Discrete & Computational Geometry

, Volume 58, Issue 4, pp 822–848 | Cite as

Incremental Voronoi Diagrams

  • Sarah R. Allen
  • Luis Barba
  • John Iacono
  • Stefan Langerman
Article
  • 153 Downloads

Abstract

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in \(\mathbb {R}^3\). We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is \(O(\sqrt{n})\). A matching \(\Omega (\sqrt{n})\) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the \(O(\log {n})\) upper bound of Aronov et al. (LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, Springer, Berlin, 2006) for farthest-point Voronoi diagrams in the special case where the points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of \(S \cup \{p\}\) from the diagram of S, in time \(O(K\,\mathrm {polylog}\ n)\) worst case, which is \(O(\sqrt{n}\;\mathrm {polylog}\ n)\) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in \(O(\log n)\) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

Keywords

Voronoi diagrams Incremental Grappa tree Link-cut 

Mathematics Subject Classification

68U05 52C45 

Notes

Acknowledgements

The first named author is supported by NSF Grants CCF-0747250, CCF-1116594, and the Graduate Research Fellowship Program under Grant No. DGE-1252522. Luis Barba is supported by the ETH Postdoctoral Fellowship.

References

  1. 1.
    Aloupis, G., Barba, L., Langerman, S.: Circle separability queries in logarithmic time. In: Proceedings of the 24th Canadian Conference on Computational Geometry (CCCG’12), pp. 121–125 (2012)Google Scholar
  2. 2.
    Aronov, B., Bose, P., Demaine, E.D., Gudmundsson, J., Iacono, J., Langerman, S., Smid, M.: Data structures for halfplane proximity queries and incremental Voronoi diagrams. LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, vol. 3887, pp. 80–92. Springer, Berlin (2006)Google Scholar
  3. 3.
    Barba, L.: Disk constrained 1-center queries. In: Proceedings of the 24th Canadian Conference on Computational Geometry (CCCG’12), pp. 15–19 (2012)Google Scholar
  4. 4.
    Bentley, J.L., Saxe, J.B.: Decomposable searching problems I. Static-to-dynamic transformation. J. Algorithm. 1(4), 301–358 (1980)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bose, P., Langerman, S., Roy, S.: Smallest enclosing circle centered on a query line segment. In: Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG’08), pp. 167–170 (2008)Google Scholar
  6. 6.
    Chan, T.M.: A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries. J. ACM 57(3), Art. No. 16 (2010)Google Scholar
  7. 7.
    Chan, T.M., Tsakalidis, K.: Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. Discrete Comput. Geom. 56(4), 866–881 (2016)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Chiang, Y.-J., Tamassia, R.: Dynamic algorithms in computational geometry. Proc. IEEE 80(9), 1412–1434 (1992)CrossRefGoogle Scholar
  9. 9.
    De Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  10. 10.
    Edelsbrunner, H., Seidel, R.: Voronoi diagrams and arrangements. Discrete Comput. Geom. 1(1), 25–44 (1986)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P., Sharir, M.: Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’17), pp. 2495–2504. SIAM, Philadelphia (2017)Google Scholar
  12. 12.
    Klein, R.: Concrete and Abstract Voronoi Diagrams. Lecture Notes in Computer Science, vol. 400. Springer, Berlin (1989)MATHGoogle Scholar
  13. 13.
    Klein, R., Langetepe, E., Nilforoushan, Z.: Abstract Voronoi diagrams revisited. Comput. Geom. 42(9), 885–902 (2009)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Overmars, M.H.: The Design of Dynamic Data Structures. Lecture Notes in Computer Science, vol. 156. Springer, Berlin (1983)MATHGoogle Scholar
  15. 15.
    Pettie, S.: Applications of forbidden 0–1 matrices to search tree and path compression-based data structures. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’10), pp. 1457–1467. SIAM, Philadelphia (2010)Google Scholar
  16. 16.
    Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Sarah R. Allen
    • 1
  • Luis Barba
    • 2
  • John Iacono
    • 3
  • Stefan Langerman
    • 4
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland
  3. 3.Department of Computer Science and Engineering, Tandon School of EngineeringNew York UniversityNew YorkUSA
  4. 4.Départment d’InformatiqueUniversité Libre de BruxellesBruxellesBelgium

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