# Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Pankaj K. Agarwal, Haim Kaplan, Micha Sharir

Research output: Contribution to journalArticlepeer-review

## Abstract

Let T= { ▵1, … , ▵n} be a set of n triangles in R3 with pairwise-disjoint interiors, and let B be a convex polytope in R3 with a constant number of faces. For each i, let Ci= ▵i⊕ riB denote the Minkowski sum of ▵i 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 on 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 2 ri. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(nlog 2n) , and it improves to O(nlog 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(nd/2log n) and the bound improves to O(nd/2) if the scaling factors are chosen from a “well-behaved” pdf. The worst-case bounds are Θ (n2) in R3, and Θ (nd/2) in higher odd dimensions.

Original language English 1136-1165 30 Discrete and Computational Geometry 65 4 https://doi.org/10.1007/s00454-020-00274-0 Published - Jun 2021

## Keywords

• Arrangements
• Expected complexity
• Semi-stochastic models
• Union complexity

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