Incremental Voronoi diagrams

Authors Sarah R. Allen, Luis Barba, John Iacono, Stefan Langerman

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Sarah R. Allen
Luis Barba
John Iacono
Stefan Langerman

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Sarah R. Allen, Luis Barba, John Iacono, and Stefan Langerman. Incremental Voronoi diagrams. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (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 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(n^(1/2)). A matching Omega(n^(1/2)) 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. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where 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 U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) 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.
  • Voronoi diagrams
  • dynamic data structures
  • Delaunay triangulation


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  1. Greg Aloupis, Luis Barba, and Stefan Langerman. Circle separability queries in logarithmic time. In Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG'12, pages 121-125, August 2012. Google Scholar
  2. Boris Aronov, Prosenjit Bose, Erik D Demaine, Joachim Gudmundsson, John Iacono, Stefan Langerman, and Michiel Smid. Data structures for halfplane proximity queries and incremental Voronoi diagrams. In LATIN 2006: Theoretical Informatics, pages 80-92. Springer, 2006. Google Scholar
  3. Luis Barba. Disk constrained 1-center queries. In Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG'12, pages 15-19, August 2012. Google Scholar
  4. Jon Louis Bentley and James B Saxe. Decomposable searching problems i. static-to-dynamic transformation. Journal of Algorithms, 1(4):301-358, 1980. Google Scholar
  5. P. Bose, S. Langerman, and S. Roy. Smallest enclosing circle centered on a query line segment. In Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG 2008), pages 167-170, 2008. Google Scholar
  6. Timothy M Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. Journal of the ACM (JACM), 57(3):16, 2010. Google Scholar
  7. Yi-Jen Chiang and Roberto Tamassia. Dynamic algorithms in computational geometry. Proceedings of the IEEE, 80(9):1412-1434, 1992. Google Scholar
  8. Mark De Berg, Marc Van Kreveld, Mark Overmars, and Otfried Cheong Schwarzkopf. Computational geometry. Springer, 2000. Google Scholar
  9. Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements. Discrete &Computational Geometry, 1(1):25-44, 1986. Google Scholar
  10. Rolf Klein. Concrete and Abstract Voronoi Diagrams, volume 400 of Lecture Notes in Computer Science. Springer, 1989. Google Scholar
  11. Rolf Klein, Elmar Langetepe, and Zahra Nilforoushan. Abstract voronoi diagrams revisited. Computational Geometry, 42(9):885-902, 2009. Google Scholar
  12. Mark H Overmars. The design of dynamic data structures, volume 156. Springer Science &Business Media, 1983. Google Scholar
  13. Seth Pettie. Applications of forbidden 0–1 matrices to search tree and path compression-based data structures. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1457-1467, 2010. Google Scholar
  14. Daniel D Sleator and Robert Endre Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26(3):362-391, 1983. Google Scholar
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