Department of Computer Science and Engineering Bangladesh University of Engineering and Technology

1

Department of Computer Science and Engineering

Bangladesh University of Engineering and Technology

M. Sc. Engg. ThesisMd. Emran Chowdhury (040805068P)Supervisor: Prof. Dr. Md. Saidur Rahman

2

▒ Problem Definition

ContentsContents

▒ Motivation

▒ Previous Results and Our Results

▒ Upward Point-Set Embedding

▒ Orthogonal Point-Set Embedding

▒ Conclusion and Future Works

3

Point-Set EmbeddingPoint-Set Embedding

a

c

b

df

e

Sa

c

b

df

e

G

Each vertex is placed at a distinct point

Input

4

Inputf

c

b

d

a

e

a

c

b

df

e

SG

Point-Set EmbeddingPoint-Set Embedding

Each vertex is placed at a distinct point

Each edge is drawn by straight or poly line

Output

Bend

5

Upward Point-Set EmbeddingUpward Point-Set Embedding

f

c

b

d

a

e

a

c

b

df

e

SG

Each edge is drawn upward

Input

Each vertex is placed at a distinct point Output

6

Each edge is drawn upward


ac

b

df

e

G’ S

f

c

b

d

a

e

ac

b

df

e

Ga

cb

df

e

G S

f

c

b

d

a

e

f

c

b

d

a

e

S

G’ has no upward point-

setembedding on

S

Not every graph hasupward point-set embedding

on a fixed point-set

7

1

3

4

2

ac

d

b

Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping

ac

d

bS

G

φ

8

ac

d

b

S

G

a

c

d

bφ

a

c

b

d

S

φ’

No upward point-setembedding with this mapping

No upward point-setembedding with this mapping

Upward Point-Set Embedding with mappingUpward Point-Set Embedding with mapping

Finding upward point-set

embedding with mapping is a

real challenge

9

a b

c d

e

j k

g h

f

planar graph G

point-set in the plane

Orthogonal Point-Set EmbeddingOrthogonal Point-Set Embedding

Input

10

a b

c d

e

j k

g h

f

Output j

k

f

h

g

e

c

d

a

b

Orthogonal Point-Set EmbeddingOrthogonal Point-Set Embedding

Each edge is drawn as a sequence ofvertical and horizontal line segments

11


ContentsContents

▒ Motivation

12

a

b

c

d

e

f

a

bc

d

f

e

InterconnectionGraph

Point-Set Embedding

In VLSI design, often the placesfor the modules are fixed, wehave to connect the modules

w. r. t. the inter connection graph

MotivationMotivation

13

a

b

c

d

e

f


InterconnectionGraph

VLSI Layout

a

bc

d

f

e

It is always desirable to reduce the

number of bends

Point-Set Embedding

14


ContentsContents

▒ Motivation


15

Previous Results and Our ResultsPrevious Results and Our Results

Problem Graph classAuthors Results

Giordano et. al. ’07

upward point-set

embedding

Upward planar

digraphs

at most two bends per

edge

Giordano, Liotta, and Whiteside

’09

upward point-set

embedding with mapping

Upward planar

digraphs

at most 2n-3 bends per

edge

This Thesis

upward point-set

embedding with mapping

Upward planar

digraphs

at most n-3 bends per

edge

upper boundon total

number of bends

Upward Point-Set Embedding

16

Rahman et. al. ’99

Orthogonal drawing

Cubic 3-connected plane

graphs

bend optimal drawing

Rahman and Nishizeki ’02

Orthogonal drawing

plane graphs

with ≤ 3

bend optimal drawing

Rahman, Nishizeki

and Naznin ’03

Orthogonal drawing

plane graphs

with ≤ 3

no bend drawing


But, they did not considerthe point-set embedding


Orthogonal Drawing

Time complexity = O(n)

17


Kaufman andWiese ’02

Point-set embedding

General planegraphs

2 bends per edge

One can draw the

edge orthogonally

But, the size ofthe vertices may increase


Poly-line Point-Set Embedding

Time complexity = O(n2)

18

This Thesis

Orthogonal point-setembedding

3-connected cubic planar

graphs

at most (5n+4)/2 bends in

total


Kaufman andWiese ’02

Point-set embedding

General planegraphs

2 bends per edge

One can draw the

edge orthogonally

This Thesis

Orthogonal point-setembedding with

mapping

4-connected planar graphs

at most 6n bends in

total

Tight upper bound


Orthogonal Point-Set Embedding

But, the size ofthe vertices may increaseTime complexity = O(n2)

Time complexity = O(n)

19


ContentsContents

▒ Motivation



20


SG Input

Upward Topological Book Embedding

v1

v3

v4v2

v5

1

2

3

4

5

1

2

3

4

5

v1

v3

v4v2

v5

1

2

3

4

5

Upward Point-set Embedding

21

a

c

b

d

Upward Topological Book EmbeddingUpward Topological Book Embedding

ac

d

b

SG

Spine

LeftPage

RightPage

The vertices on the spine

The edges on the pages

Digraph

Upward Topological Book Embedding

22

G contains directed hamiltonian pathG contains directed hamiltonian path

1

3

2

4

7

6

5

A directed path containingall the vertices

A directed path containingall the vertices

1

2

3

4

5

6

7


23

G contains directed hamiltonian pathG contains directed hamiltonian path

1

3

2

4

7

6

5

1

2

3

4

5

6

7


241

2

3

4

5

6

7


251

2

3

4

5

6

7


1

2

3

4

5

6

7

261

2

3

4

5

6

7


1

2

3

4

5

6

7The drawing …..

• has no edge crossings sinceit has the same embeddingas the original graph

• has no spine crossing• has 1 bend per edge

27

G does not contain directed Hamiltonian path

1

3

2

4

7

6

5

a

b

cd

e


28

a

b

cd

e

1

3

2

4

7

6

5



29

a

b

cd

e

1

3

2

4

7

6

5



30

a

b

cd

e

1

3

2

4

7

6

5

a

b

c

d

e

1

2

3

4

5

6

7


31

1

32

4

7

6

5

a

bc

d

e

a

b

c

d

e

1

2

3

4

5

6

7


Input digraph 1

2

3

4

5

6

7

Each spine crossingcorresponds to a dummy vertex

32

Calculation of number of BendsCalculation of number of Bends

i

i+1

i+2

j-2

j-1

j

Spine crossing from ito j is at most j-i-2


The edge (1, n) has no crossingsThe edge (1, n) has no crossings

Spine Crossings per edgeis at most n-4


33







Bends per edge is at most n-3Bends per edge is at most n-3

34







Bends per edge is at most n-3Bends per edge is at most n-3

Total number of spine crossings=2(n-4)+3(n-5)+ . . . +k(n-2-k)+p(n-3-k)where p, k are integers

Number of edges which crosses thespine={k(k+1)/2}-1+p

35


ContentsContents

▒ Motivation




36

Orthogonal Point-Set Embedding

3-connected cubic

planar graphs

4-connected planar graphs

( ≤ 4)

4-connected4-regular planar graphs

37

3-connected cubic planar graph


3-connected cubic planar graph G with HC

v2 v1

v3 v4

v5 v6

v7 v8

v9v10

38point-set in the plane

v2 v1

v3 v4

v5 v6

v7 v8

v9v10

3-connected cubic planar graph G with HCPlane embedding

G’ of graph G



39

v2 v1

v3 v4

v5 v6

v7 v8

v9v10



Plane embeddingG’ of graph G

Inner edges

Outeredges

40

v2 v1

v3 v4

v5 v6

v7 v8

v9v10

Inner vertic

es




41

p10

p9

p6

p8

p7

p5

p3

p4

p2

p1

v2

v3 v4

v5 v6

v7 v8

v9



v1 v10

42

p10

p9

p6

p8

p7

p5

p3

p4

p2

p1

v2

v3 v4

v6

v7 v8

v9



v1 v10

Case 1: Inner edges in left pageCase 1: Inner edges in left page

v5

We have to consider two cases

Case 1: Inner edges in left page

Case 2 : Inner edges in right page

43

p10

p9

p6

p8

p7

p5

p3

p4

p2

p1

v2

v3 v4

v6

v7 v8

v9



v1 v10

Case 1: Inner edges in left pageCase 1: Inner edges in left page

CountL= 6

v5

Nice points (L)

44

CountL= 6

p10

p9

p6

p8

p7

p5

p3

p4

p2

p1

v2

v3 v4

v6

v7 v8

v9



v1 v10

CountR= 2

Case 2: Inner edges in right pageCase 2: Inner edges in right page

v5

Nice points (R)

45



v2

v3 v4

v6

v7 v8

v9v1 v10

v5

CountL= 6

CountR= 2

p10

p9

p6

p8

p7

p5

p3

p4

p2

p1

46

p10

p9

p6

p8

p7

p5

p3

p4

p2

p1

v2

v3 v4

v6

v7 v8

v9



v1 v10

v5

From pigeonhole principle…..

Either count L or count R is at least = (n-2)/2which edges canbe drawn with 1 bend

Left nice points

Right nice points

Total bends= 1.(n-2)/2+2.(3n/2-(n-2)/2-1)+3= n/2-1+3n-n+2-2+3= n/2+2n+2= (5n+4)/2

Computation ofnumber of bends


47

4-connected planar graph G

v2

v1

v3

v4

v5

v6

1

2

3

4

5

6

Point-set S

4-connected planar graph


48


v2

v1

v3

v4

v5

v6

1

2

3

4

5

6

Point-set S



49

v2

v1

v3

v4

v5

v6

1

2

3

4

5

6




Inner edges

Outeredges

v1

v2

v3

v4

v5

v6

50

v2

v1

v3

v4

v5

v6



Inner edges

Outeredges

v1

v2

v3

v4

v5

v6

51

v2

v1

v3

v4

v5

v6

1

2

3

4

5

6

52

1

2

3

4

5

6

Middlevertex

Leftvertex

Right

vertex

G is 4-regular, each vertex is incident to exactly four edgesG is 4-regular, each vertex is incident to exactly four edges

v1

v2

v3

v4

v5

v6

53

1

2

3

4

5

6

Orthogonal Point-set Embedding

Total bends = 3.(2n-1) + 3 = 6n



v1

v2

v3

v4

v5

v6

54

Tight ExampleTight Example

1

2

3

4

5

6

v1

v2

v3

v4

v5

v6

55

Tight ExampleTight Example

1

2

3

4

5

6

v1

v2

v3

v4

v5

v6

Each vertex of G is mapped to a point i in SEach vertex of G is mapped to a point i in S

56


ContentsContents

▒ Motivation




▒ Conclusion and Future Works

57

ConclusionConclusion

upward planar digraph

3-connected cubic planar graphs

4-connected 4-regular planar graphs

n-3 bends per edge

(5n+4)/2 bends in total

6n bends in total

Quadratic

Linear

58

Design a fast algorithm for checkingupward point-set embedding

Minimize the number of bends inupward point-set embedding

Find necessary and sufficient conditionfor orthogonal point-set embedding

Reduce the number of bends for3-connected cubic planar graphs

Find Universal Point-Setsfor sub-classes of planar graphs

Future WorksFuture Works

59

Reference

Hal91 J. H. Halton, “On the thickness of graphs of given degree”, Information Sciences, Vol. 54, pp. 219-238, 1991.

CAR09 M. E. Chowdhury, M. J. Alam, and M. S. Rahman, “On Upward Point-Set Embedding of Upward Planar Digraphs”, Proc. of the 16th

Mathematics Conference of Bangladesh Mathematical Society, 2009.

RNN99 M. S. Rahman, S. Nakano and T. Nishizeki, “A linear algorithm for bend-optimal orthogonal drawings of triconnected cubic plane Graphs”, Journal of Graph Alg. and Appl., http://jgaa.info, 3(4), pp. 31-62, 1999.

RN02 M. S. Rahman and T. Nishizeki, “Bend-minimum orthogonal drawings of plane 3-graphs”, In Proc. International Workshop on Graph Theoretic Concepts in Computer Science (WG '02), Lect. Notes in Computer Science, Springer, Vol. 2573, pp. 367-378, 2002.

60

KW02 M. Kaufmann, and R. Wiese, “Embedding vertices at points: Few bends suffice for planar graphs”. Journal of Graph Algorithms and Applications, 6(1), pp. 115–129 (2002)

GLMS07 F. Giordano, G. Liotta, T. Mchedlidze,and A, Symvonis, “Computing Upward Topological Book Embeddings of Upward Planar Digraphs”, In proceedings of International Symposium on Algorithms and Computation (ISAAC 2007), Springer, Lecture Notes in Computer Science, Vol. 4835, pp. 172–183, 2007.

GLW09 F. Giordano, G. Liotta, and S. H. Whitesides, “Embeddability Problems for Upward Planar Digraphs”, In the proceedings of The 16th International Symposium on Graph Drawing (GD 2008), Springer, Lecture Notes in Computer Science, Vol. 5417, pp. 242–253, 2009.

RNN03 M. S. Rahman, T. Nishizeki, and M. Naznin, “Orthogonal drawings of plane graphs without bends”, Journal of Graph Alg. and Appl., http://jgaa.info, 7(4), pp. 335-362, 2003.

Reference

61

Thank You

62

visual analysis of self-modifiable code,based on computing a sequence of drawingswhose edges are defined at run-time [Hal91]


Upward Point-set Embedding with mapping

That alters its own instructions while it is executing-usually to

reduce the instruction path length and improve performance.

63


• The graphs are specified one at a time

• The vertex locations for the output graphs are determined by the first graph

64

1

2

3

4

5

6Now we draw the edgesof G’-C except long edge

Casevertextype

Drawing

xi+1 > xixi+1 < xi

1 middle URU ULU

2 left RU RULU

3 right LURU LU

65

Casevertextype

Drawing

xi+1 > xixi+1 < xi

1 middle URU ULU

2 left RU RULU

3 right LURU LU

1

2

3

4

5


66

1

2

3

4

5


vi is a middle vertex

vertextype ( vj )

middle

left

right

(vi , vj ) is inner

otherwise

otheredgeof vj

is (vj , vk )

yi > yk

otherwise

yi > yk

otherwise

Condition

LUR

RUL

LUR

LURD

RUL

RULD

Drawing

67

vertextype ( vj )

middle

left

right

(vi , vj ) is inner

otherwise

otheredgeof vj

is (vj , vk )

yi > yk

otherwise

yi > yk

otherwise

Condition

LUR

RUL

LUR

LURD

RUL

RULD

Drawing

1

2

3

4

5

6

vi is a middle vertex

68

1

2

3

4

5

6

vi is a left vertex

vertexTypeof vj

middle

left

yj > yl > yi

otherwise

otheredgeof vj

is (vj , vk )

yj > yk > yi > yl

Condition

LUR

UR

URD

LURD

UR

LUR

Drawing

(vi , vl ) isother inner

edge

yj > yk > yl > yi

yk > yj > yi > yl > yk

yk > yj > yl > yi > yk

69

yj > yk > yl > yi

vertexTypeof vj

middle

left

yj > yl > yi

otherwise

otheredgeof vj

is (vj , vk )

Condition

LUR

UR

URD

LURD

UR

LUR

Drawing


edge



1

2

3

4

5

6

vi is a left vertex

yj > yk > yi > yl

70

1

2

3

4

5

6

vi is a right vertex

vertexTypeof vj

middle

right

yj > yl > yi

otherwise

otheredgeof vj

is (vj , vk )

yj > yk > yi > yl

Condition

RUL

UL

ULD

RULD

UL

RUL

Drawing


edge

yj > yk > yl > yi



71

vertexTypeof vj

middle

right

yj > yl > yi

otherwise

otheredgeof vj

is (vj , vk )

yj > yk > yi > yl

Condition

RUL

UL

ULD

RULD

UL

RUL

Drawing


edge

yj > yk > yl > yi



1

2

3

4

5

6

vi is a right vertex

72

1

2

3

4

5

6

We now draw the long edge (v1 , vn )

vertextypeof vn

middle DRULD

Drawingvertextypeof v1

left

NA

right

otherwise

DRULDDRUL

rightright

otherwise

DLULDLUR

73

vertextypeof vn

middle DRULD

Drawingvertextypeof v1

left

NA

right

otherwise

DRULDDRUL

rightright

otherwise

DLULDLUR

1

2

3

4

5

6

Orthogonal Point-set Embedding

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