Union of Random Minkowski Sums and Network Vulnerability Analysis

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 (pdf 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 on 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 the stronger model in which we are given an arbitrary multi-set $R=\{r_1,\ldots,r_n\}$ 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. %