TY - JOUR

T1 - Union of Random Minkowski Sums and Network Vulnerability Analysis

AU - Agarwal, Pankaj K.

AU - Har-Peled, Sariel

AU - Kaplan, Haim

AU - Sharir, Micha

N1 - Publisher Copyright:
© 2014, Springer Science+Business Media New York.

PY - 2014/9/1

Y1 - 2014/9/1

N2 - Let (Formula Prseented.) be a set of n pairwise-disjoint convex sets of constant description complexity, and let π be a probability density function (density for short) over the non-negative reals. For each i, let (Formula Prseented.) be the Minkowski sum of (Formula Prseented.) with a disk of radius (Formula Prseented.), where each (Formula Prseented.) is a random non-negative number drawn independently from the distribution determined by π. We show that the expected complexity of the union of (Formula Prseented.) is (Formula Prseented.) for any (Formula Prseented.) here the constant of proportionality depends on (Formula Prseented.) and the description complexity of the sets in C, but not on π. If each (Formula Prseented.) is a convex polygon with at most s vertices, then we show that the expected complexity of the union is (Formula Prseented.). Our bounds hold in a more general model in which we are given an arbitrary multi-set (Formula Prseented.) of expansion radii, each a non-negative real number. We assign them to the members of 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.

AB - Let (Formula Prseented.) be a set of n pairwise-disjoint convex sets of constant description complexity, and let π be a probability density function (density for short) over the non-negative reals. For each i, let (Formula Prseented.) be the Minkowski sum of (Formula Prseented.) with a disk of radius (Formula Prseented.), where each (Formula Prseented.) is a random non-negative number drawn independently from the distribution determined by π. We show that the expected complexity of the union of (Formula Prseented.) is (Formula Prseented.) for any (Formula Prseented.) here the constant of proportionality depends on (Formula Prseented.) and the description complexity of the sets in C, but not on π. If each (Formula Prseented.) is a convex polygon with at most s vertices, then we show that the expected complexity of the union is (Formula Prseented.). Our bounds hold in a more general model in which we are given an arbitrary multi-set (Formula Prseented.) of expansion radii, each a non-negative real number. We assign them to the members of 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.

KW - Arrangement

KW - Minkowski sum

KW - Network vulnerability

KW - Stochastic model

UR - http://www.scopus.com/inward/record.url?scp=84930722529&partnerID=8YFLogxK

U2 - 10.1007/s00454-014-9626-1

DO - 10.1007/s00454-014-9626-1

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AN - SCOPUS:84930722529

SN - 0179-5376

VL - 52

SP - 551

EP - 582

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 3

ER -