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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

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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 Node

Bridge Edge

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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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

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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

Cheswick et al.

LGL: Adai 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 MinutesLGL: 1.4 HoursFM3: 1.7 Minutes

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Internet Mapping Results

SPF: 30 MinutesLGL: 12 HoursFM3: 11 Minutes

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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

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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