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

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

Visual complexity Planar graphs Number of geometric objects for drawing

Visual complexity Planar graphs Number of geometric objects for drawing 1

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

Known results Class Lower Upper Tree K/ 2 K/ 2 [Durocher et al, 2013] 2- and 3-trees 2 V 2 V [Dujmovi´ c et al, 2007] Segments 3-connected 2 V 5 V/ 2 [Dujmovi´ c et al, 2007] Triangulation 2 V 7 V/ 3 [Durocher, Mondal, 2014] Planar 2 V 16 V/ 3 − E [Durocher, Mondal, 2014]

Known results Class Lower Upper Tree K/ 2 K/ 2 [Durocher et al, 2013] 2- and 3-trees 2 V 2 V [Dujmovi´ c et al, 2007] Segments 3-connected 2 V 5 V/ 2 [Dujmovi´ c et al, 2007] Triangulation 2 V 7 V/ 3 [Durocher, Mondal, 2014] Planar 2 V 16 V/ 3 − E [Durocher, Mondal, 2014] Circ. arcs 3-trees E/ 6 11 E/ 18 [Schulz, 2013] 3-connected E/ 6 2 E/ 3 [Schulz, 2013]

Our results Line-segment drawings Planar cubic 3-connected graphs

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

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

Deconstruction algorithm

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

Deconstruction algorithm Algorithm 1. Draw triangular prism

Deconstruction algorithm Algorithm 1. Draw triangular prism 2. Construct graph, maintaining drawing Inner faces are convex No insertions on outer face

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

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

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

Windmill algorithm

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

Postprocessing Set of harmonic equations [Aerts & Felsner, 2013 ] u = λv + (1 − λ ) w , for λ ∈ (0 , 1) w u v

Postprocessing Set of harmonic equations [Aerts & Felsner, 2013 ] u = λv + (1 − λ ) w , for λ ∈ (0 , 1) Solve for uniform edge length, i.e. λ = 1 / 2 w w u u v v

[Mondal et al, 2013]

[Mondal et al, 2013] “Grid” n/ 2 + 4 segments 6 slopes ( n/ 2 + 1) 2 grid

[Mondal et al, 2013] “Grid” n/ 2 + 4 segments 6 slopes ( n/ 2 + 1) 2 grid Resolved flaw in algorithm

[Mondal et al, 2013] “Grid” “Min” n/ 2 + 4 segments n/ 2 + 3 segments 6 slopes 7 slopes ( n/ 2 + 1) 2 grid Not on a grid Resolved flaw in algorithm

[Mondal et al, 2013] “Grid” “Min” n/ 2 + 4 segments n/ 2 + 3 segments 6 slopes 7 slopes ( n/ 2 + 1) 2 grid Not on a grid Resolved flaw in algorithm Reduced to 6 slopes On a grid

Three algorithms Deconstruction Windmill [Mondal et al, 2013]

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

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

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

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

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

Angular resolution Average WIN DEC DEC-ALT MON-GRID MON-MIN 0 π/ 2 Minimum WIN DEC DEC-ALT MON-GRID MON-MIN 0 π/ 2

Edge length Average WIN DEC DEC-ALT MON-GRID MON-MIN 0 100% Maximum WIN DEC DEC-ALT MON-GRID MON-MIN 0 100%

Face aspect ratio Average WIN DEC DEC-ALT MON-GRID MON-MIN 0 1 Minimum WIN DEC DEC-ALT MON-GRID MON-MIN 0 1

Experiment summary WIN DEC MON WIN DEC MON WIN DEC MON “Wins” “Wins” minus “Losses” -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

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

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

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

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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