Advertisement

Discrete & Computational Geometry

, Volume 1, Issue 2, pp 155–182 | Cite as

A polynomial solution for the potato-peeling problem

  • J. S. Chang
  • C. K. Yap
Article

Abstract

The potato-peeling problem asks for the largest convex polygon contained inside a given simple polygon. We give anO(n7) time algorithm to this problem, answering a question of Goodman. We also give anO(n6) time algorithm if the desired polygon is maximized with respect to perimeter.

Keywords

Convex Polygon Polynomial Solution Admissible Pair Simple Polygon Supporting Line 
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.

References

  1. 1.
    A. Aggarwal, J. S. Chang, and C. K. Yap, Minimum area circumscribing polygons, to appear in a special issue of Visual Computer: International J. of Computer Graphics. Also, NYU-Courant Institute Robotics Lab. Report No. 42, May 1985.Google Scholar
  2. 2.
    M. Ben-Or, D. Kozen, and J. Reif, The complexity of elementary algebra and geometry, 16th STOC, 457–464, 1984.Google Scholar
  3. 3.
    J. E. Boyce, D. P. Dobkin, III, R. L. Drysdale, and L J. Guibas, Finding extremal polygons, 14th STOC, 282–289, 1982.Google Scholar
  4. 4.
    G. D. Chakerian, and L. H. Lange, Geometric extremum problems, Math. Mag. 44 (1971) 57–69.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    J. S. Chang, and C. K. Yap, A polynomial solution to potato-peeling and other polygon inclusion and enclosure problems, 25th FOCS, 408–416, May 1984.Google Scholar
  6. 6.
    B. Chazelle, R. L. Drysdale, and D. T. Lee, Computing the Largest Empty Rectangle, Proc. of the Symposium on Theoretic Aspects of Comp. Sci., Paris, April, 1984.Google Scholar
  7. 7.
    A. DePano, and A. Aggarwal, Finding restrictedK-envelopes for Convex Polygons, Proc. of the 22nd Allerton Conference on Comm. Control and Computing, 1984.Google Scholar
  8. 8.
    A. Depano, Approximations of Polygons and Polyhedra: Potentials for Research, Manuscript, 1984.Google Scholar
  9. 9.
    D. P. Dobkin and L. Snyder, On a general method for maximizing among certain geometric problems, 20th FOCS, 9–17, 1979.Google Scholar
  10. 10.
    D. Dori and M. Ben-Bassat, Circumscribing a convex polygon by a polygon of fewer sides with minimal area addition, Computer Vision, Graphics and Image Processing 24 (1983) 131–159.CrossRefMATHGoogle Scholar
  11. 11.
    J. E. Goodman, On the largest convex polygon contained in a non-convexn-gon, or How to peel a potato, Geometriae Dedicata 11 (1981) 99–106.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    V. Klee and M. C. Laskowski, Finding the smallest triangles containing a given convex polygon, J. Algorithms, 457–464, to appear.Google Scholar
  13. 13.
    D. Kozen and C. K. Yap, Algebraic cell decomposition in NC, 26th FOCS, 1985.Google Scholar
  14. 14.
    A. M. Lopshits, Computation of Areas of Oriented Figures, DC Heath, 1963.Google Scholar
  15. 15.
    M. McKenna, J. O'Rourke, and S. Suri, Finding the Largest Rectangle in an Orthogonal Polygon, Tech. Report JHU/EECS-85/09, Dept. of Elec. Eng. and Comp. Sci., The Johns Hopkins University, 1985.Google Scholar
  16. 16.
    J. O'Rourke, A. Aggarwal, S. Maddila, and M. Baldwin, An Optimal Algorithm for Finding Minimal Enclosing Triangles, Technical Report JHU/EECS-84/08, Dept. of Elec. Eng. and Comp. Sci., The Johns Hopkins University, May 1984.Google Scholar
  17. 17.
    J. O'Rourke, Counterexamples to a Minimal Circumscription Algorithm, Manuscript, June 1984.Google Scholar
  18. 18.
    J. O'Rourke, Finding Minimal Enclosing Boxes, Technical Report, Dept. of Elec. Eng. and Comp. Sci., The Johns Hopkins University, 1984.Google Scholar
  19. 19.
    M. Sharir and A. Schorr, On shortest paths in polyhedral spaces, 16th STOC, 144–153, 1984.Google Scholar
  20. 20.
    G. T. Toussaint, Pattern Recognition and geometrical complexity, 5th International Conf. Pattern Recognition, 1324–1347 1979.Google Scholar
  21. 21.
    T. Woo, The Convex Skull Problem, Manuscript, 1983.Google Scholar
  22. 22.
    D. Wood and C. K. Yap, Computing a Convex Skull of an Orthogonal Polygon, Proc. of the Symposium on Computational Geometry, Baltimore, Maryland, 311–316, June, 1985.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • J. S. Chang
    • 1
  • C. K. Yap
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York

Personalised recommendations