Union of hypercubes and 3D minkowski sums with random sizes

Pankaj K. Agarwal, Haim Kaplan, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Let T = (1 n) be a set of of n pairwise-disjoint triangles in R3, 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 ofi 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 the complexity of B. The worst-case bound can be (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 2ri. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(n log2 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 (n2) in R3, and (nd/2) in higher dimensions.

Original languageEnglish
Title of host publication45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
EditorsChristos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770767
DOIs
StatePublished - 1 Jul 2018
Event45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: 9 Jul 201813 Jul 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume107
ISSN (Print)1868-8969

Conference

Conference45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Country/TerritoryCzech Republic
CityPrague
Period9/07/1813/07/18

Keywords

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

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