TY - JOUR

T1 - Voronoi diagrams of moving points in the plane and of lines in space

T2 - Tight bounds for simple configurations

AU - Weisman, Amit

AU - Chew, L. Paul

AU - Kedem, Klara

PY - 2004/12/16

Y1 - 2004/12/16

N2 - The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the ω (n2) lower bound and the O(n3+ε) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2′) the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered "cross-sections" of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1′) and (2′) have recently been shown to have a nearly tight bound: both complexities are O(n2+ε) with lower bound ω(n2). In this paper, we determine the combinatorial complexities of these four problems fro some very simple input configurations. In particular, we study point configurations with just two degrees of freedom (DOFs), exploring both the Voronoi diagrams and the corresponding Minkowski sums. We consider the traditional versions of these problems to have 4 DOFs. We show that even for these simple configurations the combinatorial complexities have upper bounds of either O(n2) or O(n2+) and lower bounds of ω (n 2).

AB - The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the ω (n2) lower bound and the O(n3+ε) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2′) the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered "cross-sections" of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1′) and (2′) have recently been shown to have a nearly tight bound: both complexities are O(n2+ε) with lower bound ω(n2). In this paper, we determine the combinatorial complexities of these four problems fro some very simple input configurations. In particular, we study point configurations with just two degrees of freedom (DOFs), exploring both the Voronoi diagrams and the corresponding Minkowski sums. We consider the traditional versions of these problems to have 4 DOFs. We show that even for these simple configurations the combinatorial complexities have upper bounds of either O(n2) or O(n2+) and lower bounds of ω (n 2).

KW - Combinatorial problems

KW - Computational geometry

KW - Voronoi diagrams

UR - http://www.scopus.com/inward/record.url?scp=7544230506&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2004.08.004

DO - 10.1016/j.ipl.2004.08.004

M3 - Article

AN - SCOPUS:7544230506

VL - 92

SP - 245

EP - 251

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 5

ER -