## Abstract

Let T = (_{1} _{n}) be a set of of n pairwise-disjoint triangles in R^{3}, 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 the complexity of B. The worst-case bound can be (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 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^{2} n), 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, we show that the expected complexity of the union of the hypercubes is O(nd/2 log 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 (n^{2}) in R^{3}, and (nd/2) in higher dimensions.

Original language | English |
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Title of host publication | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |

Editors | Christos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770767 |

DOIs | |

State | Published - 1 Jul 2018 |

Event | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic Duration: 9 Jul 2018 → 13 Jul 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 107 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |
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Country/Territory | Czech Republic |

City | Prague |

Period | 9/07/18 → 13/07/18 |

### Funding

Funders | Funder number |
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Blavatnik Research Fund in Computer Science | |

Israel Science Fund | |

U.S.-Israel Binational Science Fund | |

National Science Foundation | CCF-10-12254, CCF-09-40671, CCF-11-61359 |

Army Research Office | W911NF-07-1-0376, W911NF-08-1-0452 |

National Sleep Foundation | |

Engineer Research and Development Center | 1841-14, W9132V-11-C-0003 |

Hermann Foundation | |

Blavatnik Family Foundation | |

German-Israeli Foundation for Scientific Research and Development | 1367-2017 |

United States-Israel Binational Science Foundation | 2012/229 |

Israel Science Foundation | 892-13 |

Tel Aviv University |

## Keywords

- Axis-parallel cubes
- Computational geometry
- Minkowski sums
- Objects with random sizes
- Union of geometric objects