Smashing Peacocks Further: Drawing QuasiTrees from Biconnected Components

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Smashing Peacocks FurtherDrawing Quasi-Trees
from Biconnected Components

• Daniel Archambault and Tamara Munzner,
• University of British Columbia
• David Auber, University of Bordeaux I, LaBRI

Imager Laboratory For Graphics, Visualization,
and HCI
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Overview

• Motivation
• What is a Quasi-Tree?
• Previous Work
• SPF Algorithm and Phases
• Decomposition
• Drawing
• Results Speed, Visual Quality, Metrics

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Where are Quasi-Trees Found?

• Found in many areas including
• Bioinformatics (protein homology maps)
• Computer networking (Internet mapping)
• Software engineering (function call graphs)
• Can be very large and difficult to draw
• In this paper (30,000 200,000)

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What is a Quasi-Tree?

• A graph which is almost a tree
• Should be able to exploit tree properties with
  the addition of a few edges
• Work concerned with drawing, not detecting

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Quasi-Tree Datasets
Cheswick et al.
LGL Adai et al.
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Quasi-Tree Datasets
Cheswick et al.
LGL Adai et al.
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Decomposition
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Decomposition
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Decomposition
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Biconnected Graph

• Removal of any node edge does not disconnect the
  graph into two components

Biconnected
Not Biconnected
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Biconnected Graph

• Removal of any node edge does not disconnect the
  graph into two components

Biconnected
Not Biconnected
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Decomposition
Bridge Edge
Bridge Node
Biconnected Component Tree (block-cut tree)
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Trees, Quasi-Trees, and Biconnected components

• Graph G(V, E) V nodes, E edges
• Tree exactly V biconnected components
• Quasi-tree O(V) biconnected components

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Previous Work

• Large general graphs
• Multi-level graph drawing
• Quasi-trees
• Spanning tree based visualization
• Domain-specific graph visualization

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Multi-Level Approaches

• Coarsen large graph into balanced hierarchy
• Apply force directed algorithms top down

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Multi-Level Approaches

• Coarser graphs representative but cheaper to lay
  out
• Harel and Koren
• GRIP Gajer et al.
• FM3 Hachul and Jünger
• better visual quality and speed

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TopoLayout

• Recursively detects
• Connected
• Trees
• Biconnected
• HDE
• Complete
• Clusters

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TopoLayout

• Use appropriate algorithm depending on feature
  type detected
• SPF can be viewed as a specialized version of
  TopoLayout for quasi-trees
• Different decomposition pipeline and drawing
  algorithms

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Spanning tree methods

• Use interaction to view subsets of graph edges.
• Different goal view full complexity of graph at
  all times

H3 Viewer Munzner
Boutin et al.
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Domain-Inspired

• Works on general Quasi-Trees
• Developed in domains where general graph drawing
  tools insufficient
• LGL Adai et al. based on Cheswick et al.
• Requires hours of drawing time
• Ambitious in terms of scale
• 200,000 nodes

LGL Adai et al.
Cheswick et al.
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LGL Algorithm

• Introduce nodes in breadth-first spanning tree
  order into the layout
• Iterations of force directed to find good position

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LGL Algorithm

• Introduce nodes in breadth-first spanning tree
  order into the layout
• Iterations of force directed to find good position

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LGL Algorithm

• Introduce nodes in breadth-first spanning tree
  order into the layout
• Iterations of force directed to find good position

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LGL Algorithm

• Layout embedded in a grid

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LGL Algorithm

• Repulsive forces for close nodes computed

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LGL Algorithm

• Repulsive nodes for distant cells ignored

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SPF Video
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SPF Algorithm Phases

• Decompose into biconnected components
• Draw each biconnected piece with previous work
  (LGL Adai et al.)
• Draw the biconnected component tree using tree
  drawing algorithm

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Decomposition

• Standard algorithm in literature O(V E)

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Drawing Biconnected Components

• Use LGL
• Make two optimizations
• Not march through grid
• Nodes placed on directed fans
• Details in paper

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Challenge of High Degree Nodes

• Biconnected component trees can have high degree
  nodes

Walker Buchheim et al.
Bubble Grivet et al.
Area-Aware RINGS
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RINGS

• RINGS
• Allow node-edge overlaps to get better density
• Teoh and Ma 2002
• Does not take node size into account

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Area-Aware RINGS

• RINGS assumes the children are the same size
• Not true for biconnected component trees
• Recursive layout of tree bottom up instead of top
  down
• Details in paper

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Protein Homology Results
SPF 43 Minutes
LGL 1.4 Hours
FM3 1.7 Minutes
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Internet Mapping Results
SPF 30 Minutes
LGL 12 Hours
FM3 11 Minutes
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Major Node/Node Overlaps

• Clear depiction of high level tree because
  minimal biconnected component overlaps

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Future Work

• Improve visual quality by reducing edge crossings
• Better area-aware tree drawing algorithm?
• Improved Area-Aware RINGS
• Improve accuracy of LGL repulsive force
  calculations
• Multipole method used in FM3
• Automatic quasi-tree detection

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Conclusion

• A new algorithm for drawing quasi-trees
• Exploits tree-like structure (biconnected
  components) for better visual quality
• Significant running time improvement
• Demonstrated on two examples in domain literature