Discrete & Computational Geometry

, Volume 53, Issue 3, pp 547–568 | Cite as

On the Complexity of Randomly Weighted Multiplicative Voronoi Diagrams

Article

Abstract

We provide an \(O(n \,\hbox {polylog}\, n)\) bound on the expected complexity of the randomly weighted multiplicative Voronoi diagram of a set of \(n\) sites in the plane, where the sites can be either points, interior disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not on their location. This compares favorably with the worst-case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal et al. (Discrete Comput Geom 54:551–582, 2014) of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems.

Keywords

Arrangements Randomized incremental construction Voronoi diagrams 

References

  1. 1.
    Agarwal, P.K., Har-Peled, S., Kaplan, H., Sharir, M.: Union of random Minkowski sums and network vulnerability analysis. Discrete Comput. Geom. 54, 551–582 (2014). doi:10.1007/s00454-014-9626-1
  2. 2.
    Agarwal, P.K., Kaplan, H., Sharir, M.: Union of random Minkowski sums and network vulnerability analysis. In: Proceedings of 29th Annual Symposium on Computational Geometry (SoCG), pp. 177–186 (2013)Google Scholar
  3. 3.
    Aurenhammer, F.: Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16(1), 78–96 (1987)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Aurenhammer, F.: Voronoi diagrams: a survey of a fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991)CrossRefGoogle Scholar
  5. 5.
    Aurenhammer, F., Edelsbrunner, H.: An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recognition 17(2), 251–257 (1984)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Aurenhammer, F., Klein, R., Lee, D.T.: Voronoi Diagrams and Delaunay Triangulations. World Scientific, Singapore (2013)CrossRefGoogle Scholar
  7. 7.
    Chang, H.C., Har-Peled, S., Raichel, B.: From proximity to utility: A Voronoi partition of Pareto optima. arXiv:1404.3403 (2014)
  8. 8.
    Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387–421 (1989)Google Scholar
  9. 9.
    Driemel, A., Har-Peled, S., Raichel, B.: On the expected complexity of Voronoi diagrams on terrains. In: Proceedings of 28th Annual Symposium on Computational Geometry (SoCG), pp. 101–110 (2012)Google Scholar
  10. 10.
    Dwyer, R.: Higher-dimensional Voronoi diagrams in linear expected time. In: Proceedings of 5th Annual Symposium on Computational Geometry (SoCG), pp. 326–333 (1989)Google Scholar
  11. 11.
    Har-Peled, S.: An output sensitive algorithm for discrete convex hulls. Comput. Geom. Theory Appl. 10, 125–138 (1998)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Har-Peled, S.: Geometric Approximation Algorithms, Mathematical Surveys and Monographs, vol. 173. American Mathematical Society, Providence, RI (2011)CrossRefGoogle Scholar
  13. 13.
    Har-Peled, S.: On the expected complexity of random convex hulls. CoRR abs/1111.5340 (2011)Google Scholar
  14. 14.
    Har-Peled, S., Raichel, B.: On the complexity of randomly weighted Voronoi diagrams. In: Proceedings of 30th Annual Symposium on Computational Geometry (SoCG), pp. 232–241 (2014)Google Scholar
  15. 15.
    Kaplan, H., Ramos, E., Sharir, M.: The overlay of minimization diagrams in a randomized incremental construction. Discrete Comput. Geom. 45(3), 371–382 (2011)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Klein, R.: Abstract Voronoi diagrams and their applications. In: Workshop on Computational Geometry, pp. 148–157 (1988)Google Scholar
  17. 17.
    Klein, R., Langetepe, E., Nilforoushan, Z.: Abstract Voronoi diagrams revisited. Comput. Geom. 42(9), 885–902 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Mulmuley, K.: Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs (1994)Google Scholar
  19. 19.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. Probability and Statistics. Wiley, New York (2000)Google Scholar
  20. 20.
    Pettie, S.: Sharp bounds on Davenport–Schinzel sequences of every order. In: Proceedings of 29th Annual Symposium on Computational Geometry (SoCG), SoCG ’13, pp. 319–328 (2013). doi:10.1145/2462356.2462390
  21. 21.
    Raynaud, H.: Sur l’enveloppe convex des nuages de points aleatoires dans \(R^{n}\). J. Appl. Probab. 7, 35–48 (1970)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rényi, A., Sulanke, R.: Über die konvexe Hülle von \(n\) zufällig gerwähten Punkten I. Z. Wahrsch. Verw. Gebiete 2, 75–84 (1963)CrossRefMATHGoogle Scholar
  23. 23.
    Santalo, L.: Introduction to Integral Geometry. Hermann, Paris (1953)MATHGoogle Scholar
  24. 24.
    Schneider, R., Wieacker, J.A.: Integral geometry. In: P.M. Gruber, J.M. Wills (eds.) Handbook of Convex Geometry, vol. B, chap. 5.1, pp. 1349–1390. North-Holland, Amsterdam (1993)Google Scholar
  25. 25.
    Sharir, M.: Almost tight upper bounds for lower envelopes in higher dimensions. Discrete Comput. Geom. 12, 327–345 (1994)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995)Google Scholar
  27. 27.
    Weil, W., Wieacker, J.A.: Stochastic geometry. In: P.M. Gruber, J.M. Wills (eds.) Handbook of Convex Geometry, vol. B, chap. 5.2, pp. 1393–1438. North-Holland, Amsterdam (1993)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of IllinoisUrbanaUSA

Personalised recommendations