Union of Random Minkowski Sums and Network Vulnerability Analysis

Abstract

Let $$\mathcal {C}=\{C_1,\ldots ,C_n\}$$C={C1,?,Cn} be a set of $$n$$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 $$i$$i, let $$K_i$$Ki be the Minkowski sum of $$C_i$$Ci with a disk of radius $$r_i$$ri, where each $$r_i$$ri is a random non-negative number drawn independently from the distribution determined by $$\pi $$?. We show that the expected complexity of the union of $$K_1, \ldots , K_n$$K1,?,Kn is $$O(n^{1+{\varepsilon }})$$O(n1+?) for any $${\varepsilon }> 0$$?>0; here the constant of proportionality depends on $${\varepsilon }$$? and the description complexity of the sets in $$\mathcal {C}$$C, but not on $$\pi $$?. If each $$C_i$$Ci is a convex polygon with at most $$s$$s vertices, then we show that the expected complexity of the union is $$O(s^2n\log n)$$O(s2nlogn). Our bounds hold in a more general model in which we are given an arbitrary multi-set $$\varTheta =\{\theta _1,\ldots ,\theta _n\}$$?={?1,?,?n} of expansion radii, each a non-negative real number. We assign them to the members of $$\mathcal {C}$$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.