TY - GEN

T1 - Union of hypercubes and 3D minkowski sums with random sizes

AU - Agarwal, Pankaj K.

AU - Kaplan, Haim

AU - Sharir, Micha

N1 - Publisher Copyright:
© Pankaj K. Agarwal, Haim Kaplan, and Micha Sharir;.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

KW - Axis-parallel cubes

KW - Computational geometry

KW - Minkowski sums

KW - Objects with random sizes

KW - Union of geometric objects

UR - http://www.scopus.com/inward/record.url?scp=85049801207&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2018.10

DO - 10.4230/LIPIcs.ICALP.2018.10

M3 - פרסום בספר כנס

AN - SCOPUS:85049801207

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018

A2 - Kaklamanis, Christos

A2 - Marx, Daniel

A2 - Chatzigiannakis, Ioannis

A2 - Sannella, Donald

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 9 July 2018 through 13 July 2018

ER -