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1
Smashing Peacocks Further:Drawing 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
2
Overview
Motivation What is a Quasi-Tree? Previous Work SPF Algorithm and Phases
– Decomposition– Drawing
Results: Speed, Visual Quality, Metrics
3
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)
4
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
10
Biconnected Graph
Removal of any node edge does not disconnect the graph into two components
Biconnected Not Biconnected
11
Biconnected Graph
Removal of any node edge does not disconnect the graph into two components
Biconnected Not Biconnected
13
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
14
Previous Work
Large general graphs– Multi-level graph drawing
Quasi-trees– Spanning tree based visualization– Domain-specific graph visualization
15
Multi-Level Approaches
Coarsen large graph into balanced hierarchy Apply force directed algorithms top down
16
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
18
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
19
H3 Viewer: MunznerBoutin et al.
Spanning tree methods
Use interaction to view subsets of graph edges.
Different goal: view full complexity of graph at all times
20
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
Cheswick et al.
LGL: Adai et al.
21
LGL Algorithm
Introduce nodes in breadth-first spanning tree order into the layout
Iterations of force directed to find good position
22
LGL Algorithm
Introduce nodes in breadth-first spanning tree order into the layout
Iterations of force directed to find good position
23
LGL Algorithm
Introduce nodes in breadth-first spanning tree order into the layout
Iterations of force directed to find good position
28
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
30
Drawing Biconnected Components
Use LGL Make two optimizations
– Not march through grid– Nodes placed on directed fans
Details in paper
31
Challenge of High Degree Nodes
Biconnected component trees can have high degree nodes
Walker: Buchheim et al. Bubble: Grivet et al. Area-Aware RINGS
32
RINGS
RINGS– Allow node-edge overlaps to get better density– Teoh and Ma 2002
Does not take node size into account
33
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
36
Major Node/Node Overlaps
Clear depiction of high level tree because minimal biconnected component overlaps
Algorithm Protein Hom. Internet Mapping
FM3 2,400 162,620
LGL 2,657 170,073
SPF 0 8
37
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