A Note on the SelfSimilarity of some Orthogonal Drawings

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A Note on the Self-Similarity of some Orthogonal
Drawings

• Maurizio Titto Patrignani

Roma Tre University, Italy
GD2004 NYC 28 Sept 2 Oct 2004
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Orthogonal drawings
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Are orthogonal drawings self-similar?
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Purpose of this note

• Prove that orthogonal drawings with a reduced
  number of bends are actually self-similar
• How?
• Explore the implications of self-similarity
• Find some measurable property of self-similar
  objects
• Perform measures on a suitable number of
  orthogonal drawings obtained with different
  approaches and different types of graphs

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Self-similarity and dimension
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Recursively defined self-similar objects

• Koch curve recursively replace each segment with
  four segments whose length is 1/3 of the original

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Recursively defined self-similar objects

• Koch curve recursively replace each segment with
  four segments whose length is 1/3 of the original

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Dimension of the Koch curve
number of copies
scaling factor
dimension
d
4
3

d
16
9

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Strategy

• Self-similarity implies fractal dimension
• To prove that orthogonal drawings are
  self-similar it suffices to show that they have a
  fractal dimension
• We may choose between a number of fractal
  dimensions
• Similarity dimension
• Hausdorf dimension
• Box-counting dimension
• Correlation dimension

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Box-counting fractal dimension
slope -d
log(non empty boxes)
log(box side length)
15 non empty boxes
98 non empty boxes
N ? l -d
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Box-counting fractal dimension
similarity dimension d given by
scaling factor a
c ad
Hp
number of copies c
box-side length l 1
box-side length l 1/a
non empty boxes N0
non empty boxes N cN0
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Box-counting fractal dimension

• PROS
• Easy to compute
• Also accounts for statistical self-similarity
• CONS
• Defined for finite geometric objects only
• Defined for plane geometric objects only

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Graph drawing and box-counting

• We used FracDim Package L. Wu and C. Faloutsos

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Graph drawing and box-counting
A
B
C
Doubling the size of the boxes the number of
non-empty boxes doesnt change
N ? l 0
D
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Graph drawing and box-counting
A
B
C
Doubling the size of the boxes the number of
non-empty boxes is divided by two
N ? l -1
D
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Graph drawing and box-counting
A
B
C
Doubling the size of the boxes the number of
non-empty boxes is divided by four
N ? l -2
D
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Graph drawing and box-counting
A
B
C
Doubling the size of the boxes the number of
non-empty boxes doesnt change
N ? l 0
D
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Graph drawing and box-counting
A
If this segment exists then the geometrical
object is a fractal
B
C
D
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A test-suite of planar graphs

• Using P.I.G.A.L.E. H. de Fraysseix, P. Ossona de
  Mendez, we generated three test suites of random
  graphs
• planar connected, planar biconnected and planar
  triconnected
• ranging from 500 to 3,000 edges, increasing each
  time by 500 edges
• 10 graphs for each type
• After the generation we removed multiple edges
  and self-loops

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Three Orthogonal drawing approaches

• Orthogonal From Visibility approach
  (OFV)Construct a visibility representation of a
  biconnected graphTransform it into an orthogonal
  drawing Di Battista et al. 99
• Relative Coordinates Scenario (RCS) We used the
  simple algorithm described in Papakostas
  Tollis 2000 for biconnected graphs
• Topology-Shape-Metrics approach
  (TSM)Planarization we used Boyer Myrvold 99
  Orthogonalization Tamassia 87, Fossmeier
  Kaufmann 96Compaction rectangularization of
  the faces Tamassia 87

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The Fractal Dimension of Orthogonal Drawings
(OFV Orth. From Visibility, RCS Rel. Coord.
Scenario, TSM Topology-Shape-Metrics)
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Conclusions and open problems

• We assessed a fractal dimension (box-counting) of
  about 1.7 for orthogonal drawings with a reduced
  number of bends
• Open problems
• Do other graph drawing standards also produce
  self-similar drawings of large graphs?
• Can alternative measures of fractal dimension,
  like the correlation dimension, help deepening
  our understanding of this phenomenon?
• Can we lose self-similarity without adding too
  many bends to the drawings?

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Biconnected graph with 1500 vert.
drawn with OFV approach
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Biconnected graph with 1500 vert.
drawn with RCS approach
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Maximal Planar (LEDA) 5000 vert.
drawn with TSM approach
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A test-suite of planar graphs