Drawing Planar Cubic 3-Connected Graphs with Few Segments: - - PowerPoint PPT Presentation

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Drawing Planar Cubic 3-Connected Graphs with Few Segments: - - PowerPoint PPT Presentation

Feb 28, 2023 155 likes •745 views

Drawing Planar Cubic 3-Connected Graphs with Few Segments: Algorithms & Experiments Alex Igamberdiev Wouter Meulemans Andr e Schulz Graph complexity Complexity of a graph G = ( V, E ) Usually | V | , | E | , etc. Graph complexity

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Drawing Planar Cubic 3-Connected Graphs with

Andr´ e Schulz

Few Segments: Algorithms & Experiments

Alex Igamberdiev Wouter Meulemans

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Graph complexity

Complexity of a graph G = (V, E) Usually |V |, |E|, etc.

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Graph complexity

Complexity of a graph G = (V, E) Usually |V |, |E|, etc. Says nothing about how complex a drawing is

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Visual complexity

Planar graphs Number of geometric objects for drawing

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Visual complexity

Planar graphs Number of geometric objects for drawing

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Visual complexity

Planar graphs Number of geometric objects for drawing 1

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Visual complexity

Planar graphs Number of geometric objects for drawing 2

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Visual complexity

Planar graphs Number of geometric objects for drawing 3

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Visual complexity

Planar graphs Number of geometric objects for drawing 4

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Visual complexity

Planar graphs Number of geometric objects for drawing 5

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Visual complexity

Planar graphs Number of geometric objects for drawing 6

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Visual complexity

Planar graphs Number of geometric objects for drawing 7

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Visual complexity

Planar graphs Number of geometric objects for drawing 8

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Visual complexity

Planar graphs Number of geometric objects for drawing 9

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Visual complexity

Planar graphs Number of geometric objects for drawing 9 line segments for 18 edges

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Known results

Lower Upper Segments Class Triangulation Tree 2- and 3-trees 2V 2V 3-connected 2V 5V/2 Planar 16V/3 − E 7V/3 2V 2V K/2 K/2

[Durocher, Mondal, 2014] [Durocher, Mondal, 2014] [Durocher et al, 2013] [Dujmovi´ c et al, 2007] [Dujmovi´ c et al, 2007]

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Known results

Lower Upper Segments

  • Circ. arcs

Class Triangulation Tree 2- and 3-trees 2V 2V 3-connected 2V 5V/2 Planar 16V/3 − E 7V/3 2V 2V K/2 K/2 3-trees 3-connected E/6 E/6 11E/18 2E/3

[Schulz, 2013] [Schulz, 2013] [Durocher, Mondal, 2014] [Durocher, Mondal, 2014] [Durocher et al, 2013] [Dujmovi´ c et al, 2007] [Dujmovi´ c et al, 2007]

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Our results

Line-segment drawings Planar cubic 3-connected graphs

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Our results

Line-segment drawings Two new algorithms n/2 + 3 segments Planar cubic 3-connected graphs [Mondal et al, 2013] Resolve flaw & improved

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Our results

Line-segment drawings Two new algorithms n/2 + 3 segments Planar cubic 3-connected graphs [Mondal et al, 2013] Resolve flaw & improved Experimental comparison

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Deconstruction algorithm

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Deconstruction algorithm

Theorem. Every graph can be constructed maintaining a given outer face. Insertion with insertions from the triangular prism

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Deconstruction algorithm

Algorithm

  • 1. Draw triangular prism
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Deconstruction algorithm

Algorithm

  • 1. Draw triangular prism
  • 2. Construct graph, maintaining drawing

Inner faces are convex No insertions on outer face

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Deconstruction algorithm

Algorithm

  • 1. Draw triangular prism
  • 2. Construct graph, maintaining drawing

Insertion

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Deconstruction algorithm

Algorithm

  • 1. Draw triangular prism
  • 2. Construct graph, maintaining drawing
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Deconstruction algorithm

Algorithm

  • 1. Draw triangular prism
  • 2. Construct graph, maintaining drawing
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Deconstruction algorithm

Algorithm

  • 1. Draw triangular prism
  • 2. Construct graph, maintaining drawing

Insertion

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Deconstruction algorithm

Algorithm

  • 1. Draw triangular prism
  • 2. Construct graph, maintaining drawing

Insertion

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Windmill algorithm

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Windmill algorithm

Algorithm Pre: Post: inside of C drawn cycle C drawn convex

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Postprocessing

Set of harmonic equations

[Aerts & Felsner, 2013 ]

u = λv + (1 − λ)w, for λ ∈ (0, 1) v w u

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Postprocessing

Set of harmonic equations Solve for uniform edge length, i.e. λ = 1/2

[Aerts & Felsner, 2013 ]

u = λv + (1 − λ)w, for λ ∈ (0, 1) v w u v w u

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[Mondal et al, 2013]

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[Mondal et al, 2013]

“Grid” n/2 + 4 segments 6 slopes (n/2 + 1)2 grid

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[Mondal et al, 2013]

“Grid” n/2 + 4 segments 6 slopes (n/2 + 1)2 grid Resolved flaw in algorithm

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[Mondal et al, 2013]

“Grid” n/2 + 4 segments 6 slopes (n/2 + 1)2 grid “Min” n/2 + 3 segments 7 slopes Not on a grid Resolved flaw in algorithm

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[Mondal et al, 2013]

“Grid” n/2 + 4 segments 6 slopes (n/2 + 1)2 grid “Min” n/2 + 3 segments 7 slopes Not on a grid Resolved flaw in algorithm Reduced to 6 slopes On a grid

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Three algorithms

Deconstruction Windmill [Mondal et al, 2013]

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Measuring layout quality

2000 graphs with 24 . . . 30 vertices Six measures for each graph-algorithm pair using plantri

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Measuring layout quality

2000 graphs with 24 . . . 30 vertices Six measures for each graph-algorithm pair Angular resolution using plantri

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Measuring layout quality

2000 graphs with 24 . . . 30 vertices Six measures for each graph-algorithm pair Angular resolution Edge length using plantri

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Measuring layout quality

2000 graphs with 24 . . . 30 vertices Six measures for each graph-algorithm pair Angular resolution Edge length Face aspect ratio using plantri

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Measuring layout quality

2000 graphs with 24 . . . 30 vertices Six measures for each graph-algorithm pair Angular resolution Edge length Face aspect ratio using plantri Average and worst-case

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Angular resolution

DEC-ALT WIN MON-GRID MON-MIN

Average Minimum

π/2 π/2 DEC DEC-ALT WIN MON-GRID MON-MIN DEC

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Edge length

DEC-ALT WIN MON-GRID MON-MIN

Average Maximum

100% 100% DEC DEC-ALT WIN MON-GRID MON-MIN DEC

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Face aspect ratio

DEC-ALT WIN MON-GRID MON-MIN

Average Minimum

1 1 DEC DEC-ALT WIN MON-GRID MON-MIN DEC

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Experiment summary

WIN DEC MON WIN DEC MON “Wins” “Wins” minus “Losses”

  • 6

6

  • 5 -4 -3 -2 -1 0

1 2 3 4 5

WIN DEC MON

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Conclusion

Minimal visual complexity Two new algorithms Experiments Fixed and improved [Mondal et al, 2013] Best depends on measure

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Conclusion

Minimal visual complexity Future work Closing gap for other classes Circular arcs Visual complexity ∼ observer’s assessment? Visual complexity ∼ cognitive load? Two new algorithms Experiments Fixed and improved [Mondal et al, 2013] Best depends on measure