Necklaces, Convolutions, and X+Y

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.

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Notes

  1. 1.

    “Tropical convolution” would also make sense, by direct analogy with tropical geometry, but we have never seen this terminology used in print.

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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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Correspondence to Perouz Taslakian.

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In memory of our colleague Mihai Pǎtraşcu.

D. Bremner and T.M. Chan are supported by NSERC. E.D. Demaine and J. Iacono are supported in part by NSF grants CCF-0430849 and OISE-0334653 and by an Alfred P. Sloan Fellowship. F. Hurtado is supported in part by projects MICINN MTM2009-07242, Gen. Cat. DGR 2009SGR1040, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain.

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Bremner, D., Chan, T.M., Demaine, E.D. et al. Necklaces, Convolutions, and X+Y . Algorithmica 69, 294–314 (2014). https://doi.org/10.1007/s00453-012-9734-3

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Keywords

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