We analyze structural properties of the order-k Voronoi dia- gram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-k Voronoi regions of line segments may be disconnected; in fact a single order- k Voronoi region may consist of Ω(n) disjoint faces. Nevertheless, the structural complexity of the order-k Voronoi diagram of non-intersecting segments remains O(k(n − k)) similarly to points. For intersecting line segments the structural complexity remains O(k(n − k)) for k ≥ n/2.

1. 23rd
Interna3onal
Symposium
on
Algorithms
and
Computa3on,
ISAAC
2012
Taipei,
Taiwan,
December
2012
On
higher
order
Voronoi
diagrams
of
line
segments
Maksym Zavershynskyi
Evanthia Papadopoulou
University
of
Lugano,
Switzerland
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.

2. 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

3. 1. Introduction

4. 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.

5. 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

6. 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!

7. 2. Disjoint Line-Segments
a) Disconnected regions

8. Disconnected Regions
A single order-k Voronoi region may disconnect t
faces
2-­‐order
Voronoi
diagram

9. Disconnected Regions
An order-k Voronoi region may disconnect
to bounded faces.
For
Order-­‐2
Voronoi
diargam
of
6
line-­‐segments

10. Disconnected Regions
An order-k Voronoi region may disconnect
to unbounded faces.
For
k
n-k
Order-­‐4
Voronoi
diargam
of
7
line-­‐segments
*
Generalizing
[Aurenhammer
et
al
06].

11. Disconnected Regions
An order-k Voronoi region may disconnect
to unbounded faces.
For F1
Line-­‐segments
can
be
untangled!
F2
F3
Order-­‐4
Voronoi
diargam
of
7
line-­‐segments
F4
*
Generalizing
[Aurenhammer
et
al
06].

12. 2. Disjoint Line-Segments
b) Differences with points

13. 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

14. 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

15. 2. Disjoint Line-Segments
c) Structural complexity

16. 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 (S)
Vk−1 (S)

17. Structural complexity
¤ Generalizing Lee’s approach we prove:
¤ For this formula already implies
¤ For we need to bound

18. Bounding
¤ We use well-known point-line duality transformation.
¤ We transform every line-segment to a wedge [Aurenhammer et al 06]

19. Bounding
¤ Consider an arrangement of dual wedges
w5
w4
w3
w2
p q
w1
¤ An unbounded edge in order-k Voronoi diagram corresponds to a
vertex of
*
for
direc3ons
from
π
to
2π#

20. Bounding
¤ maximum complexity of
¤ maximum complexity of
¤ The previous observation implies:

21. 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]

22. 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]

23. Structural complexity
¤ The number of unbounded faces:
¤ The total number of unbounded faces [this paper]:
¤ We can bound:

24. Structural complexity
¤ The number of faces in the order-k Voronoi diagram of n
disjoint line-segments:

25. 3. Planar Straight-Line Graph

26. 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 s1
b(s1, s2)
b(s1, s2) v b(s1, s2) v b(s1, s2) b
s2 s2
(a) (b)

27. Definition
¤ We extend the notion of the order-k Voronoi diargam.
¤ A set H is called an order-k subset iff
¤ type-1: I(p)
¤ type-2: , where and , p
is the set of line-segments incident to . x
Representative
¤ Order-k Voronoi region:

28. Planar straight-line graph
Order-1 Voronoi Diagram
of Planar Straight-Line Graph
V (6, 5) V (3, 4)
V (5) V (4)
V (6) 5 4
6
V (1, 6, 7) V (7, 8)
7 V (4, 5) 3
V (7)
8 V (3)
1
V (1)
V (8)
2
V (1, 2) V (2, 3, 8)
V (2)

29. Planar straight-line graph
Order-2 Voronoi Diagram
of Planar Straight-Line Graph
V (6, 5) V (4, 5) V (3, 4)
V (5, 7, 8)
6 5 4
V (1, 6, 7)
V (7, 5)
V (6, 7)
7 V (3, 4, 5)
V (1, 7) V (7, 8) 3
1 8
V (3, 8)
V (8, 4, 5)
V (2, 7) V (2, 8)
2 V (2, 3, 8)
V (1, 2)

30. Planar straight-line graph
V (1, 5, 6, 7)
V (4, 5, 6) Order-3 Voronoi Diagram
of Planar Straight-Line Graph
V (5, 6, 7) V (3, 4, 5)
V (4, 5, 7)
V (4, 5, 7, 8)
5 4
6
V (6, 7, 8)
V (1, 6, 7)
V (5, 7, 8) V (4, 5, 8)
7 V (3, 4, 8)
V (1, 7, 8)
V (3, 7, 8)
V (1, 2, 6, 7) V (3, 4, 5, 8)
V (2, 7, 8) 3
8
1
V (2, 3, 8)
V (1, 2, 7) 2
V (1, 2, 8)

31. 4. Intersecting Line-Segments

32. 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.

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

34. 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:

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36. Thank you!
h[p://zavermax.github.com