Union of Random Minkowski Sums and Network Vulnerability Analysis

Abstract

Let C = {C1, . . . , Cn} be a set of n pairwise-disjoint convex sgons, for some constant s, and let π be a probability density function (pdf) over the non-negative reals. For each i, let Ki be the Minkowski sum of Ci with a disk of radius ri, where each ri is a random non-negative number drawn independently from the distribution determined by π. We show that the expected complexity of the union of K1, . . . , Kn is O(n log n), for any pdf π; the constant of proportionality depends on s, but not on the pdf.Next, we consider the following problem that arises in analyzing the vulnerability of a network under a physical attack. Let G = (V, E) be a planar geometric graph where E is a set of n line segments with pairwise-disjoint relative interiors. Let ϕ : R ≥0 → [0, 1] be an edge failure probability function, where a physical attack at a location x ∈ R 2 causes an edge e of E at distance r from x to fail with probability ϕ(r); we assume that ϕ is of the form 1 − Π(x), where Π is a cumulative distribution function on the non-negative reals. The goal is to compute the most vulnerable location for G, i.e., the location of the attack that maximizes the expected number of failing edges of G. Using our bound on the complexity of the union of random Minkowski sums, we present a near-linear Monte-Carlo algorithm for computing a location that is an approximately most vulnerable location of attack for G. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems-Geometrical problems and computations General TermsAlgorithms, Theory Keywords Arrangements, union complexity, network analysis Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

Minimum shared‐power edge cut

“…Proof. We prove that the copies of v in K * form a prefix of v(0), v (1), …v(c − 1). Then the result follows since there are k * v copies of v in K * .…”

“…The idea of our approximation algorithm is to convert the MSPEC problem to a minimum vertex cut problem. Observe that if we can only assign power 0 or 1 to every vertex in V, then our problem is minimum vertex cut-remove a minimum number of vertices to disconnect s and t. We will discretize our problem by replacing each vertex v ∈ V by multiple copies of v such that removing one copy corresponds to assigning a small fraction of the maximum power to v. A similar approach was used, in a geometric setting, by Agarwal et al [1]. We want to ensure that the discretization introduces an error of at most n for each vertex with respect to the optimum solution.…”

“…Let the values in D v be v 0 , v 1 , … , v c v in increasing order. We modify the construction of graph G ′ from Algorithm 1, to create |D v | = c v + 1 copies of each vertex v, which we number v(0), v (1)…”

Union of Hypercubes and 3D Minkowski Sums with Random Sizes

“…As noted in [2], the permutation model is more general than the density model, in the sense that an upper bound on ψ(n) in the permutation model immediately implies the same bound on ψ(n) in the density model. Although we mostly focus on the permutation model, we also show that some of our bounds can be improved, by a logarithmic factor, in the density model if the underlying pdf is "well behaved," in a sense to be made precise below.…”

“…We want to compute a point of attack in which we maximize the expected number of triangles that we hit. The same technique to approximately solve this problem, as done in the planar case [1,2], leads to questions about the complexity of the union of Minkowski sums of the input triangles with balls of random radii. The case studied here replaces the ball by a convex polytope, which we choose to approximate the Euclidean ball.…”