Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Abstract

Let $$T=\{\triangle _1,\ldots ,\triangle _n\}$$ be a set of n triangles in $$\mathbb {R}^3$$ with pairwise-disjoint interiors, and let B be a convex polytope in $$\mathbb {R}^3$$ with a constant number of faces. For each i, let $$C_i = \triangle _i \oplus r_i B$$ denote the Minkowski sum of $$\triangle _i$$ with a copy of B scaled by $$r_i>0$$ . We show that if the scaling factors $$r_1, \ldots , r_n$$ are chosen randomly then the expected complexity of the union of $$C_1, \ldots , C_n$$ is $$O(n^{2+{\varepsilon }})$$ , for any $${\varepsilon }> 0$$ ; the constant of proportionality depends on $${\varepsilon }$$ and on the complexity of B. The worst-case bound can be $$\Theta (n^3)$$ . We also consider a special case of this problem in which T is a set of points in $$\mathbb {R}^3$$ and B is a unit cube in $$\mathbb {R}^3$$ , i.e., each $$C_i$$ is a cube of side-length $$2r_i$$ . We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is $$O(n\log ^2n)$$ , and it improves to $$O(n\log n)$$ 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 $$d>3$$ , we show that the expected complexity of the union of the hypercubes is $$O(n^{{\lfloor d/2\rfloor }}\log n)$$ and the bound improves to $$O(n^{{\lfloor d/2\rfloor }})$$ if the scaling factors are chosen from a “well-behaved” pdf. The worst-case bounds are $$\Theta (n^2)$$ in $$\mathbb {R}^3$$ , and $$\Theta (n^{\lceil d/2\rceil })$$ in higher odd dimensions.