. Timothy M. Chan
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  3. Erik D. Demaine
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  4. Jeff Erickson
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  5. Ferran Hurtado
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  6. John Iacono
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  7. Stefan Langerman
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  8. Perouz Taslakian
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Editor information

Editors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA and School of Computer Science, Tel-Aviv University, 69978, Tel-Aviv, Israel

    Yossi Azar

  2. Department of Computer Science, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Thomas Erlebach

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© 2006 Springer-Verlag Berlin Heidelberg

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Bremner, D. et al. (2006). Necklaces, Convolutions, and X + Y . In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_17

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  • DOI: https://doi.org/10.1007/11841036_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38875-3

  • Online ISBN: 978-3-540-38876-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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