Smooth Orthogonal Drawings of Planar Graphs Philipp Kindermann - - PowerPoint PPT Presentation

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Smooth Orthogonal Drawings of Planar Graphs Philipp Kindermann - - PowerPoint PPT Presentation

Aug 23, 2023 26 likes •1.48k views

Smooth Orthogonal Drawings of Planar Graphs Philipp Kindermann Chair of Computer Science I Universit at W urzburg Joint work with Md. Jawaherul Alam, Michael A. Bekos, Michael Kaufmann, Stephen G. Kobourov & Alexander Wolff

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SLIDE 1

Smooth Orthogonal Drawings

  • f Planar Graphs

Philipp Kindermann Chair of Computer Science I Universit¨ at W¨ urzburg

Joint work with

  • Md. Jawaherul Alam, Michael A. Bekos, Michael Kaufmann,

Stephen G. Kobourov & Alexander Wolff

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SLIDE 2

Orthogonal Layouts

  • all edge segments are horizontal or vertical
  • a well-studied drawing convention
  • many examples in applications
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SLIDE 3

Orthogonal Layouts

  • all edge segments are horizontal or vertical
  • a well-studied drawing convention
  • many examples in applications

ER diagram in OGDF

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SLIDE 4

Orthogonal Layouts

  • all edge segments are horizontal or vertical
  • a well-studied drawing convention
  • many examples in applications

ER diagram in OGDF UML diagram by Oracle

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SLIDE 5

Orthogonal Layouts

  • all edge segments are horizontal or vertical
  • a well-studied drawing convention
  • many examples in applications

ER diagram in OGDF UML diagram by Oracle Organigram of HS Limburg

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SLIDE 6

Orthogonal Layouts

  • all edge segments are horizontal or vertical
  • a well-studied drawing convention
  • many examples in applications

ER diagram in OGDF UML diagram by Oracle Organigram of HS Limburg Circuit diagram by Jeff Atwood

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SLIDE 7

Orthogonal Layouts

  • all edge segments are horizontal or vertical
  • a well-studied drawing convention
  • many examples in applications

UML diagram by Oracle Circuit diagram by Jeff Atwood Fused Grid city layouts [By Fgrammen, via Wikimedia Commons]

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SLIDE 8

Orthogonal Layouts

  • all edge segments are horizontal or vertical
  • a well-studied drawing convention
  • many examples in applications

VLSI/PCI chip design [Mora et al. 2013] Fused Grid city layouts [By Fgrammen, via Wikimedia Commons]

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SLIDE 9

Orthogonal Layouts – Well-Known Results

Can minimize number of bends for fixed embedding. [Tamassia, SIAM J Comp’87] [Bl¨ asius et al., ’11] Given an embedding and a function flex: E → N≥1, can compute a drawing with ≤ flex(e) bends/edge (if one exists). [Garg & Tamassia, SIAM J Comp’01] Without fixed embedding, bend minimization is hard to approx. [Biedl & Kant, CGTA’98], [Liu et al., DAM’98] Can compute drawing on the (n × n)-grid with ≤ 2n + 2 bends for any embedding (and ≤ 2 bends/edge – except octahedron)

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SLIDE 10

Smooth Drawings

Mark Lombardi (1951–2000)

Lombardi drawings

  • circular arc edges
  • perfect angular

resolution

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SLIDE 11

Smooth Drawings

Mark Lombardi (1951–2000)

Lombardi drawings

  • circular arc edges
  • perfect angular

resolution k-Lombardi drawings

  • each edge sequence of k

circular arcs

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SLIDE 12

Smooth Drawings

The New York Times info diagram

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SLIDE 13

Smooth Drawings

Terra Nova - Design Overdrive by Carlos Barbosa

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SLIDE 14

Smooth Drawings

Public transportation map by O¨ OVG

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SLIDE 15

Smooth Orthogonal Layouts

Combine both worlds:

  • rthogonal

smooth orthogonal

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SLIDE 16

Smooth Orthogonal Layouts

Combine both worlds:

  • edges leave and enter vertices horizontally or vertically
  • rthogonal

smooth orthogonal

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SLIDE 17

Smooth Orthogonal Layouts

Combine both worlds:

  • edges leave and enter vertices horizontally or vertically
  • each edge is drawn as a sequence of axis-aligned line

segments and circular-arc segments without bends

  • rthogonal

smooth orthogonal

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SLIDE 18

Smooth Orthogonal Layouts

Combine both worlds:

  • edges leave and enter vertices horizontally or vertically
  • each edge is drawn as a sequence of axis-aligned line

segments and circular-arc segments without bends

  • there are no edge-crossings (for planar graphs)
  • rthogonal

smooth orthogonal

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SLIDE 19

Edge Complexity

  • rthogonal

smooth orthogonal

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SLIDE 20

Edge Complexity

  • rthogonal

smooth orthogonal complexity of an edge: number of arcs

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SLIDE 21

Edge Complexity

  • rthogonal

smooth orthogonal complexity of an edge: number of arcs 2

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SLIDE 22

Edge Complexity

  • rthogonal

smooth orthogonal complexity of an edge: number of arcs 2 2

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SLIDE 23

Edge Complexity

  • rthogonal

smooth orthogonal complexity of an edge: number of arcs edge complexity: maximum complexity over all edges 2 2

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SLIDE 24

Edge Complexity

  • rthogonal

smooth orthogonal complexity of an edge: number of arcs edge complexity: maximum complexity over all edges 2 2 1 2

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SLIDE 25

Liu et al. Algorithm

biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 26

Liu et al. Algorithm

  • choose vertices s and t

biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 27

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering

biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 28

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 29

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 30

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 31

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 32

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2 3

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 33

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2 3

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 34

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2 3

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 35

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2 3

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 36

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2 3

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 37

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2 3

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 38

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2 3

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← → biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 39

Liu et al. Algorithm

3

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

1 n 2

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← →

  • Eliminate S-shapes

biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 40

Liu et al. Algorithm

  • choose vertices s and t
  • place vertices by their st-numbering
  • Invariant:

When we fix an endpoint of edge e, we associate e with a column.

  • Use ports in this order:
  • ut:

↑ → ← in: ↓ ← →

  • Eliminate S-shapes

3 1 n 2

biconnected 4 -planar graph → orthogonal complexity-3 layout

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SLIDE 41

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

3 1 n 2 smooth

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SLIDE 42

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ 3 1 n 2 smooth

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SLIDE 43

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ 3 1 n 2 smooth

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SLIDE 44

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → 3 1 n 2 smooth

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SLIDE 45

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → 3 1 n 2 smooth

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SLIDE 46

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → 3 1 n 2 smooth

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SLIDE 47

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → 3 1 n 2 smooth

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SLIDE 48

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → → 3 1 n 2 smooth

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SLIDE 49

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → → 3 1 n 2 smooth

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SLIDE 50

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → → 3 1 n 2 → smooth

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SLIDE 51

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → → → 3 1 n 2 smooth

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SLIDE 52

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → → → → 3 1 n 2 smooth

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SLIDE 53

SC2-Layouts

biconnected 4 -planar graph → orthogonal complexity-2 layout

→ → → → → → 3 1 n 2 smooth

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SLIDE 54

Crossings

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SLIDE 55

Crossings

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SLIDE 56

Crossings

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SLIDE 57

Crossings

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SLIDE 58

Crossings

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SLIDE 59

Crossings

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SLIDE 60

Cut

Def.

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SLIDE 61

Cut

Def.

  • y-monotone curve

u

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SLIDE 62

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments

u u

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SLIDE 63

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments
  • divides the current drawing into a left and a right part

u u

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SLIDE 64

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments
  • divides the current drawing into a left and a right part
  • intersects only horizontal segments

u u u

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SLIDE 65

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments
  • divides the current drawing into a left and a right part
  • intersects only horizontal segments

u u u u

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SLIDE 66

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments
  • divides the current drawing into a left and a right part
  • intersects only horizontal segments

u u u u u

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SLIDE 67

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments
  • divides the current drawing into a left and a right part
  • intersects only horizontal segments

u u u u u

Problems:

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SLIDE 68

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments
  • divides the current drawing into a left and a right part
  • intersects only horizontal segments

u u u u u

Problems:

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SLIDE 69

Cut

Def.

  • y-monotone curve
  • consists of horizontal, vertical and circular segments
  • divides the current drawing into a left and a right part
  • intersects only horizontal segments

u u u u u

Problems:

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SLIDE 70

Invariants

(I1) Every open edge is associated with a column

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SLIDE 71

Invariants

(I1) An L-shape always contains a horizontal segment; it never contains a vertical segment. Every open edge is associated with a column (I2)

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SLIDE 72

Invariants

(I1) An L-shape always contains a horizontal segment; it never contains a vertical segment. Every open edge is associated with a column (I2) (I3) A C-shape always has a horizontal segment incident to its bottom vertex.

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SLIDE 73

Maintain invariants

L-shape

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SLIDE 74

Maintain invariants

L-shape C-shape

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SLIDE 75

Maintain invariants

L-shape C-shape Double C-shape

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SLIDE 76

Maintain invariants

L-shape C-shape Double C-shape protected

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SLIDE 77

Invariants, updated

(I1) An L-shape always contains a horizontal segment; it never contains a vertical segment. Every open edge is associated with a column (I2) (I3) An unprotected C-shape always has a horizontal segment incident to its bottom vertex.

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SLIDE 78

Eliminate crossings

u v v u

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SLIDE 79

Eliminate crossings

u v v u

  • 1. move v up
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SLIDE 80

Eliminate crossings

u v

  • 1. move v up

v u

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SLIDE 81

Eliminate crossings

u v

  • 1. move v up

v u

  • 2. find a cut
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SLIDE 82

Eliminate crossings

u v

  • 1. move v up

v u

  • 2. find a cut
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SLIDE 83

Eliminate crossings

u v

  • 1. move v up

v u

  • 2. find a cut
  • 3. move vertices to the left
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SLIDE 84

Eliminate crossings

u v

  • 1. move v up
  • 2. find a cut
  • 3. move vertices to the left

v u

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SLIDE 85

Example Run

1

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SLIDE 86

Example Run

1

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SLIDE 87

Example Run

1

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SLIDE 88

Example Run

1

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SLIDE 89

Example Run

1

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SLIDE 90

Example Run

1

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SLIDE 91

Example Run

1

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SLIDE 92

Example Run

1

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SLIDE 93

Example Run

1

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SLIDE 94

Example Run

1

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SLIDE 95

Example Run

1

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SLIDE 96

Example Run

1

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SLIDE 97

Example Run

1

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SLIDE 98

Example Run

1

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SLIDE 99

Example Run

1

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SLIDE 100

Example Run

1

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SLIDE 101

Example Run

1

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SLIDE 102

Extension to Arbitrary Graphs

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SLIDE 103

Extension to Arbitrary Graphs

cutvertices

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SLIDE 104

Extension to Arbitrary Graphs

bridges cutvertices

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SLIDE 105

Extension to Arbitrary Graphs

bridges cutvertices

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SLIDE 106

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face cutvertices

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SLIDE 107

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices

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SLIDE 108

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices

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SLIDE 109

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices

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SLIDE 110

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices

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SLIDE 111

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices

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SLIDE 112

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices

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SLIDE 113

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices Connect the pieces

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SLIDE 114

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices Connect the pieces

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SLIDE 115

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices Connect the pieces

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SLIDE 116

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices Connect the pieces

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SLIDE 117

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices Connect the pieces

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SLIDE 118

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices Connect the pieces

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SLIDE 119

Extension to Arbitrary Graphs

bridges Draw specific vertex on outer face Draw cut vertices with right angles cutvertices Connect the pieces 4-planar graph → smooth orthogonal complexity-2 layout

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SLIDE 120

SC1-Layouts

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SLIDE 121

SC1-Layouts

Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 122

SC1-Layouts

Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 123

SC1-Layouts

Consider the dual tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 124

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 125

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 126

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 127

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 128

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 129

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 130

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 131

SC1-Layouts

Consider the dual tree Pick a root, traverse the tree Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 132

SC1-Layouts

Any triconnected 3-planar graph admits an SC1-layout. Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 133

SC1-Layouts

Any Hamiltonian 3-planar graph admits an SC1-layout. Any triconnected 3-planar graph admits an SC1-layout. Any biconnected 4-outerplanar graph admits an SC1-layout.

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SLIDE 134

Area Requirement of SC1-Layouts

There is an infinite class of graphs that require exponential area if they are drawn with SC1.

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SLIDE 135

Area Requirement of SC1-Layouts

There is an infinite class of graphs that require exponential area if they are drawn with SC1.

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SLIDE 136

Area Requirement of SC1-Layouts

There is an infinite class of graphs that require exponential area if they are drawn with SC1.

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SLIDE 137

Area Requirement of SC1-Layouts

There is an infinite class of graphs that require exponential area if they are drawn with SC1.

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SLIDE 138

Area Requirement of SC1-Layouts

There is an infinite class of graphs that require exponential area if they are drawn with SC1.

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SLIDE 139

Biconnected Graphs without SC1-Layout

There exists a biconnected 4-planar graph that admits an OC2-layout, but does not admit an SC1-layout.

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SLIDE 140

Biconnected Graphs without SC1-Layout

There exists a biconnected 4-planar graph that admits an OC2-layout, but does not admit an SC1-layout.

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SLIDE 141

Biconnected Graphs without SC1-Layout

There exists a biconnected 4-planar graph that admits an OC2-layout, but does not admit an SC1-layout.

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SLIDE 142

Open Problems

  • Do all 4-planar graphs admit an SC2-layout

in polynomial area?

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SLIDE 143

Open Problems

  • Do all 4-planar graphs admit an SC2-layout

in polynomial area?

  • Do all 4-outerplanar graphs admit an SC1-layout?
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SLIDE 144

Open Problems

  • Do all 4-planar graphs admit an SC2-layout

in polynomial area?

  • Do all 4-outerplanar graphs admit an SC1-layout?
  • Do all 3-planar graphs admit an SC1-layout?
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SLIDE 145

Open Problems

  • Do all 4-planar graphs admit an SC2-layout

in polynomial area?

  • Do all 4-outerplanar graphs admit an SC1-layout?
  • Do all 3-planar graphs admit an SC1-layout?
  • Is it NP-hard to decide whether a 4-planar graph

admits an SC1-layout?