Maksym Zavershynskyi

Evanthia Papadopoulou

On  higher  order  Voronoi  diagrams    of  line  segments  

Supported  in  part  by  the  Swiss  Na3onal  Science  Founda3on  (SNF)  grant  200021-­‐127137.    Also  by  SNF  grant  20GG21-­‐134355  within  the  collabora3ve  research  project  EuroGIGA/VORONOI  of  the  European  Science  Founda3on.  

23rd  Interna3onal  Symposium  on  Algorithms  and  Computa3on,  ISAAC  2012  Taipei,  Taiwan,  December  2012  

University  of  Lugano,  Switzerland  

Overview

1.  Introduction

2.  Disjoint line-segments a)  Disconnected regions

b)  Differences with points

c)  Structural complexity

3.  Planar straight-line graph

4.  Intersecting line-segments

1. Introduction

Nearest Neighbor Voronoi Diagram

The nearest neighbor Voronoi diagram is the partitioning of the plane into maximal regions, such that all points within a region have the same closest site.

Higher Order Voronoi Diagram

The order-k Voronoi diagram is the partitioning of the plane into maximal regions, such that all points within an order-k region have the same k nearest sites.

2-­‐order  Voronoi    diagram  

Related Work

¤  Higher order Voronoi diagram of points:

¤  Structural complexity [Lee 82, Edelsbrunner 87]

¤  Construction algorithms

¤  Iterative in time [Lee 82]

¤  Randomized incremental in

expected time [Agarwal et al 98]

¤  Farthest Voronoi diagram of line-segments [Aurenhammer et al 06]

¤  Higher order Voronoi diagram of line-segments NOT STUDIED!

2. Disjoint Line-Segments

a) Disconnected regions

Disconnected Regions

A single order-k Voronoi region may disconnect t faces

2-­‐order  Voronoi    diagram  

Disconnected Regions

An order-k Voronoi region may disconnect to bounded faces. For

Order-­‐2  Voronoi  diargam  of  6  line-­‐segments  

Disconnected Regions

An order-k Voronoi region may disconnect to unbounded faces. For

Order-­‐4  Voronoi  diargam  of  7  line-­‐segments  *  Generalizing  [Aurenhammer  et  al  06].  

k

n-k

Disconnected Regions

An order-k Voronoi region may disconnect to unbounded faces. For

F1

F2

F3

F4

Order-­‐4  Voronoi  diargam  of  7  line-­‐segments  *  Generalizing  [Aurenhammer  et  al  06].  

Line-­‐segments  can  be  untangled!  

2. Disjoint Line-Segments

b) Differences with points

For Points:

¤  Order-k Voronoi regions are connected convex polygons

¤  The number of faces [Lee82]

¤  The symmetry property for the number of unbounded faces:

¤  The k-set theory [Edelsbrunner 87, Alon et al 86] implies bounds:

¤  The structural complexity is

For Line-Segments:

¤  A single order-k Voronoi region may disconnect to faces

¤  The number of faces [this paper]

¤  NO symmetry property for unbounded faces!

¤  NO k-set theory available for unbounded faces!

¤  The structural complexity is

2. Disjoint Line-Segments

c) Structural complexity

Structural complexity

¤  Let F be a face of region in

¤  The graph structure of enclosed in F

is a connected tree that consists of at least one edge

Vk−1(S)

Vk(S)

Structural complexity

¤  Generalizing Lee’s approach we prove:

¤  For this formula already implies

¤  For we need to bound

Bounding ¤  We use well-known point-line duality transformation.

¤  We transform every line-segment to a wedge [Aurenhammer et al 06]

Bounding ¤  Consider an arrangement of dual wedges

w1

w2

w3

w4

w5

p q

¤  An unbounded edge in order-k Voronoi diagram corresponds to a vertex of

*  for  direc3ons  from  π  to  2π#

Bounding ¤  maximum complexity of

¤  maximum complexity of

¤  The previous observation implies:

Bounding ¤  maximum complexity of

¤  maximum complexity of

¤  The previous observation implies:

¤  Formula for complexity of of Jordan curves [Sharir, Agarwal 95]

¤  Complexity of lower envelope of wedges [Edelsbrunner et al 82]

Bounding ¤  maximum complexity of

¤  maximum complexity of

¤  The previous observation implies:

¤  Formula for complexity of of Jordan curves [Sharir, Agarwal 95]

¤  Complexity of lower envelope of wedges [Edelsbrunner et al 82]

Structural complexity

¤  The number of unbounded faces:

¤  The total number of unbounded faces [this paper]:

¤  We can bound:

Structural complexity

¤  The number of faces in the order-k Voronoi diagram of n disjoint line-segments:

3. Planar Straight-Line Graph

Planar straight-line graph

¤  Challenge: Define order-k line-segment Voronoi diagram of a planar straight-line graph consistently ¤  Avoid artificial splitting of equidistant regions for abutting segments

that cause degeneracies

¤  Do not alter the definition for disjoint line-segments (using 3 sites per line-segment)

s1

s2v

b(s1, s2)

b(s1, s2)

s1

s2v b(s1, s2)b(s1, s2)

(a) (b)

s1

s2

v

b(s1, s2)

b(s1, s2)

(c)

Definition

¤ We extend the notion of the order-k Voronoi diargam.

¤ A set H is called an order-k subset iff

¤  type-1:

¤  type-2: , where and ,

is the set of line-segments incident to .

Representative

¤ Order-k Voronoi region:

x

p

I(p)

Planar straight-line graph

1

2

3

456

7

8

V (6, 5)

V (1, 6, 7)

V (1, 2) V (2, 3, 8)

V (3, 4)V (5) V (4)

V (3)

V (2)

V (1)

V (6)

V (7)

V (8)

V (4, 5)

V (7, 8)

Order-1 Voronoi Diagram of Planar Straight-Line Graph

Planar straight-line graph

Order-2 Voronoi Diagram of Planar Straight-Line Graph

1

2

3

456

7

8

V (1, 2)

V (6, 5) V (3, 4)

V (3, 8)

V (2, 8)

V (4, 5)

V (7, 5)

V (7, 8)

V (3, 4, 5)

V (8, 4, 5)

V (5, 7, 8)

V (2, 7)

V (1, 6, 7)

V (2, 3, 8)

V (1, 7)

V (6, 7)

Planar straight-line graph

Order-3 Voronoi Diagram of Planar Straight-Line Graph

1

2

3

456

7

8

V (2, 3, 8)

V (3, 4, 8)

V (3, 4, 5)

V (4, 5, 6)

V (1, 5, 6, 7)

V (5, 6, 7)

V (1, 6, 7)

V (1, 2, 6, 7)

V (1, 2, 3, 8)

V (1, 2, 8)V (1, 2, 7)

V (2, 7, 8)

V (1, 7, 8)V (3, 7, 8)

V (4, 5, 8)

V (4, 5, 7)

V (5, 7, 8)

V (6, 7, 8)

V (4, 5, 7, 8)

V (3, 4, 5, 8)

4. Intersecting Line-Segments

Intersecting line-segments

¤  Number of faces: ¤  Nearest neighbor Voronoi diagram of line-segments

¤  Farthest Voronoi diagram of line-segments

where - # of intersections

Intuitively, intersections influence small orders and the influence grows weaker as k increases.

Intersecting line-segments

¤  The number of faces in the order-k Voronoi diagram of n intersecting line-segments

Summary ¤  Lower bounds for disconnected regions

¤  Structural complexity for disjoint line-segments:

¤  Consistent definition for a planar straight-line graph.

¤  Structural complexity for intersecting line-segments:

References

1.  P. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27(3): 654-667 (1998)

2.  N. Alon and E. Gyori. The number of small semispaces of a finite set of points in the plane. J. Comb. Theory, Ser. A 41(1): 154-157 (1986)

3.  F. Aurenhammer, R. Drysdale, and H. Krasser. Farthest line segment Voronoi diagrams. Inf. Process. Lett. 100(6): 220-225 (2006)

4.  F. Aurenhammer and R. Klein Voronoi Diagrams in ”Handbook of computational geometry.” J.-R. Sack and J. Urrutia, North-Holland Publishing Co., 2000

5.  J.-D. Boissonnat, O. Devillers and M. Teillaud A Semidynamic Construction of Higher-Order Voronoi Diagrams and Its Randomized Analysis. Algorithmica 9(4): 329-356 (1993)

6.  H. Edelsbrunner. Algorithms in combinatorial geometry. EATCS monographs on theoretical computer science. Springer, 1987., Chapter 13.4

7.  H. Edelsbrunner, H. A. Maurer, F. P. Preparata, A. L. Rosenberg, E. Welzl and D. Wood. Stabbing Line Segments. BIT 22(3): 274-281 (1982)

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9.  D. T. Lee. On k-Nearest Neighbor Voronoi Diagrams in the Plane. IEEE Trans. Computers 31(6): 478-487 (1982)

10.  D. T. Lee, R. L. S. Drysdale. Generalization of Voronoi Diagrams in the Plane. SIAM J. Comput. 10(1): 73-87 (1981)

11.  E. Papadopoulou Net-Aware Critical Area Extraction for Opens in VLSI Circuits Via Higher-Order Voronoi Diagrams. IEEE Trans. on CAD of Integrated Circuits and Systems 30(5): 704-717 (2011)

12.  M. I. Shamos and D. Hoey. Closest-point problems. In Proc. 16th IEEE Symp. on Foundations of Comput. Sci.: 151-162 (1975)

13.  M. Sharir and P. Agarwal. Davenport-Schinzel Sequences and their Geometric Applications. Cambridge University Press, 1995., Chapter 5.4

14.  C.-K. Yap. An O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments. Discrete & Computational Geometry 2: 365-393 (1987)

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