## Abstract

Let T= { ▵_{1}, … , ▵_{n}} be a set of n triangles in R^{3} with pairwise-disjoint interiors, and let B be a convex polytope in R^{3} with a constant number of faces. For each i, let C_{i}= ▵_{i}⊕ r_{i}B denote the Minkowski sum of ▵_{i} with a copy of B scaled by r_{i}> 0. We show that if the scaling factors r_{1}, … , r_{n} are chosen randomly then the expected complexity of the union of C_{1}, … , C_{n} is O(n^{2}^{+}^{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 (n^{3}). We also consider a special case of this problem in which T is a set of points in R^{3} and B is a unit cube in R^{3}, i.e., each C_{i} is a cube of side-length 2 r_{i}. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(nlog ^{2}n) , 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(n^{⌊}^{d}^{/}^{2}^{⌋}log n) and the bound improves to O(n^{⌊}^{d}^{/}^{2}^{⌋}) if the scaling factors are chosen from a “well-behaved” pdf. The worst-case bounds are Θ (n^{2}) in R^{3}, and Θ (n^{⌈}^{d}^{/}^{2}^{⌉}) in higher odd dimensions.

Original language | English |
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Pages (from-to) | 1136-1165 |

Number of pages | 30 |

Journal | Discrete and Computational Geometry |

Volume | 65 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2021 |

## Keywords

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