Higher Order City Voronoi Diagrams Andreas Gemsa 1 , D.T. Lee 2 , 3 , - - PowerPoint PPT Presentation

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Higher Order City Voronoi Diagrams Andreas Gemsa 1 , D.T. Lee 2 , 3 , - - PowerPoint PPT Presentation

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Higher Order City Voronoi Diagrams Andreas Gemsa 1 , D.T. Lee 2 , 3 , Chih-Hung Liu 1 , 2 , Dorothea Wagner 1 1 Karlsruhe Institute of Technology (KIT), 2 Academia Sinica, 3 National Chung Hsing University. Institute of Theoretical Informatics

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SLIDE 1

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Andreas Gemsa1, D.T. Lee2,3, Chih-Hung Liu1,2, Dorothea Wagner1

1Karlsruhe Institute of Technology (KIT), 2Academia Sinica, 3National Chung Hsing University.

Higher Order City Voronoi Diagrams

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SLIDE 2

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order City Voronoi Diagrams

Higher Order Voronoi Diagrams? City Voronoi Diagrams?

Q: Q:

slide-3
SLIDE 3

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

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SLIDE 4

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

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SLIDE 5

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: [first order] Voronoi diagram V1(S)

slide-6
SLIDE 6

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: [first order] Voronoi diagram V1(S)

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SLIDE 7

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: [first order] Voronoi diagram V1(S)

slide-8
SLIDE 8

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: [first order] Voronoi diagram V1(S)

first-order farthest-site

kth-order

O(n) O(n)

Structural Complexity

O(k(n − k))

slide-9
SLIDE 9

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: [first order] Voronoi diagram V1(S)

first-order farthest-site

kth-order

O(n) O(n)

Structural Complexity

O(k(n − k))

first-order farthest-site

kth-order

Time Complexity

O(n log n) O(n log n) O(k2n log n)

slide-10
SLIDE 10

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: [first order] Voronoi diagram V1(S)

slide-11
SLIDE 11

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: second order Voronoi diagram V2(S)

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SLIDE 12

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: second order Voronoi diagram V2(S)

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SLIDE 13

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: second order Voronoi diagram V2(S)

slide-14
SLIDE 14

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: second order Voronoi diagram V2(S)

slide-15
SLIDE 15

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given: second order Voronoi diagram V2(S)

slide-16
SLIDE 16

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

k-th order Voronoi diagram Vk(S)

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SLIDE 17

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

k-th order Voronoi diagram Vk(S)

first-order farthest-site

kth-order

O(n) O(n)

Structural Complexity

O(k(n − k))

first-order farthest-site

kth-order

Time Complexity

O(n log n) O(n log n) O(k2n log n)

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SLIDE 18

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

(n − 1)-th order Voronoi diagram Vn−1(S)

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SLIDE 19

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

(n − 1)-th order Voronoi diagram Vn−1(S)

slide-20
SLIDE 20

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

(n − 1)-th order Voronoi diagram Vn−1(S)

slide-21
SLIDE 21

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

(n − 1)-th order Voronoi diagram Vn−1(S)

slide-22
SLIDE 22

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order Voronoi Diagrams Set S of n points in the plane Given:

(n − 1)-th order Voronoi diagram Vn−1(S)

first-order farthest-site

kth-order

O(n) O(n)

Structural Complexity

O(k(n − k))

first-order farthest-site

kth-order

Time Complexity

O(n log n) O(n log n) O(k2n log n)

slide-23
SLIDE 23

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order City Voronoi Diagrams

Higher Order Voronoi Diagrams? City Voronoi Diagrams?

Q: Q:

slide-24
SLIDE 24

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

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SLIDE 25

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

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SLIDE 26

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

shortest path:

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SLIDE 27

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk shortest path:

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SLIDE 28

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk shortest path: quickest path?

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SLIDE 29

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk walk shortest path: quickest path?

slide-30
SLIDE 30

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk walk shortest path: quickest path?

slide-31
SLIDE 31

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk walk shortest path: quickest path?

slide-32
SLIDE 32

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk walk walk shortest path: quickest path?

slide-33
SLIDE 33

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk walk walk shortest path: quickest path?

L1 metric

city metric

slide-34
SLIDE 34

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk walk walk shortest path: quickest path?

L1 metric

city metric

slide-35
SLIDE 35

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

walk walk walk shortest path: quickest path?

L1 metric

city metric

L1 metric and transportation network

city metric:

slide-36
SLIDE 36

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

slide-37
SLIDE 37

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric: Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-38
SLIDE 38

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-39
SLIDE 39

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

speed off the network: 1 speed on the network: v > 1 Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-40
SLIDE 40

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

speed off the network: 1 speed on the network: v > 1 Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-41
SLIDE 41

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

speed off the network: 1 speed on the network: v > 1

c := |VC|

Complexity of G: O(c) Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-42
SLIDE 42

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

distance between two points? speed off the network: 1 speed on the network: v > 1

p q

c := |VC|

Complexity of G: O(c) Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-43
SLIDE 43

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

distance between two points?

L1

speed off the network: 1 speed on the network: v > 1

p q

c := |VC|

Complexity of G: O(c) Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-44
SLIDE 44

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

distance between two points?

L1

speed off the network: 1 speed on the network: v > 1

p q

c := |VC|

Complexity of G: O(c) Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-45
SLIDE 45

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

City Metric

L1 metric and transportation network

city metric:

G

distance between two points? quickest path

L1

speed off the network: 1 speed on the network: v > 1

p q

dC(p, q) c := |VC|

Complexity of G: O(c) Def.: transportation network graph G = (VC, EC) planar, straight-line embedding

  • nly horizontal and vertical edges
slide-46
SLIDE 46

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Higher Order City Voronoi Diagrams

Higher Order Voronoi Diagrams? City Voronoi Diagrams?

Q: Q:

slide-47
SLIDE 47

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Our Contribution

first-order farthest-site

kth-order

Θ(n) Θ(n)

Structural Complexity

Θ(k(n − k))

first-order farthest-site

kth-order

Time Complexity

O(n log n) O(n log n) O(k2n log n)

Structural Complexity Time Complexity

O((n + c) log n) O(nc log n log2(n + c))

first-order farthest-site

kth-order

first-order farthest-site

kth-order

Euclidean-Metric City-Metric

[Bae et al., 2012] [Bae et al., 2012]

Θ(n + c) Θ(nc) O(k(n − k) + kc) Ω(n + kc) O(k2(n + c) log(n + c))

slide-48
SLIDE 48

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Our Contribution

first-order farthest-site

kth-order

Θ(n) Θ(n)

Structural Complexity

Θ(k(n − k))

first-order farthest-site

kth-order

Time Complexity

O(n log n) O(n log n) O(k2n log n)

Structural Complexity Time Complexity

O((n + c) log n) O(nc log n log2(n + c))

first-order farthest-site

kth-order

first-order farthest-site

kth-order

Euclidean-Metric City-Metric

[Bae et al., 2012] [Bae et al., 2012]

Θ(n + c) Θ(nc) O(k(n − k) + kc) Ω(n + kc) O(k2(n + c) log(n + c))

slide-49
SLIDE 49

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – Euclidean Metric

slide-50
SLIDE 50

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – Euclidean Metric

slide-51
SLIDE 51

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – Euclidean Metric

slide-52
SLIDE 52

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – Euclidean Metric

slide-53
SLIDE 53

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – Euclidean Metric

slide-54
SLIDE 54

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – Euclidean Metric

slide-55
SLIDE 55

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-56
SLIDE 56

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-57
SLIDE 57

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-58
SLIDE 58

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-59
SLIDE 59

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-60
SLIDE 60

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-61
SLIDE 61

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-62
SLIDE 62

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – L1 Metric

slide-63
SLIDE 63

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-64
SLIDE 64

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-65
SLIDE 65

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-66
SLIDE 66

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-67
SLIDE 67

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-68
SLIDE 68

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-69
SLIDE 69

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-70
SLIDE 70

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-71
SLIDE 71

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-72
SLIDE 72

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-73
SLIDE 73

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-74
SLIDE 74

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-75
SLIDE 75

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-76
SLIDE 76

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-77
SLIDE 77

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-78
SLIDE 78

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-79
SLIDE 79

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-80
SLIDE 80

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-81
SLIDE 81

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-82
SLIDE 82

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-83
SLIDE 83

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-84
SLIDE 84

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-85
SLIDE 85

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-86
SLIDE 86

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-87
SLIDE 87

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-88
SLIDE 88

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-89
SLIDE 89

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

[Aichholzer et al., 2004]

O(c) activation points

slide-90
SLIDE 90

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

slide-91
SLIDE 91

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

arrow

slide-92
SLIDE 92

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

arrow

slide-93
SLIDE 93

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

activation point arrow

slide-94
SLIDE 94

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

activation point arrow

slide-95
SLIDE 95

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

arrow

slide-96
SLIDE 96

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

arrow

slide-97
SLIDE 97

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

arrow

slide-98
SLIDE 98

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

arrow

slide-99
SLIDE 99

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

arrow

slide-100
SLIDE 100

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r

arrow

slide-101
SLIDE 101

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r dC(p, q)

’quickest path’ from q to r arrow

slide-102
SLIDE 102

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Wavefront Propagation – City Metric

p q′ q r dC(p, q)

’quickest path’ from q to r

[Bae et al., 2009]

Needle: Weighted network segment

q′ q

arrow

slide-103
SLIDE 103

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

slide-104
SLIDE 104

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q) |BC(p, q)| = Θ(c)

slide-105
SLIDE 105

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q) |BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

slide-106
SLIDE 106

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices

slide-107
SLIDE 107

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices

slide-108
SLIDE 108

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices # of mixed vertices

slide-109
SLIDE 109

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-110
SLIDE 110

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-111
SLIDE 111

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-112
SLIDE 112

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-113
SLIDE 113

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-114
SLIDE 114

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-115
SLIDE 115

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-116
SLIDE 116

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-117
SLIDE 117

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-118
SLIDE 118

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-119
SLIDE 119

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p

slide-120
SLIDE 120

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p r

slide-121
SLIDE 121

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p r

slide-122
SLIDE 122

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Shortest Path Map – Example

q p p r

slide-123
SLIDE 123

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-124
SLIDE 124

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-125
SLIDE 125

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

BC(p, q) p q

slide-126
SLIDE 126

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

BC(p, q) p q

slide-127
SLIDE 127

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

BC(p, q) p q m4 m2 m3 m1

slide-128
SLIDE 128

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

Mixed Vertices in BC(p, q)

BC(p, q) p q m4 m2 m3 m1

slide-129
SLIDE 129

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

Mixed Vertices in BC(p, q)

BC(p, q) p q m4 m2 m3 m1

intersection of three shortest path regions

slide-130
SLIDE 130

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

Mixed Vertices in BC(p, q)

BC(p, q) p q m4 m2 m3 m1

intersection of three shortest path regions Path between mi and mi+1?

Q:

slide-131
SLIDE 131

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

Mixed Vertices in BC(p, q)

BC(p, q) p q m4 m2 m3 m1

intersection of three shortest path regions

→ L1 bisector between two needles

Path between mi and mi+1?

Q:

slide-132
SLIDE 132

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

Mixed Vertices in BC(p, q)

BC(p, q) p q m4 m2 m3 m1

intersection of three shortest path regions

→ L1 bisector between two needles

bisector has complexity O(1) Path between mi and mi+1?

Q:

[Bae and Chwa, 2005]

slide-133
SLIDE 133

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

Mixed Vertices in BC(p, q)

BC(p, q)

Lemma 1

p q m4 m2 m3 m1

intersection of three shortest path regions

Voronoi edge e has m mixed vertices.

→ L1 bisector between two needles

bisector has complexity O(1) Path between mi and mi+1?

Q:

[Bae and Chwa, 2005]

slide-134
SLIDE 134

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

Mixed Vertices in BC(p, q)

BC(p, q)

Lemma 1

p q m4 m2 m3 m1

intersection of three shortest path regions

Voronoi edge e has m mixed vertices. Then e consists of O(m + 1) segments.

→ L1 bisector between two needles

bisector has complexity O(1) Path between mi and mi+1?

Q:

[Bae and Chwa, 2005]

slide-135
SLIDE 135

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Mixed Voronoi Vertices and An Upper Bound

BC(p, q)

Lemma 1 Lemma 2

|Vk(S)| = O(k(n − k) + M), M = #mixed vertices of Vk(S)

[Lee, 1982]

p q m4 m2 m3 m1

Voronoi edge e has m mixed vertices. Then e consists of O(m + 1) segments.

#edges is O(k(n − k))

|ei| = O(mi + 1) (Lemma 1) M = mi

slide-136
SLIDE 136

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-137
SLIDE 137

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-138
SLIDE 138

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-139
SLIDE 139

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S)

q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-140
SLIDE 140

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S)

q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-141
SLIDE 141

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S) V1(Q)

q1 q2 q3 q4 q5 q6

Q = S \ {p}

q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-142
SLIDE 142

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S) Q = S \ {p} V1(Q)

q1 q2 q3 q4 q5 q6 q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-143
SLIDE 143

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S) Q = S \ {p} V1(Q)

q1 q2 q3 q4 q5 q6 q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-144
SLIDE 144

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S) Q = S \ {p} V1(Q)

q1 q2 q3 q4 q5 q6 q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-145
SLIDE 145

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S) Q = S \ {p} V1(Q)

q1 q2 q3 q4 q5 q6 q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-146
SLIDE 146

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S) Q = S \ {p} V1(Q)

q1 q2 q3 q4 q5 q6 q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-147
SLIDE 147

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

Q = S \ {p} V1(S) ∩ V1(Q)

q1 q2 q3 q4 q5 q6 p

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-148
SLIDE 148

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

Q = S \ {p} V1(S) ∩ V1(Q)

q1 q2 q3 q4 q5 q6 p

V2(S)

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-149
SLIDE 149

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk

V1(S) V1(Q)

q1 q2 q3 q4 q5 q6 q1 q2 q3 q4 q5 q6 p

V2(S)

Euclidean Metric

Vj+1(S) can be constructed from Vj(S).

[Lee, 1982]

slide-150
SLIDE 150

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Wavefront

q1 q2 p q6 q5 q4 q3

How to transfer this approach to wavefront propagation? Propagate a wavefront from every qi and not from p

Q:

Euclidean Metric

slide-151
SLIDE 151

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Wavefront

q1 q2 p q6 q5 q4 q3

How to transfer this approach to wavefront propagation? Propagate a wavefront from every qi and not from p

Q:

Euclidean Metric

slide-152
SLIDE 152

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Wavefront

q1 q2 p q6 q5 q4 q3

How to transfer this approach to wavefront propagation? Propagate a wavefront from every qi and not from p

Q:

Euclidean Metric

slide-153
SLIDE 153

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Wavefront

q1 q2 p q6 q5 q4 q3

How to transfer this approach to wavefront propagation? Propagate a wavefront from every qi and not from p

Q:

Euclidean Metric

slide-154
SLIDE 154

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Wavefront

q1 q2 p q6 q5 q4 q3

How to transfer this approach to wavefront propagation? Propagate a wavefront from every qi and not from p

Q:

Euclidean Metric

slide-155
SLIDE 155

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Wavefront

q1 q2 p q6 q5 q4 q3

How to transfer this approach to wavefront propagation? Propagate a wavefront from every qi and not from p

Q:

Euclidean Metric

slide-156
SLIDE 156

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

BC(p, q) p q

wavefront propagation only from q before:

slide-157
SLIDE 157

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q before:

slide-158
SLIDE 158

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

slide-159
SLIDE 159

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

slide-160
SLIDE 160

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

slide-161
SLIDE 161

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

slide-162
SLIDE 162

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

slide-163
SLIDE 163

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

slide-164
SLIDE 164

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

Q:Connection with mixed

vertices?

slide-165
SLIDE 165

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

Q:Connection with mixed

vertices? #mixed vertices in Vj(S)

≤

slide-166
SLIDE 166

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

Q:Connection with mixed

vertices? #mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

≤

slide-167
SLIDE 167

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

Q:Connection with mixed

vertices? #mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

2#(new wavefronts) by q ∈S

+

≤

slide-168
SLIDE 168

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – City Metric

q1

BC(p, q) p q

q′

1

q2

wavefront propagation only from q now: wavefront propagation from needles before:

Q:Connection with mixed

vertices? #mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

2#(new wavefronts) by q ∈S

+ +

complexity

  • f

the trans- portation network: O(c)

≤

slide-169
SLIDE 169

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Upper Bound

#mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

+ +

complexity

  • f

the trans- portation network: O(c)

≤ 2#(new wavefronts) by q ∈S

slide-170
SLIDE 170

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Upper Bound

#mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

+ +

complexity

  • f

the trans- portation network: O(c)

≤

#mixed vertices in Vk(S)

= 2#(new wavefronts) by q ∈S

slide-171
SLIDE 171

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Upper Bound

#mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

+ +

complexity

  • f

the trans- portation network: O(c)

≤

  • #(new wavefronts) = O(n)

#mixed vertices in Vk(S)

= 2#(new wavefronts) by q ∈S

slide-172
SLIDE 172

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Upper Bound

#mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

+ +

complexity

  • f

the trans- portation network: O(c)

≤ O(c) = O(kc)

  • #(new wavefronts) = O(n)

#mixed vertices in Vk(S)

=

+

2#(new wavefronts) by q ∈S

slide-173
SLIDE 173

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Upper Bound

#mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

+ +

complexity

  • f

the trans- portation network: O(c)

≤ O(c) = O(kc)

  • #(new wavefronts) = O(n)

#mixed vertices in Vk(S)

=

+

Lemma 2

|Vk(S)| = O(k(n − k) + M), M = #mixed vertices of Vk(S) 2#(new wavefronts) by q ∈S

slide-174
SLIDE 174

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Upper Bound

#mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

+ +

complexity

  • f

the trans- portation network: O(c)

≤ O(c) = O(kc)

  • #(new wavefronts) = O(n)

#mixed vertices in Vk(S)

=

+

Lemma 2

|Vk(S)| = O(k(n − k) + M), M = #mixed vertices of Vk(S)

Theorem 1

|Vk(S)| = O(k(n − k) + kc) 2#(new wavefronts) by q ∈S

slide-175
SLIDE 175

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Upper Bound

#mixed vertices in Vj(S) #mixed vertices in Vj−1(S)

+ +

complexity

  • f

the trans- portation network: O(c)

≤ O(c) = O(kc)

  • #(new wavefronts) = O(n)

#mixed vertices in Vk(S)

=

+

Lemma 2

|Vk(S)| = O(k(n − k) + M), M = #mixed vertices of Vk(S)

Theorem 1

|Vk(S)| = O(k(n − k) + kc) 2#(new wavefronts) by q ∈S

slide-176
SLIDE 176

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-177
SLIDE 177

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Structural Complexity – Overview

p q

BC(p, q)

Mixed Vertices

structural complexity of Vk(S)

Wavefront Propagation

Iterative Construction of Vk(S)

|BC(p, q)| = Θ(c)

Shortest Path Map (SPM)

# of mixed vertices structural complexity of Vk(S) # of mixed vertices

slide-178
SLIDE 178

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Algorithm

2009).

slide-179
SLIDE 179

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Algorithm

2009).

compute V1(S)

slide-180
SLIDE 180

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Algorithm

2009).

compute V1(S) From Vj to Vj+1:

(Vj+1(S) from Vj(S))

q1 q2 p q6 q5 q4 q3

slide-181
SLIDE 181

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Algorithm

2009).

compute V1(S) From Vj to Vj+1:

p q q′

1

q2 q1

slide-182
SLIDE 182

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Algorithm

2009).

compute V1(S) For every Voronoi region R: compute set N of relevant needles determine V1(N) calculate intersection V1(N) ∩ R with this: determine Vj+1(S) ∩ R From Vj to Vj+1:

p q q′

1

q2 q1

slide-183
SLIDE 183

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Algorithm

2009).

compute V1(S) For every Voronoi region R: compute set N of relevant needles determine V1(N) calculate intersection V1(N) ∩ R with this: determine Vj+1(S) ∩ R Computing V1(·) with algorithm by Bae et al. From Vj to Vj+1:

[Bae, Kim, Chwa, 2009]

p q q′

1

q2 q1

slide-184
SLIDE 184

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Iterative Construction of Vk – Algorithm

2009).

compute V1(S) For every Voronoi region R: compute set N of relevant needles determine V1(N) calculate intersection V1(N) ∩ R with this: determine Vj+1(S) ∩ R Computing V1(·) with algorithm by Bae et al. From Vj to Vj+1: Theorem 2 The k-th order city Voronoi diagram can be computed in O(k2(n + c) log(n + c)) time.

[Bae, Kim, Chwa, 2009]

p q q′

1

q2 q1

slide-185
SLIDE 185

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Conclusion

Structural Complexity Time Complexity

O((n + c) log n) O(nc log n log2(n + c)) O(k2(n + c) log(n + c))

first-order farthest-site

kth-order

first-order farthest-site

kth-order

City-Metric

Θ(n + c) O(k(n − k) + kc) Ω(n + kc) Θ(nc)

Transportation network under the Euclidean metric? Tight bound for the structural complexity? Open questions: first-order: impact of c is an additive constant farthest site: impact of c is not an additive constant

slide-186
SLIDE 186

Andreas Gemsa, D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Institute of Theoretical Informatics

  • Prof. Dr. Dorothea Wagner

Conclusion

Structural Complexity Time Complexity

O((n + c) log n) O(nc log n log2(n + c)) O(k2(n + c) log(n + c))

first-order farthest-site

kth-order

first-order farthest-site

kth-order

City-Metric

Θ(n + c) O(k(n − k) + kc) Ω(n + kc) Θ(nc)

Transportation network under the Euclidean metric?

Thanks!

Tight bound for the structural complexity? Open questions: first-order: impact of c is an additive constant farthest site: impact of c is not an additive constant