Skip to main content

Voronoi Diagrams for Moving Disks and Applications

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2125))

Included in the following conference series:

Abstract

In this paper we discuss the kinetic maintenance of the Euclidean Voronoi diagram and its dual, the Delaunay triangulation, for a set of moving disks. The most important aspect in our approach is that we can maintain the Voronoi diagram even in the case of intersecting disks. We achieve that by augmenting the Delaunay triangulation with some edges associated with the disks that do not contribute to the Voronoi diagram. Using the augmented Delaunay triangulation of the set of disks as the underlying structure, we discuss how to maintain, as the disks move, (1) the closest pair, (2) the connectivity of the set of disks and (3) in the case of non-intersecting disks, the near neighbors of a disk.

Supported by NSF grant CCR-9910633.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. J. Basch, L. J. Guibas, and J. Hershberger. Data structures for mobile data. J. Algorithms, 1:1–28, 1999.

    Article  MathSciNet  Google Scholar 

  2. M. Gavrilova and J. Rokne. Swap conditions for dynamic Voronoi diagrams for circles and line segments. CAGD, 16:89–106, 1999.

    MATH  MathSciNet  Google Scholar 

  3. L. J. Guibas, J. Hershberger, S. Suri, and L. Zhang. Kinetic connectivity for unit disks. In Proc. 16th ACM Symp. on Computat. Geom., pages 331–339, 2000.

    Google Scholar 

  4. L. J. Guibas, J. S. B. Mitchell, and T. Roos. Voronoi diagrams of moving points in the plane. In G. Schmidt and R. Berghammer, editors, Proc. 17th International Workshop on Graph-Theoretic Concepts in Computer Science, volume 570 of LNCS, pages 113–125. Springer, 1991.

    Google Scholar 

  5. L. J. Guibas, J. Snoeyink, and L. Zhang. Compact Voronoi diagrams for moving convex polygons. In Magnús M. Halldórsson, editor, Proc. 7th SWAT, volume 1851 of LNCS, pages 339–352. Springer, 2000.

    Google Scholar 

  6. L. J. Guibas and L. Zhang. Euclidean proximity and power diagram. In Proc. 10th CCCG, 1998.

    Google Scholar 

  7. J. Holm, K. de Lichtenberg, and M. Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and bicon-nectivity. In Proc. 30th ACM STOC, pages 79–89, 1998.

    Google Scholar 

  8. H. Imai, M. Iri, and K. Murota. Voronoi diagram in the Laguerre geometry and its applications. SIAM J. Comput., 14(1):93–105, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. I. Karavelas and L. J. Guibas. Static and kinetic geometric spanners with applications. In Proc. 12th ACM-SIAM SODA, pages 168–176, 2001.

    Google Scholar 

  10. J.-C. Latombe. Robot Motion Planning. Kluwer Academic Publishers, Boston, 1991.

    Google Scholar 

  11. M. C. Lin and J. F. Canny. Efficient algorithms for incremental distance computation. In Proc. IEEE Intern. Conf. Robot. Autom., volume 2, pages 1008–1014, 1991.

    Article  Google Scholar 

  12. M. Sharir. Intersection and closest-pair problems for a set of planar discs. SIAM J. of Comput., 14(2):448–468, May 1985.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Karavelas, M.I. (2001). Voronoi Diagrams for Moving Disks and Applications. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_7

Download citation

  • DOI: https://doi.org/10.1007/3-540-44634-6_7

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42423-9

  • Online ISBN: 978-3-540-44634-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics