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Algorithmica

, Volume 21, Issue 1, pp 119–136 | Cite as

On Minimum-Area Hulls

  • E. M. Arkin
  • Y. -J. Chiang
  • M. Held
  • J. S. B. Mitchell
  • V. Sacristan
  • S. S. Skiena
  • T. -C. Yang

Abstract.

We study some minimum-area hull problems that generalize the notion of convex hull to star-shaped and monotone hulls. Specifically, we consider the minimum-area star-shaped hull problem: Given an n -vertex simple polygon P , find a minimum-area, star-shaped polygon P * containing P . This problem arises in lattice packings of translates of multiple, nonidentical shapes in material layout problems (e.g., in clothing manufacture), and has been recently posed by Daniels and Milenkovic. We consider two versions of the problem: the restricted version, in which the vertices of P * are constrained to be vertices of P , and the unrestricted version, in which the vertices of P * can be anywhere in the plane. We prove that the restricted problem falls in the class of ``3sum-hard'' (sometimes called ``n 2 -hard'') problems, which are suspected to admit no solutions in o(n 2 ) time. Further, we give an O(n 2 ) time algorithm, improving the previous bound of O(n 5 ) . We also show that the unrestricted problem can be solved in O(n 2 p(n)) time, where p(n) is the time needed to find the roots of two equations in two unknowns, each a polynomial of degree O(n) .

We also consider the case in which P * is required to be monotone, with respect to an unspecified direction; we refer to this as the minimum-area monotone hull problem. We give a matching lower and upper bound of Θ(n log n) time for computing P * in the restricted version, and an upper bound of O(n q(n)) time in the unrestricted version, where q(n) is the time needed to find the roots of two polynomial equations in two unknowns with degrees 2 and O(n) .

Key words. Minimum-area hulls, Star-shaped polygons, Monotone polygons, 3sum-Hardness, Lower bounds, Computational geometry, Design and analysis of algorithms. 

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Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • E. M. Arkin
    • 1
  • Y. -J. Chiang
    • 2
  • M. Held
    • 3
  • J. S. B. Mitchell
    • 4
  • V. Sacristan
    • 5
  • S. S. Skiena
    • 6
  • T. -C. Yang
    • 7
  1. 1.Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794-3600, USA. estie@ams.sunysb.edu.US
  2. 2.Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794-3600, USA. yjc@ams.sunysb.edu.US
  3. 3.Universität Salzburg, Institut für Computerwissenschaften, Jakob-Haringer Strasse 2, A-5020 Salzburg, Austria. held@cosy.sbg.ac.at.AT
  4. 4.Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794-3600, USA. jsbm@ams.sunysb.edu.US
  5. 5.Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028 Barcelona, Spain. vera@ma2.upc.es.ES
  6. 6.Department of Computer Science, State University of New York, Stony Brook, NY 11794-4400, USA. skiena@cs.sunysb.edu.US
  7. 7.Department of Computer Science and Statistics, Kyungsung University, 110, DaeYeon-dong, Nam-gu, Pusan, 608-736, Korea. tcyang@csd.kyungsung.ac.kr.KR

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