Advertisement

Mollified Zone Diagrams and Their Computation

  • Sergio C. de Biasi
  • Bahman Kalantari
  • Iraj Kalantari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6970)

Abstract

The notion of the zone diagram of a finite set of points in the Euclidean plane is an interesting and rich variation of the classical Voronoi diagram, introduced by Asano, Matoušek, and Tokuyama [1]. In this paper, we define mollified versions of zone diagram named territory diagram and maximal territory diagram. A zone diagram is a particular maximal territory diagram satisfying a sharp dominance property. The proof of existence of maximal territory diagrams depends on less restrictive initial conditions and is established via Zorn’s lemma in contrast to the use of fixed-point theory in proving the existence of the zone diagram. Our proof of existence relies on a characterization which allows embedding any territory diagram into a maximal one. Our analysis of the structure of maximal territory diagrams is based on the introduction of a pair of dual concepts we call safe zone and forbidden zone. These in turn give rise to computational algorithms for the approximation of maximal territory diagrams. Maximal territory diagrams offer their own interesting theoretical and computational challenges, as well as insights into the structure of zone diagrams. This paper extends and updates previous work presented in [4].

Keywords

computational geometry Voronoi diagram zone diagram mollified zone diagram territory diagram maximal territory diagram 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Asano, T., Matoušek, J., Tokuyama, T.: Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge. SIAM Journal on Computing 37, 1182–1198 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asano, T., Brimkov, V.E., Barneva, R.P.: Some theoretical challenges in digital geometry, A perspective. Discrete Applied Mathematics 157(16), 3362–3371 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aurenhammer, F.: Voronoi diagrams - a survey of a fundamental geometric data structure. ACM Computing Surveys 23, 345–405 (1991)CrossRefGoogle Scholar
  4. 4.
    de Biasi, S.C., Kalantari, B., Kalantari, I.: Maximal Zone Diagrams and their Computation. In: ISVD 2010, pp. 171–180. IEEE Computer Society (2010)Google Scholar
  5. 5.
    Kalantari, B.: Voronoi Diagrams and Polynomial Root-Finding. In: ISVD 2009, pp. 31–40. IEEE Computer Society (2009)Google Scholar
  6. 6.
    Kalantari, B.: Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram. Discrete & Computational Geometry (2011), doi:10.1007/s00454-011-9330-3Google Scholar
  7. 7.
    Reem, D.: An Algorithm for Computing Voronoi Diagrams of General Generators in General Normed Spaces. In: ISVD 2009, pp. 144–152. IEEE Computer Society (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Sergio C. de Biasi
    • 1
  • Bahman Kalantari
    • 1
  • Iraj Kalantari
    • 2
  1. 1.Department of Computer ScienceRutgers UniversityNew BrunswickUSA
  2. 2.Department of MathematicsWestern Illinois UniversityMacombUSA

Personalised recommendations