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
Article

Abstract

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.

Keywords

Minkowski sum Arrangement Network vulnerability Stochastic model 

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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

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