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Algorithmic Folding Complexity

  • Jean Cardinal
  • Erik D. Demaine
  • Martin L. Demaine
  • Shinji Imahori
  • Tsuyoshi Ito
  • Masashi Kiyomi
  • Stefan Langerman
  • Ryuhei Uehara
  • Takeaki Uno
Proceedings Paper

Abstract

How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a length-n pleat (alternating mountain/valley), is O(lg2 n). We also show that a lower bound of the complexity of the problems is Ω(lg2 n/lg lg n). Next we show that almost all mountain-valley patterns require Ω(n/lg n) folds, which means that the uniform and pleat foldings are relatively easy problems. We also give a general algorithm for folding an arbitrary sequence of length n in O(n/lg n) folds, meeting the lower bound up to a constant factor.

Keywords

Algorithm design Origami 

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Copyright information

© Springer 2011

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Erik D. Demaine
    • 2
  • Martin L. Demaine
    • 2
  • Shinji Imahori
    • 3
  • Tsuyoshi Ito
    • 4
  • Masashi Kiyomi
    • 5
  • Stefan Langerman
    • 1
  • Ryuhei Uehara
    • 5
  • Takeaki Uno
    • 6
  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  2. 2.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of Computational Science and Engineering, Graduate School of EngineeringNagoya UniversityNagoyaJapan
  4. 4.Institute for Quantum Computing and School of Computer ScienceUniversity of WaterlooWaterlooCanada
  5. 5.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan
  6. 6.National Institute of InformaticsChiyoda-ku, TokyoJapan

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