Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

Abstract Voronoi Diagrams

  • Rolf Klein
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_603-1

Years and Authors of Summarized Original Work

2009; Klein, Langetepe, Nilforoushan

Problem Definition

Concrete Voronoi diagrams are usually defined for a set S of sites p that exert influence over the points z of a surrounding space M. Often, influence is measured by distance functions d p( z) that are associated with the sites. For each p, its Voronoi region is given by
$$\displaystyle{\text{VR}(p,S) =\{\, z \in M;\,d_{p}(z) < d_{q}(z)\text{ for all }q \in S\setminus \{p\}\,\},}$$


Abstract Voronoi diagram Bisector Computational geometry Metric Voronoi diagram 
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Recommended Reading

  1. 1.
    Aurenhammer F, Klein R, Lee DT (2013) Voronoi diagrams and Delaunay triangulations. World Scientific, SingaporeCrossRefGoogle Scholar
  2. 2.
    Bohler C, Cheilaris P, Klein R, Liu CH, Papadopoulou E, Zavershynskyi M (2013) On the complexity of higher order abstract Voronoi diagrams. In: Proceedings of the 40th international colloquium on automata languages and programming, Riga. Lecture notes in computer science, vol 7965, pp 208–219Google Scholar
  3. 3.
    Bohler C, Klein R (2013) Abstract Voronoi diagrams with disconnected regions. In: Proceedings of the 24th international symposium on algorithms and computation, Hong Kong. Lecture notes in computer science, vol 8283, pp 306–316Google Scholar
  4. 4.
    Bohler C, Klein R, Liu CH (2014) Forest-like abstract Voronoi diagrams in linear time. In: 26th Canadian conference on computational geometry, HalifaxGoogle Scholar
  5. 5.
    Boissonnat JD, Wormser C, Yvinec M (2006) Curved Voronoi diagrams. In: Boissonnat JD, Teillaud M (eds) Effective computational geometry for curves and surfaces. Mathematics and visualization. Springer, Berlin/New York, pp 67–116CrossRefGoogle Scholar
  6. 6.
    Edelsbrunner H, Seidel R (1986) Voronoi diagrams and arrangements. Discret Comput Geom 1:387–421MathSciNetCrossRefGoogle Scholar
  7. 7.
    Klein R (1989) Concrete and abstract Voronoi diagrams. Lecture notes in computer science, vol 400. Springer, Berlin/New YorkzbMATHGoogle Scholar
  8. 8.
    Klein R, Langetepe E, Nilforoushan Z (2009) Abstract Voronoi diagrams revisited. Comput Geom Theory Appl 42(9):885–902zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Klein R, Lingas A (1994) Hamiltonian abstract Voronoi diagrams in linear time. In: Proceedings of the 5th international symposium on algorithms and computation, Beijing. Lecture notes in computer science, vol 834, pp 11–19Google Scholar
  10. 10.
    Klein R, Mehlhorn K, Meiser S (1993) Randomized incremental construction of abstract Voronoi diagrams. Comput Geom Theory Appl 3:157–184zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Lê NM (1995) Randomized incremental construction of simple abstract Voronoi diagrams in 3-space. In: Proceedings of the 10th international conference on fundamentals of computation theory, Dresden. Lecture notes in computer science, vol 965, pp 333–342Google Scholar
  12. 12.
    Malinauskas KK (2008) Dynamic construction of abstract Voronoi diagrams. J Math Sci 154(2):214–222zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Mehlhorn K, Meiser S, Rasch R (2009) Furthest site abstract Voronoi diagrams. Int J Comput Geom Appl 11:583–616MathSciNetCrossRefGoogle Scholar
  14. 14.
    Okabe A, Boots B, Sugihara K, Chiu SN (2000) Spatial tessellations: concepts and applications of Voronoi diagrams. Wiley, ChichesterCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Rolf Klein
    • 1
  1. 1.Institute for Computer Science IUniversity of BonnBonnGermany