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.