Skip to main content
SpringerLink
Log in

A Fast Robust Algorithm for Computing Discrete Voronoi Diagrams

  • Published:
Journal of Mathematical Modelling and Algorithms

Abstract

We describe an algorithm for the construction of discretized Voronoi diagrams on a CPU within the context of a large scale numerical fluid simulation. The Discrete Voronoi Chain (DVC) algorithm is fast, flexible and robust. The algorithm stores the Voronoi diagram on a grid or lattice that may be structured or unstructured. The Voronoi diagram is computed by alternatively updating two lists of grid cells per particle to propagate a growth boundary of cells from the particle locations. Distance tests only occur when growth boundaries from different particles collide with each other, hence the number of distance tests is effectively minimized. We give some empirical results for two and three dimensions. The method generalizes to any dimension in a straight forward manner. The distance tests can be based any metric.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Moresi, L., Dufour, F., Mühlhaus, H.B.: A lagrangian integration point finite element method for large deformation modeling of viscoelastic geomaterials. J. Comput. Phys. 184, 476–497 (2003)

    Article  MATH  Google Scholar 

  2. Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  3. Tzionas, P.G., Thanailakis, A., Tsalides, P.G.: Collision-free path planning for a diamond-shaped robot using two-dimensional cellular automata. IEEE Trans. Robot. Autom. 13(2), 237–250 (1997)

    Article  Google Scholar 

  4. Tsai, Y.: Rapid and accurate computation of the distance function using grids. J. Comput. Phys. 178(1), 175–195 (2002). ISSN 0021-9991. doi:10.1006/jcph.2002.7028

    Article  MATH  MathSciNet  Google Scholar 

  5. Bespamyatnikh, S., Segal, M.: Fast algorithms for approximating distances. Algorithmica 33, 263–269 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mauch, S.: A fast algorithm for computing the closest point and distance transform. Technical report, Caltech (2000)

  7. Rong, G., Tan, T.S.: Jump flooding in GPU with applications to Voronoi diagram and distance transform. In: I3D ’06: Proceedings of the 2006 Symposium on Interactive 3D Graphics and Games, pp. 109–116. ACM, New York (2006). ISBN 1-59593-295-X. doi:10.1145/1111411.1111431

  8. Rong, G.: Jump flooding algorithm on graphics hardware and it’s applications. Ph.D. thesis, National University of Singapore (2007)

  9. Sud, A., Govindaraju, N., Manocha, D.: Interactive computation of discrete generalized Voronoi diagrams using range culling. In: Proc. 2nd International Symposium on Voronoi Diagrams in Science and Engineering. Hanyang University, October 2005

  10. Mirtich, B.: V-clip: fast and robust polyhedral collision detection. ACM Trans. Graph. 17(3), 177–208, July (1998)

    Article  Google Scholar 

  11. Sud, A., Govindaraju, N., Gayle, R., Kabul, I., Manocha, D.: Fast proximity computation among deformable models using discrete Voronoi diagrams. ACM Trans. Graph. 25(3), 1144–1153 (2006). ISSN 0730-0301. doi:10.1145/1141911.1142006

    Article  Google Scholar 

  12. Hu, S.Y., Liao, G.M.: Scalable peer-to-peer networked virtual environment. In: NetGames ’04: Proceedings of 3rd ACM SIGCOMM Workshop on Network and System Support for Games, pp. 129–133. ACM, New York (2004). ISBN 1-58113-942-X. doi:10.1145/1016540.1016552

  13. Adamatzky, A.I.: Voronoi-like partition of lattice in cellular automata. Math. Comput. Model. 23(4), 51–66 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Adamatzky, A.I., Holland, O.: Voronoi-like nondeterministic partition of lattice by collectives of finite automata. Math. Comput. Model. 28(10), 73–93 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Aurenhammer, F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991). ISSN 0360-0300. doi:10.1145/116873.116880

    Article  Google Scholar 

  16. Okabe, A., Boots, B., Sugihara, K.: Spatial Tessellations: Concepts and Applications of Voronoi diagrams. Wiley, New York (1992). ISBN 0-471-93430-5

    MATH  Google Scholar 

  17. Sambridge, M.S., Braun, J., McQueen H.: Geophysical parametrization and interpolation of irregular data using natural neighbours. Geophys. J. Int. 122, 837–857 (1995)

    Article  Google Scholar 

  18. May, D.A.: Ph.D. thesis, Monash University (2008)

  19. Vleugels, J., Overmars, M.: Approximating Voronoi diagrams of convex sites in any dimension. Int. J. Comput. Geom. Appl. 8(2), 201–221 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kenneth, E., Hoff, I., Keyser, J., Lin, M., Manocha, D., Culver, T.: Fast computation of generalized Voronoi diagrams using graphics hardware. In: SIGGRAPH ’99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, pp. 277–286. ACM/Addison-Wesley, New York (1999). ISBN 0-201-48560-5. doi:10.1145/311535.311567

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Building 28, Clayton, Victoria, 3800, Australia

    Mirko Velić, Dave May & Louis Moresi

Authors
  1. Mirko Velić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dave May
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Louis Moresi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mirko Velić.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Velić, M., May, D. & Moresi, L. A Fast Robust Algorithm for Computing Discrete Voronoi Diagrams. J Math Model Algor 8, 343–355 (2009). https://doi.org/10.1007/s10852-008-9097-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10852-008-9097-6

Keywords

Search

Navigation