Discrete & Computational Geometry

, Volume 52, Issue 3, pp 551–582 | Cite as

Union of Random Minkowski Sums and Network Vulnerability Analysis

  • Pankaj K. Agarwal
  • Sariel Har-Peled
  • Haim Kaplan
  • Micha Sharir


Let \(\mathcal {C}=\{C_1,\ldots ,C_n\}\) be a set of \(n\) pairwise-disjoint convex sets of constant description complexity, and let \(\pi \) be a probability density function (density for short) over the non-negative reals. For each \(i\), let \(K_i\) be the Minkowski sum of \(C_i\) with a disk of radius \(r_i\), where each \(r_i\) is a random non-negative number drawn independently from the distribution determined by \(\pi \). We show that the expected complexity of the union of \(K_1, \ldots , K_n\) is \(O(n^{1+{\varepsilon }})\) for any \({\varepsilon }> 0\); here the constant of proportionality depends on \({\varepsilon }\) and the description complexity of the sets in \(\mathcal {C}\), but not on \(\pi \). If each \(C_i\) is a convex polygon with at most \(s\) vertices, then we show that the expected complexity of the union is \(O(s^2n\log n)\). Our bounds hold in a more general model in which we are given an arbitrary multi-set \(\varTheta =\{\theta _1,\ldots ,\theta _n\}\) of expansion radii, each a non-negative real number. We assign them to the members of \(\mathcal {C}\) by a random permutation, where all permutations are equally likely to be chosen; the expectations are now with respect to these permutations. We also present an application of our results to a problem that arises in analyzing the vulnerability of a network to a physical attack.


Minkowski sum Arrangement Network vulnerability Stochastic model 


  1. 1.
    Agarwal, P.K., Efrat, A., Ganjugunte, S.K., Hay, D., Sankararaman, S., Zussman, G.: Network vulnerability to single, multiple, and probabilistic physical attacks. In: Proceedings of Military Communication Conference 2010, pp. 1824–1829 (2010)Google Scholar
  2. 2.
    Agarwal, P.K., Efrat, A., Ganjugunte, S. K., Hay, D., Sankararaman, S., Zussman, G.: The resilience of WDM networks to probabilistic geographical failures. In: IEEE/ACM Transactions of Network 21 pp. 1525–1538 (2013)Google Scholar
  3. 3.
    Agarwal, P.K., Ezra, E., Sharir, M.: Near-linear approximation algorithms for geometric hitting sets. Algorithmica 63, 1–25 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Agarwal, P. K., Hagerup, T., Ray, R., Sharir, M., Smid, M., Welzl, E.: Translating a planar object to maximize point containment. In: Proceedings of 10th Annual European Symposium on Algorithms, pp. 42–53 (2002)Google Scholar
  5. 5.
    Agarwal, P.K., Pach, J., Sharir, M.: State of the union (of geometric objects). In: Goodman, J., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry, pp. 9–48. American Mathematical Society, Providence, RI (2008)CrossRefGoogle Scholar
  6. 6.
    Aronov, B., Har-Peled, S.: On approximating the depth and related problems. SIAM J. Comput. 38, 899–921 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 201–290. Elsevier, Amsterdam (1999)Google Scholar
  8. 8.
    Bhandari, R.: Survivable Networks: Algorithms for Diverse Routing. Kluwer, Norwell, MA (1998)Google Scholar
  9. 9.
    Chang, H.-C., Har-Peled, S., Raichel, B.: From proximity to utility: a Voronoi partition of Pareto optima, CoRR. http://arxiv.org/abs/1404.3403 (2014)
  10. 10.
    Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry II. Discrete Comput. Geom. 4, 387–421 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    de Berg, M., Katz, M.J., van der Stappen, A.F., Vleugels, J.: Realistic input models for geometric algorithms. Algorithmica 34, 81–97 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Edelsbrunner, H., Fasy, B.T., Rote, G.: Add isotropic Gaussian kernels at own risk: more and more resilient modes in higher dimensions. In: Proceedings of 28th Annual Symposium Computational Geometry, pp. 91–100 (2012)Google Scholar
  13. 13.
    Foster, J.S., Gjelde, E., Graham, W.R., Hermann, R.J., Kluepfel, H.M., Lawson, R.L., Soper, G.K., Wood, L.L., Woodard, J.B.: Report of the Commission to Assess the Threat to the United States from Electromagnetic Pulse (EMP) Attack, Critical National Infrastructures (2008)Google Scholar
  14. 14.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison Wesley, Boston (1994)MATHGoogle Scholar
  15. 15.
    Har-Peled, S., Raichel, B.: On the complexity of randomly weighted Voronoi diagrams. In: Proceedings of 30th Annual Symposium Computational Geometry, pp. 232–241 (2014)Google Scholar
  16. 16.
    Har-Peled, S., Sharir, M.: Relative \((p,\varepsilon )\)-approximations in geometry. Discrete Comput. Geom. 45, 462–496 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Johnson, N.L., Kemp, A.W., Kotz, S.: Univariate Discrete Distributions, 3rd edn. Wiley, New York (2005)CrossRefMATHGoogle Scholar
  18. 18.
    Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1, 59–71 (1986)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lifshitz, L.M., Pizer, S.M.: A multiresolution hierarchical approach to image segmentation based on intensity extrema. IEEE Trans. Pattern Anal. Mach. Intell. 12, 529–540 (1990)CrossRefGoogle Scholar
  20. 20.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
  21. 21.
    Neumayer, S., Modiano, E.: Network reliability with geographically correlated failures. In: Proceedings 29th IEEE International Conference on Computer Communications, pp. 1658–1666 (2010)Google Scholar
  22. 22.
    Neumayer, S., Zussman, G., Cohen, R., Modiano, E.: Assessing the vulnerability of the fiber infrastructure to disasters. IEEE/ACM Trans. Netw. 19, 1610–1623 (2011)CrossRefGoogle Scholar
  23. 23.
    Ou, C., Mukherjee, B.: Survivable Optical WDM Networks. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  24. 24.
    Pach, J., Sharir, M.: On the boundary of the union of planar convex sets. Discrete Comput. Geom. 21, 321–328 (1999)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  26. 26.
    Spielman, D.A., Teng, S.H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practise. Commun. ACM 52, 76–84 (2009)CrossRefGoogle Scholar
  27. 27.
    Wu, W., Moran, B., Manton, J., Zukerman, M.: Topology design of undersea cables considering survivability under major disasters. In: Proceedings of International Conference on Advanced Information Networking and Applications Workshops, pp. 1154–1159 (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Sariel Har-Peled
    • 2
  • Haim Kaplan
    • 3
  • Micha Sharir
    • 3
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Department of Computer ScienceUniversity of IllinoisUrbanaUSA
  3. 3.School of Computer ScienceTel Aviv UniversityTel AvivIsrael

Personalised recommendations