1. Order-k Voronoi Diagram in the Plane
Computational Geometry, WS 2006/07
Lecture 16
Dr. Dominique Schmitt
Laboratoire MAGE - FST
Université de Haute-Alsace
Algorithmen & Datenstrukturen, Institut für Informatik
Fakultät für Angewandte Wissenschaften
Albert-Ludwigs-Universität Freiburg

2. (Very) Basic properties on circles

3. Voronoi diagram (reminder)
S : a set of n sites in the plane
V(s) : the set of points of the plane
closer to s than to any other site
= the Voronoi region of s
V(3)
Vor(S) : the partition of the plane
formed by the Voronoi regions,
their edges, and vertices

4. Voronoi regions, edges, and vertices
c(s,t) : the set of centers of all the
empty circles passing through
s and t
u(Q) : the center of the empty circle
passing through a set Q of at
least 3 co-circular sites
V(s) : the set of centers of all the
empty circles passing through s

5. Generalizations of the Voronoi diagram
d-dimensional space,
non-Euclidean space,
non-Euclidean distance,
weighted sites,
sites are not points,
k nearest neighbors
- ...

6. Farthest point Voronoi diagram
V-1(s) : the set of points
of the plane farther
from s than from any
other site
Vor-1(S) : the partition of the plane
formed by the farthest point Voronoi
regions, their edges, and vertices

7. Farthest point Voronoi regions
Construction of V-1(7)
Property
The farthest point Voronoi
regions are convex

8. Farthest point Voronoi regions
Property
If the farthest point Voronoi
region of s is non empty then
s is a vertex of conv(S)

9. Farthest point Voronoi regions
Property
If s is a vertex of conv(S)
then the farthest point
Voronoi region of s is
non empty
Property
The farthest point
Voronoi regions are
unbounded

10. Farthest point Voronoi regions
Property
If s is a vertex of conv(S)
then the farthest point
Voronoi region of s is
non empty
Property
The farthest point
Voronoi regions are
unbounded
Corollary
The farthest point
Voronoi edges and
vertices form a tree

11. Farthest point Voronoi edges and vertices
edge : set of points equidistant
from 2 sites and closer
to all the others
V-1(4) x
V-1(1)
=> c-1(s,t)

12. Farthest point Voronoi edges and vertices
edge : set of points equidistant
from 2 sites and closer
to all the others
V-1(4)
=> c-1(s,t)
V-1(7)
V-1(2)
vertex : point equidistant
from at least 3 sites and
closer to all the others
=> u-1(Q)

13. Duality in planar partitions (reminder)
Delaunay diagram : a partition
dual to the Voronoi diagram
=> denoted by Del(S)

14. Farthest point Delaunay diagram
Property
The farthest point
Delaunay edges do
not intersect
c-1(2,4)

15. Farthest point Delaunay regions
Property
The farthest point
Delaunay diagram of S
is a partition of conv(S)
in polygons inscribable
in “full” circles
u-1(2,4,7)
=> Construction in O(n log n)

16. Order-2 Voronoi diagram
V(s,t) : the set of points
of the plane closer
to each of s and t
than to any other site
Property
The order-2 Voronoi
regions are convex

17. Order-2 Voronoi regions
V(s,t) : the set of points
of the plane closer
to each of s and t
than to any other site
Property
The order-2 Voronoi
regions are convex
Construction of V(3,5)

18. Order-2 Voronoi edges edge : set of centers of circles passing through
2 sites s and t and
containing 1 site p
V(2,3)
c3(1,2)
=> cp(s,t)
V(1,3)
Question
Which are the regions
on both sides of cp(s,t) ?
=> V(p,s) and V(p,t)

19. Order-2 Voronoi vertices
vertex : center of a circle
passing through at least
3 sites and containing
either 1 or 0 site
vertex : center of a circle
passing through at least
3 sites and containing
1 site
u5(2,3,7)
=> up(Q)
or u(Q)
u(3,6,7,5)

20. Order-2 Voronoi vertices
vertex : center of a circle
passing through at least
3 sites and containing
either 1 or 0 site
V(5,2)
u5(2,3,7)
V(5,7)
=> up(Q)
or u(Q)
V(5,3)
Question
Which are the regions
incident to up(Q) ?
=> V(p,q) with q  Q

21. Order-2 Voronoi vertices
vertex : center of a circle
passing through at least
3 sites and containing
either 1 or 0 site
V(5,7)
V(3,5)
=> up(Q)
or u(Q)
u(3,6,7,5)
Question
Which are the regions
incident to up(Q) ?
V(3,6)
V(6,7)
=> V(p,q) with q  Q
Question
Which are the regions
incident to u(Q) ?
=> V(q,q’) with q and q’ consecutive on the circle circumscribed to Q

22. Construction of Vor2(S) from Vor(S)

23. Construction of Vor2(S) from Vor(S)
V(3) in Vor(S)
The set of points of the plane
closer to 3 than to any other
site of S
 V(3,5)
V(5) in Vor(S\{3})
The set of points of the plane
closer to 5 than to any other
site of S\{3}
=> The intersection of V(3) and
Vor(S\{3}) is a subset of Vor2(S)

24. Construction of Vor2(S) from Vor(S)
V(5) in Vor(S)
The set of points of the plane
closer to 5 than to any other
site of S
 V(3,5)
V(3) in Vor(S\{5})
The set of points of the plane
closer to 3 than to any other
site of S\{5}
=> The intersection of V(5) and
Vor(S\{5}) is a second subset
of Vor2(S)

25. Construction of Vor2(S) from Vor(S)

26. Construction of Vor2(S) from Vor(S)

27. Construction of Vor2(S) from Vor(S)
The algorithm
For every region V(s) of Vor(S)
- construct Vor(S\{s})
- compute the intersection of Vor(S\{s}) with V(s)
Stick the different pieces together
The time complexity
Construction of one Voronoi diagram of n sites => O(n log n)
Construction of n such diagrams => O(n2 log n)

28. Construction of Vor2(S) from Vor(S)

29. Vor(S) and Vor2(S)
V(3,5)
c(3,5)
V(6,7)
c(1,3)
V(1,3)
c(6,7)

30. Vor(S) and Vor2(S) Property
Every order-1 Voronoi edge c(s,t) is included in one and only one
order-2 Voronoi region V(s,t).
Conversely, every order-2 Voronoi region contains one and only
one order-1 Voronoi edge.
=> c(s,t)  V(s,t)

31. Vor(S) and Vor2(S) Property
Every order-1 Voronoi edge c(s,t) is included in one and only one
order-2 Voronoi region V(s,t).
Conversely, every order-2 Voronoi region contains one and only
one order-1 Voronoi edge. x y
y  c(s,t)
x  c(s,t)

32. Vor(S) and Vor2(S) Corollary The order-1 and order-2
Voronoi edges do not
intersect
Corollary
V(s,t) is an order-2 region iff
c(s,t) is an order-1 edge
<=>
V(s,t) is a region in Vor2(S) iff
V(s) and V(t) share an edge
in Vor(S)

33. Construction of Vor2(S) from Vor(S)
A better algorithm
For every region V(s) of Vor(S)
- let S’ be the set of sites whose regions in Vor(S)
share an edge with V(s)
- construct Vor(S’)
- compute the intersection of Vor(S’) with V(s)
Stick the different pieces together
Space complexity
Number of regions of Vor2(S) = number of edges of Vor(S) => O(n)

34. Construction of Vor2(S) from Vor(S)
Time complexity
ns = number of edges of V(s)
construction of 1 Vor(S’) => O(ns log ns)
construction of all Vor(S’) => O( (ns log ns))
< O(( ns ) log n)
< O(n log n)

35. Construction of Vor2(S) from Vor(S)
Theorem
The ordre-2 Voronoi diagram of n sites can be computed
from the order-1 Voronoi diagram in O(n) time.

36. Order-2 Delaunay diagram
Dual vertices
The dual of an order-2 Voronoi
region V(s,t) is the midpoint of
st.
=> g(s,t)

37. Order-2 Delaunay vertices
Property
The order-2 Delaunay
vertices are distinct points
Proof
g(s,t) a vertex of Del2(S)
<=> V(s,t) a region of Vor2(S)
<=> c(s,t) an edge of Vor(S)
<=> st an edge of Del(S)

38. Order-2 Delaunay edges Dual edges The dual of an order-2
Voronoi edge cp(s,t) is
the segment connecting
g(p,s) and g(p,t).
=> ep(s,t)
g(4,1)
g(4,6)
e4(1,6)
V(4,1)
V(4,6)
c4(1,6)
ep(s,t) : image of st by an homothety
of center p and ratio 1/2

39. Order-2 Delaunay edges Property
The ordre-2 Delaunay edges do not intersect.
Proof
Let ep(s,t) and ep’(s’,t’) be 2 distinct order-2 Delaunay edges.

40. Order-2 Delaunay edges Property
The ordre-2 Delaunay edges do not intersect. 
Exercise
Deal with the other cases.

41. Order-2 Delaunay regions
Region dual of up(Q)
The polygon with vertices
g(p,q) with q  Q.
Region dual of u(Q)
The polygon with vertices
g(q,q’) with {q,q’} consecutive
on the circle circumscribed
to Q.

42. Constructing Del2(S) from Del(S)
For every region of Del(S)
compute its inscribed polygon
For every site s of S
- determine the polygon of the
neighbors of s in Del(S)
- construct the order-1 Delaunay
diagram of this polygon
- compute its image by an homothety
with center s and ratio 1/2
Stick the different pieces together

43. Order-k Voronoi and Delaunay diagrams
V(T) : the set of points
of the plane closer
to each site of T
than to any other site
(with |T| = k)
g(T) : the center of gravity
of T

44. Order-k Voronoi and Delaunay diagrams
Theorem
The size of the order-k
diagrams is O(k(n-k))
Theorem
The order-k diagrams
can be constructed from
the order-(k-1) diagrams
in O(k(n-k)) time
Corollary
The order-k diagrams can
be iteratively constructed
in O(n log n + k2(n-k)) time