Transcript Clustering Cycles into Cycles of Clusters

A Note on the
Self-Similarity of some
Orthogonal Drawings
Maurizio “Titto” Patrignani
Roma Tre University, Italy
GD2004
NYC 28 Sept – 2 Oct 2004
Orthogonal drawings
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
Are orthogonal drawings self-similar?
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
2=2
4=4
1
1
4=2
16 = 4
2
2
scaling factor
dimension
number of copies
scaling factor
dimension
number of copies
scaling factor
dimension
number of copies
Self-similarity and dimension
8=2
64 = 4
3
3
Recursively defined self-similar objects

Koch curve: recursively replace each segment
with four segments whose length is 1/3 of the
original
Recursively defined self-similar objects

Koch curve: recursively replace each segment
with four segments whose length is 1/3 of the
original
scaling factor
dimension
number of copies
Dimension of the Koch curve
4=3
d
4=3
d
d
log(4) = log(3 )
log(4) = d log(3)
16 = 9
d
log(4)
d=
= 1.26
log(3)
Strategy



Self-similarity implies fractal dimension
To prove that orthogonal drawings are selfsimilar 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
…
log(#non empty boxes)
Box-counting fractal dimension
15 non empty boxes
98 non empty boxes
slope -d
log(box side length)
N
l -d
Box-counting fractal dimension
Hp
similarity dimension d given by
scaling factor = a
c = ad
number of copies = c
N = cN0
c = N/N0
N/N0=ad
Nad
l
box-side length = 1
non empty boxes = N0
l
box-side length = 1/a
non empty boxes = N = cN0
N
l
-d
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
Graph drawing and box-counting

We used FracDim Package [L. Wu and C.
Faloutsos]
Graph drawing and box-counting
A
B
C
D
N
l0
Doubling the size of
the boxes the number
of non-empty boxes
doesn’t change
Graph drawing and box-counting
A
B
C
D
N
l
Doubling the size of
the boxes the number
-1
of non-empty boxes is
divided by two
Graph drawing and box-counting
A
B
C
D
N
l
Doubling the size of
the boxes the number
-2
of non-empty boxes is
divided by four
Graph drawing and box-counting
A
B
C
D
N
l0
Doubling the size of
the boxes the number
of non-empty boxes
doesn’t change
Graph drawing and box-counting
A
If this segment exists
then the geometrical object
is a fractal
B
C
D
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
Three Orthogonal drawing approaches
Orthogonal From Visibility approach (OFV)
Construct a visibility representation of a biconnected graph
Transform 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
Orthogonal Drawings
(OFV = Orth. From Visibility, RCS = Rel. Coord. Scenario, TSM = Topology-Shape-Metrics)
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 selfsimilar 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 1500 vert.
Biconnected graph with 1500 vert.
Maximal Planar (LEDA) 5000 vert.
A test-suite of planar graphs