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