Scandinavian Thins on Top of Cake: On the Smallest One-Size-Fits-All Box

  • Esther M. Arkin
  • Alon Efrat
  • George Hart
  • Irina Kostitsyna
  • Alexander Kröller
  • Joseph S. B. Mitchell
  • Valentin Polishchuk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7288)

Abstract

We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and give a PTAS for minimizing the perimeter of the convex enclosure.

Keywords

Convex Polygon Simple Polygon Lower Envelope Small Rectangle Variable Item 
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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References

  1. 1.
  2. 2.
  3. 3.
  4. 4.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. ACM 51(4), 606–635 (2004)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ahn, H.-K., Cheong, O.: Aligning two convex figures to minimize area or perimeter. Algorithmica 62(1-2), 464–479 (2012)MATHCrossRefGoogle Scholar
  6. 6.
    Chan, T.M.: Faster core-set constructions and data stream algorithms in fixed dimensions. In: SoCG 2004 (2004)Google Scholar
  7. 7.
    Chazelle, B.M.: The polygon containment problem. Advances in Computing Research 1, 1–33 (1983)Google Scholar
  8. 8.
    Martin, R.R., Stephenson, P.C.: Containment algorithms for objects in rectangular boxes. In: Theory and Practice of Geometric Modeling, pp. 307–325 (1989)Google Scholar
  9. 9.
    Mattila, A.-L.: Piparikirja. Atena, Jyväskylä (2001) (in Finnish)Google Scholar
  10. 10.
    Pirzadeh, H.: Computational geometry with the rotating calipers. Master’s thesis, McGill U (1999)Google Scholar
  11. 11.
    Toussaint, G.T.: Solving geometric problems with the rotating calipers. In: Proceedngs of IEEE MELECON 1983, pp. 1–4 (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Esther M. Arkin
    • 1
  • Alon Efrat
    • 2
  • George Hart
    • 3
  • Irina Kostitsyna
    • 4
  • Alexander Kröller
    • 5
  • Joseph S. B. Mitchell
    • 1
  • Valentin Polishchuk
    • 6
  1. 1.AMS Dept.Stony Brook UniversityUSA
  2. 2.CS Dept.The University of ArizonaUSA
  3. 3.The Museum of MathematicsUSA
  4. 4.CS Dept.Stony Brook UniversityUSA
  5. 5.CS Dept.Technische Universität BraunschweigGermany
  6. 6.CS Dept.University of Helsinki, HIITFinland

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