Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions
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Published:2023-05-25
Issue:
Volume:
Page:
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ISSN:0179-5376
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Container-title:Discrete & Computational Geometry
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language:en
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Short-container-title:Discrete Comput Geom
Author:
Barequet Gill, Papadopoulou EvanthiaORCID, Suderland MartinORCID
Abstract
AbstractWe study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions $$\mathbb {S}^{d-1}$$
S
d
-
1
. We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is $$O(\min \{k,n-k\}n^{d-1})$$
O
(
min
{
k
,
n
-
k
}
n
d
-
1
)
, which is tight for $$n-k=O(1)$$
n
-
k
=
O
(
1
)
. This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its $$(d-1)$$
(
d
-
1
)
-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of $$n \ge 2$$
n
≥
2
lines in general position has exactly $$n(n-1)$$
n
(
n
-
1
)
three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in $$O(n^{d-1} \alpha (n))$$
O
(
n
d
-
1
α
(
n
)
)
time, for $$d\ge 4$$
d
≥
4
, while if $$d=3$$
d
=
3
, the time drops to worst-case optimal $$\Theta (n^2)$$
Θ
(
n
2
)
. We extend the obtained results to bounded polyhedra and clusters of points as sites.
Funder
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung United States - Israel Binational Science Foundation
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Theoretical Computer Science
Reference34 articles.
1. Abellanas, M., Hurtado, F., Icking, Ch., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: The farthest color Voronoi diagram and related problems (extended abstract). In: 17th European Workshop Comput. Geom. (Berlin 2001), pp. 113–116. http://www.pi6.fernuni-hagen.de/downloads/publ/ahiklmps-fcvdr-01.pdf 2. Agarwal, P.K., de Berg, M., Matoušek, J., Schwarzkopf, O.: Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27(3), 654–667 (1998) 3. Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Handbook of Computational Geometry, pp. 49–119. North-Holland, Amsterdam (2000) 4. Aronov, B.: A lower bound on Voronoi diagram complexity. Inform. Process. Lett. 83(4), 183–185 (2002) 5. Aurenhammer, F., Drysdale, R.L.S., Krasser, H.: Farthest line segment Voronoi diagrams. Inform. Process. Lett. 100(6), 220–225 (2006)
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