Flip Distance between Triangulations of a Simple Polygon is NP-Complete

  • Oswin Aichholzer
  • Wolfgang Mulzer
  • Alexander Pilz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)

Abstract

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study.

We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abel, Z., Ballinger, B., Bose, P., Collette, S., Dujmović, V., Hurtado, F., Kominers, S., Langerman, S., Pór, A., Wood, D.: Every large point set contains many collinear points or an empty pentagon. Graphs Combin. 27, 47–60 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aichholzer, O., Mulzer, W., Pilz, A.: Flip Distance Between Triangulations of a Simple Polygon is NP-Complete. ArXiv e-prints (2012) arXiv:1209.0579 [cs.CG]Google Scholar
  3. 3.
    Bose, P., Hurtado, F.: Flips in planar graphs. Comput. Geom. 42(1), 60–80 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Culik II, K., Wood, D.: A note on some tree similarity measures. Inf. Process. Lett. 15(1), 39–42 (1982)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Eppstein, D.: Happy endings for flip graphs. JoCG 1(1), 3–28 (2010)MathSciNetGoogle Scholar
  6. 6.
    Hanke, S., Ottmann, T., Schuierer, S.: The edge-flipping distance of triangulations. J. UCS 2(8), 570–579 (1996)MathSciNetGoogle Scholar
  7. 7.
    Hurtado, F., Noy, M., Urrutia, J.: Flipping edges in triangulations. Discrete Comput. Geom. 22, 333–346 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Hwang, F., Richards, D., Winter, P.: The Steiner Tree Problem. Annals of Discrete Mathematics (1992)Google Scholar
  9. 9.
    Lawson, C.L.: Transforming triangulations. Discrete Math. 3(4), 365–372 (1972)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Lawson, C.L.: Software for C 1 surface interpolation. In: Rice, J.R. (ed.) Mathematical Software III, pp. 161–194. Academic Press, NY (1977)Google Scholar
  11. 11.
    Lubiw, A., Pathak, V.: Flip distance between two triangulations of a point-set is NP-complete. In: Proc. 24th CCCG, pp. 127–132 (2012)Google Scholar
  12. 12.
    Pilz, A.: Flip distance between triangulations of a planar point set is APX-hard. ArXiv e-prints (2012) arXiv:1206.3179 [cs.CG]Google Scholar
  13. 13.
    Rao, S.K., Sadayappan, P., Hwang, F.K., Shor, P.W.: The rectilinear Steiner arborescence problem. Algorithmica 7, 277–288 (1992)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Shi, W., Su, C.: The rectilinear Steiner arborescence problem is NP-complete. In: Proc. 11th SODA, pp. 780–787 (2000)Google Scholar
  15. 15.
    Sleator, D., Tarjan, R., Thurston, W.: Rotation distance, triangulations and hyperbolic geometry. J. Amer. Math. Soc. 1, 647–682 (1988)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Trubin, V.: Subclass of the Steiner problems on a plane with rectilinear metric. Cybernetics 21, 320–324 (1985)MATHCrossRefGoogle Scholar
  17. 17.
    Urrutia, J.: Algunos problemas abiertos. In: Proc. IX Encuentros de Geometría Computacional, pp. 13–24 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Wolfgang Mulzer
    • 2
  • Alexander Pilz
    • 1
  1. 1.Institute for Software TechnologyGraz University of TechnologyAustria
  2. 2.Institute of Computer ScienceFreie Universität BerlinGermany

Personalised recommendations