Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Let T = {41, . . . ,4n} be a set of of n pairwise-disjoint triangles in R3, and let B be a convex polytope in R3 with a constant number of faces. For each i, let Ci = 4i ⊕ riB denote the Minkowski sum of 4i with a copy of B scaled by ri > 0. We show that if the scaling factors r1, . . . , rn are chosen randomly then the expected complexity of the union of C1, . . . , Cn is O(n2+ε), for any ε > 0; the constant of proportionality depends on ε and the complexity of B. The worst-case bound can be Θ(n3). We also consider a special case of this problem in which T is a set of points in R3 and B is a unit cube in R3, i.e., each Ci is a cube of side-length 2ri. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(n log2 n), and it improves to O(n logn) if the scaling factors are chosen randomly from a “well-behaved” probability density function (pdf). We also extend the latter results to higher dimensions. For any fixed odd value of d, we show that the expected complexity of the union of the hypercubes is O(nbd/2c logn) and the bound improves to O(nbd/2c) if the scaling factors are chosen from a “well-behaved” pdf. The worst-case bounds are Θ(n2) in R3, and Θ(ndd/2e) in higher dimensions. 2012 ACM Subject Classification Theory of computation → Computational geometry, Theory of computation → Generating random combinatorial structures