a note on the self-similarity of some orthogonal drawings maurizio “titto” patrignani roma tre...
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A Note on the A Note on the Self-Similarity of Self-Similarity of some Orthogonal some Orthogonal
DrawingsDrawings
Maurizio “Titto” PatrignaniRoma Tre University, Italy
GD2004 NYC 28 Sept – 2 Oct 2004
Orthogonal drawingsOrthogonal drawings
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Are orthogonal drawings Are orthogonal drawings self-similar?self-similar?
Purpose of this notePurpose 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
Self-similarity and Self-similarity and dimensiondimension
nu
mber
of
copie
s
scalin
g f
act
or
2 2
4 4
nu
mber
of
copie
s
scalin
g f
act
or
4 2
16 4
nu
mber
of
copie
s
scalin
g f
act
or
8 2
64 4
=
dim
en
sion
1
=1
=
dim
en
sion
2
=2
=
dim
en
sion
3
=3
Recursively defined self-Recursively defined self-similar objectssimilar objects
Koch curve: recursively replace each segment with four segments whose length is 1/3 of the original
Recursively defined self-Recursively defined self-similar objectssimilar objects
Koch curve: recursively replace each segment with four segments whose length is 1/3 of the original
Dimension of the Koch Dimension of the Koch curvecurve
nu
mber
of
copie
s
scalin
g f
act
or
4 3
16 9
=
dim
en
sion
d
=d
log(4) = d log(3)
log(4)
log(3)d = = 1.26
4 = 3d
log(4) = log(3 )d
StrategyStrategy
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 …
Box-counting fractal Box-counting fractal dimensiondimension
log(box side length)
log(#
non
em
pty
boxes)
N l -d
slope -d
15 non empty boxes 98 non empty boxes
Box-counting fractal Box-counting fractal dimensiondimension
box-side length l = 1non empty boxes = N0
box-side length l = 1/non empty boxes = N = cN0
scaling factor =
number of copies = c c = dHp
similarity dimension d given by
N = cN0
Nd
c = N/N0
N/N0=d
N l -d
Box-counting fractal Box-counting fractal dimensiondimension
PROS Easy to compute Also accounts for “statistical” self-similarity
CONS Defined for finite geometric objects only Defined for plane geometric objects only
Graph drawing and box-Graph drawing and box-countingcounting
We used FracDim Package [L. Wu and C. Faloutsos]
Graph drawing and box-Graph drawing and box-countingcounting
Doubling the size of the boxes the number of non-empty boxes doesn’t change
N l 0
A
B
C
D
Graph drawing and box-Graph drawing and box-countingcounting
Doubling the size of the boxes the number of non-empty boxes is divided by two
N l -1
A
B
C
D
Graph drawing and box-Graph drawing and box-countingcounting
Doubling the size of the boxes the number of non-empty boxes is divided by four
N l -2
A
B
C
D
Graph drawing and box-Graph drawing and box-countingcounting
Doubling the size of the boxes the number of non-empty boxes doesn’t change
N l 0
A
B
C
D
Graph drawing and box-Graph drawing and box-countingcounting
If this segment existsthen the geometrical objectis a fractal
A
B
C
D
A test-suite of planar A test-suite of planar graphsgraphs
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
Three Orthogonal drawing Three Orthogonal drawing approachesapproaches
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 96]Compaction: rectangularization of the faces [Tamassia 87]
The Fractal Dimension of The Fractal Dimension of Orthogonal DrawingsOrthogonal Drawings
(OFV = Orth. From Visibility, RCS = Rel. Coord. Scenario, TSM = Topology-Shape-Metrics)
Conclusions and open Conclusions and open problemsproblems
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?
Biconnected graph with Biconnected graph with 1500 vert.1500 vert.
draw
n w
ith
OFV
app
roac
h
Biconnected graph with Biconnected graph with 1500 vert.1500 vert.
drawn with
RCS approach
Maximal Planar (LEDA) Maximal Planar (LEDA) 5000 vert.5000 vert.
drawn with
TSM
approach
A test-suite of planar A test-suite of planar graphsgraphs