1. Voronoi Diagram Presenter: GI1 11號 蔡逸凡
Reference: Computational Geometry ch7
ISBN:

2. Outline Introduction Definitions Incremental Algorithm
Fortune’s Algorithm
Hardware Algorithm
Applications

3. Where do people go to shop?
Which location is suitable for new store?
7-11 Shoppers …
Assumptions
price is the same at every site
cost = price + transportation
transportation = (Euclidean distance)  (price per unit distance)
consumers try to minimize the cost.
Figure 7.1
The trading areas of the capitals of the twelve provinces in the Netherlands, as predicted by the Voronoi assignment model

4. Introduction (cont) Fire observation towers Towers on fire
Facility location: center of largest empty circle
Path planning: stay as far away from all obstacles as possible

5. Definitions
Vor(P) [Voronoi diagram of P]: subdivision of the plane into n cells, one for each site in P,
[Voronoi cell of pi]:the cell of Vor(P) that corresponds to a site pi
V(P): the set of all points that have more than one nearest neighbor

6. Definition (Cont) h (p, q) h (q, p) bisector p3 p2 p1
Circumcircle of p1p2p3
Two sites
Three sites

7. Observation 7.1

8. Theorem 7.2
Let P be a set of n point sites in the plane. If all the sites are collinear then Vor(P) consists of n-1 parallel lines. Otherwise, Vor(P) is connected and its edges are either segments or half-lines.

9. Give a bound to number of Voronoi edges and vertices
Theorem 7.3
For n≧3, the number of vertices in the Voronoi diagram of a set of n point sites in the plane is at most 2n-5 and the number of edges is at most 3n-6.
Give a bound to number of Voronoi edges and vertices

10. CP(q): largest empty circle of q w.r.t. P
To characterize which bisectors and intersections define features of the Voronoi diagram we make the following:
not contain any
site of P

11. Theorem 7.4
For the Voronoi diagram Vor(P) of a set of points P the following holds:
(i) A point q is a vertex
of Vor(P) if and only if
its largest CP(q) empty
circle contains three or
more sites on its
boundary. q

12. Theorem 7.4 (Cont)
(ii) The bisector between sites pi and pj defines an edge of Vor(P) if and only if there is a point q on the bisector such that CP(q) contains both pi and pj on its boundary but no other site. q

13. Theorem 7.4 (Cont)
(ii) The bisector between sites pi and pj defines an edge of Vor(P) if and only if there is a point q on the bisector such that CP(q) contains both pi and pj on its boundary but no other site. q

14. Incremental Algorithm
Suppose that we have already built the Voronoi diagram Vp-1, and would like to add a new sites sp. First, find the site, say si, whose Voronoi polygon contains sp, and draw the perpendicular bisector between sp and si, denoted by B(sp, si).
The bisector crosses the boundary of V(si) at two points, point x1 and point x2. Site sp is to the left of the directed line segment x1x2. The line segment x1x2 divides the Voronoi polygon V(si) into two pieces, the one on the left belonging to the Voronoi polygon of sp. Thus, we get a Voronoi edge on the boundary of the Voronoi polygon of si.

15. Incremental Algorithm-2
Starting with the edge x1x2, expand the boundary of the Voronoi polygon of sl by the following procedure. The B(si, sl) crosses the boundary of V(si) at x2, entering the adjacent Voronoi polygon, say V(sj). Therefore, next draw the B(si, sj), and find the point, x3, at which the bisector crosses the boundary of V(sj). Similarly, find the sequence of segments of perpendicular bisectors of s and the neighboring sites until we reach the starting point x1.
Let this sequence be (x1x2, x2x3, …, xm-1xm, xmx1). This sequence forms a CCW boundary of the Voronoi polygon of the new site s.
Finally, we delete from Vp-1 the substructure inside the new Voronoi polygon, and thus get Vp.

16. Example

17. Example

18. Example

19. Example

20. Example
O(n) per point insertion
Total complexity: O(n2)

21. Complexity The method describe before: plane sweep algorithm—
per Voronoi cell:
total:
plane sweep algorithm—
Fortune’s algorithm:

22. Fortune’s Algorithm (1985)
Use plane sweep to explore the Voronoi structure
The loci of equal distance points between the focus and the directrix is a parabola
Sweep line

23. Fortune (cont) As the sweep line moves, more parabolas got generated
The intersection between parabola is the trace of equal distance point between the two sites
Beach line: monotone union of parabolic arcs
Break point: where two arcs meet

24. Site Event (Voronoi Edge)
The branch starts when the sweep line first touches the site, forming a degenerate parabola (a line)

25. Circle Event (Voronoi Vertex)
The Voronoi edge comes to an end when the circumcircle pass the sweep line
At such points, the corresponding arc got removed from the beach line

26. Plane Sweep Algorithm (Cont)
:sweep line
:The closed half-plane above 12 q q s1 s2 s3

27. Plane Sweep Algorithm (Cont)
beach line
breakpoint

28. Observation 7.5
The beach line is x-monotone, that is, every vertical line intersects it in exactly one point. p1 p2 p3 p4

29. Process – Site event
When reaching a new site, we consider the events where a new arc appears on the beach line

30. Process – Site event (Cont)
What happen to the Voronoi diagram at a site event?

31. Lemma 7.6
The only way in which a new arc can appear on the beach line is through a site event.

32. Process – Circle event

33. Lemma 7.7
The only way in which an existing arc can disappear from the beach line is through a circle event.

34. The detection of circle event
Does not define a potential circle event
False alarm

35. This circle event helps to terminate the Voronoi edges
Sweep line intersects the circumcircle

36. This, however, does not generate circle event: cir
This, however, does not generate circle event: cir.circle does not intersect sweep line [site coincide w/ circle event]

37. Summary Site event: encounter a new site - get new arc Circle event:
A new triple has converging breakpoint
– has a circle event insert into the event queue
The new arc is in the middle
– never cause
Check disappear triples that have circle event
– false alarm, delete it from event queue

38. Structure Event queue: the priority of an event is its y-coordinate
Site event: store the site
Circle event: lowest point of the circle
Doubly-connected edge list
Binary search tree T

39. Structure (Cont) Binary search tree T breakpoint
Site that define the arc
breakpoint
Binary search tree T

40. Example 1 p1 p2 p3
Q={p1,p2,p3}
D={}
Q={p2,p3}
D={} T: p1

41. Example 1 (Cont)
Q={p3}
D={e1} T: p2 p1
<p2,p1>->e1

42. Example 1 (Cont) Q={} D={e1,e2} T: <p1,p3>->e2 p3 p2 p1

43. Example 1 (Cont) Q={c1} D={e1,e2} T: <p1,p3>->e2 c1 p3 p2 p1
e1<-<p2,p1>

44. Example 1 (Cont)
Q={}
D={e1,e2,e3} T: p2 p3
e3<-<p2,p3>

45. Example 1 (Cont) Q={} D={e1,e2,e3} T: p2 p3 e3<-<p2,p3>
A visual implementation of Fortune's Voronoi algorithm

46. Example 2
Q={p1,p2,p3,p4}
D={}
Q={p2,p3,p4}
D={} T: p1

47. Example 2 (Cont)
Q={p3,p4}
D={e1} T: p1 p2
<p1,p2>->e1

48. Example 2 (Cont) Q={c1,p4} D={e1,e2} Q={p4} D={e1,e2} T:
<p1,p2>->e1 T:
<p1,p2>->e1 c1 p2 p3 p1
e2<-<p3,p1> p2 p3 p1
e2<-<p3,p1>

49. Example 2 (Cont) Q={p4} D={e1,e2,e3} T: Q={} D={e1,e2,e3 e4,e5} T: p3
e3<-<p3,p2>
<p4,p2>->e5 p2 p3 p4
e4<-<p3,p4>

50. Example 2 (Cont) Q={} D={e1,e2,e3 e4,e5} T: <p4,p2>->e5 p2 p3

51. Example 3
Q={p1,p2,p3,p4}
D={}
Q={p2,p3,p4}
D={} T: p1

52. Example 3 (Cont)
Q={p3,p4}
D={e1} T: p2 p1
<p2,p1>->e1

53. Example 3 (Cont) Q={p4,c1} D={e1,e2} Q={p4} D={e1,e2} T:
<p1,p3>->e2 T:
<p1,p3>->e2 c1 p3 p2 p1
e1<-<p2,p1> p3 p2 p1
e1<-<p2,p1>

54. Example 3 (Cont)

55. Example 3 (Cont) Q={p4} D={e1,e2} T: <p1,p3>->e2 p3 p2 p1

56. Example 3 (Cont)

57. Example 3 (Cont)

58. Proofs

59. Theorem 7.2 (Cont)
Proof:
(1) Collinear: Vor(P) consists of n parallel lines
．．． 1 2 3
n - 1 n
n – 1 parallel lines

60. Theorem 7.2 (Cont)
Proof:
(2) Vor(P) is connected and its edges are either segments or half-lines.
→ suppose : an edge e of Vor(P) that is a full line f

61. Theorem 7.2 (Cont)
Proof:
(2) Vor(P) is connected and its edges are either segments or half-lines.
→ suppose : an edge e of Vor(P) that is a full line

62. Theorem 7.3 (Cont)

63. Lemma 7.6 (Cont)
Proof:
First possibility: suppose an already existing parabola break through the beach line, is defined by a site

64. Lemma 7.6 (Cont) Proof: Let denote the y-coordinate of the sweep line12
The parabola is given by:

65. Lemma 7.6 (Cont)
Proof:
Second possibility: appear in between two arcs and

66. Lemma 7.6 (Cont)
Proof:

67. Lemma 7.6 (Cont) Consequence:
The beach line consists of at most 2n-1 parabolic arcs.