Advertisement

Algorithmica

, Volume 74, Issue 1, pp 415–439 | Cite as

The Higher-Order Voronoi Diagram of Line Segments

  • Evanthia Papadopoulou
  • Maksym Zavershynskyi
Article

Abstract

The order-\(k\) Voronoi diagram of line segments has properties surprisingly different from its counterpart for points. For example, a single order-\(k\) Voronoi region may consist of \(\varOmega (n)\) disjoint faces. In this paper, we analyze the structural properties of this diagram and show that its combinatorial complexity is \(O(k(n-k))\), for \(n\) non-crossing line segments, despite the presence of disconnected regions. The same bound holds for \(n\) intersecting line segments, for \(k\ge n/2\). We also consider the order-\(k\) Voronoi diagram of line segments that form a planar straight-line graph, and augment the definition of an order-\(k\) diagram to cover sites that are not disjoint. On the algorithmic side, we extend the iterative approach to construct this diagram, handling complications caused by the presence of disconnected regions. All bounds are valid in the general \(L_p\) metric, \(1\le p\le \infty \). For non-crossing segments in the \(L_\infty \) and \(L_1\) metrics, we show a tighter \(O((n-k)^2)\) bound for \(k>n/2\).

Keywords

Computational geometry Voronoi diagram Line segments Planar straight line graph Order-k Voronoi diagram \(k\) nearest neighbors \(L_p\) metric 

Notes

Acknowledgments

The authors are grateful to Gill Barequet for reading through an earlier draft and providing useful comments. This work was supported in part by the Swiss National Science Foundation, SNF 200021-127137, and the ESF EUROCORES program EuroGIGA/VORONOI, SNF 20GG21-134355.

References

  1. 1.
    Agarwal, P.K., de Berg, M., Matousek, J., Schwarzkopf, O.: Constructing levels in arrangements and higher-order Voronoi diagrams. SIAM J. Comput. 27(3), 654–667 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Aggarwal, A., Guibas, L.J., Saxe, J.B., Shor, P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete Comput. Geom. 4, 591–604 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aurenhammer, F., Schwarzkopf, O.: A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams. Int. J. Comput. Geom. Appl. 2(4), 363–381 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Aurenhammer, F., Drysdale, R.L.S., Krasser, H.: Farthest line segment Voronoi diagrams. Inf. Process. Lett. 100(6), 220–225 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Aurenhammer, F., Klein, K., Lee, D.-T.: Voronoi Diagrams and Delaunay Triangulations, World Scientific, 1–337, ISBN 978-981-4447-63-8, pp. I-VIII (2013)Google Scholar
  6. 6.
    Bohler, C., Klein, R., Liu, C.-H., Papadopoulou, E., Cheilaris, P., Zavershynskyi, M.: On the complexity of higher order abstract Voronoi diagrams. In: Fomin, F.V., et al. (eds.) ICALP 2013. Lecture Notes in Computer Science, vol. 7965, pp. 208–219. Springer, Heidelberg (2013)Google Scholar
  7. 7.
    Bohler, C., Liu, C.-H., Papadopoulou, E., Zavershynskyi, M.: A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams. 25th International Symposium on Algorithms and Computation, ISAAC 2014. Lecture Notes in Computer Science, vol. 8889, (to appear)Google Scholar
  8. 8.
    Chan, T.M.: Random sampling, halfspace range reporting, and construction of \(\le k\)-levels in three dimensions. SIAM J. Comput. 30(2), 561–575 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chazelle, B., Edelsbrunner, H.: An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput. 36(11), 1349–1454 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.-S.: Farthest-polygon Voronoi diagrams. Comput. Geom. 44(4), 234–247 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Clarkson, K.L.: New applications of random sampling in computational geometry. Discrete Comput. Geom. 2, 195–222 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Clarkson, K.L., Shor, P.W.: Application of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387–421 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Edelsbrunner, H.: The complexity of higher-order Voronoi diagrams. In: Edelsbrunner, H.: Algorithms in Combinatorial Geometry, pp. 319–324. EATCS monographs on theoretical computer science. Springer, Nre York (1987)Google Scholar
  14. 14.
    Edelsbrunner, H., Maurer, H.A., Preparata, F.P., Rosenberg, A.L., Welzl, E., Wood, D.: Stabbing line segments. BIT 22(3), 274–281 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kirkpatrick, D.G.: Efficient computation of continuous skeletons. FOCS: 18–27, 20th Annual Symposium on Foundations of Computer Science (1979)Google Scholar
  16. 16.
    Klein, R., Langetepe, E., Nilforoushan, Z.: Abstract Voronoi diagrams revisited. Comput. Geom. 42(9), 885–902 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Lee, D.T.: Two-dimensional Voronoi diagrams in the \(\text{ L }_{p}\)-metric. J. ACM 27(4), 604–618 (1980)zbMATHCrossRefGoogle Scholar
  18. 18.
    Lee, D.T.: On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput. 31(6), 478–487 (1982)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Liu, C.-H., Papadopoulou, E., Lee, D.T.: The k-nearest-neighbor Voronoi diagram revisited. Algorithmica (2013). doi: 10.1007/s00453-013-9809-9
  20. 20.
    Mehlhorn, K., Meiser, S., Rasch, R.: Furthest site abstract Voronoi diagrams. Int. J. Comput. Geom. Appl. 11(6), 583–616 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Papadopoulou, E.: Net-aware critical area extraction for opens in VLSI circuits via higher-order Voronoi diagrams. IEEE Trans. CAD Integr. Circuits Syst. 30(5), 704–717 (2011)CrossRefGoogle Scholar
  22. 22.
    Papadopoulou, E., Dey, S.K.: On the farthest line segment Voronoi diagram. Int. J. Comput. Geom. Appl. 23(6), 443–460 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Papadopoulou, E., Zavershynskyi, M.Z.: On higher order Voronoi diagrams of line segments. In: Chao, K.-M. et al. (eds.) ISAAC 2012, Lecture Notes in Computer Science, 7676, 177–186 (2012)Google Scholar
  24. 24.
    Ramos, E.A.: On range reporting, ray shooting and k-level construction. In: Proceedings of 15th Annual Symposium on Computational Geometry, 390–399 (1999)Google Scholar
  25. 25.
    Rappaport, D.: Computing the furthest site Voronoi diagram for a set of discs (preliminary report). In: Dehne, F.K.H.A., et al. (eds.) WADS 1989. Lecture Notes in Computer Science, vol. 382, pp. 57–66. Springer, Heidelberg (1989)Google Scholar
  26. 26.
    Rosenberger, H.: Order-k Voronoi diagrams of sites with additive weights in the plane. Algorithmica 6(4), 490–521 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Seidel, R.: The nature and meaning of perturbations in geometric computing. Discrete Comput. Geom. 19(1), 1–17 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  29. 29.
    Sharir, M.: The Clarkson–Shor technique revisited and extended. Comb. Probab. Comput. 12(2), 191–201 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Zavershynskyi, M., Papadopoulou, E.: A sweepline algorithm for higher order Voronoi diagrams. In: Proceedings of 10th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2013. IEEE-CS, pp. 16–22 (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Faculty of InformaticsUniversità della Svizzera italiana (USI)LuganoSwitzerland

Personalised recommendations