octagonal drawings of plane graphs with prescribed face areas takao nishizeki (tohoku univ.)...

Octagonal Drawings of Plane Graphs with Prescribed Face Areas

Takao Nishizeki (Tohoku Univ.)

指定面積的平面図的八角形描画

西関　隆夫　（東北大学）

ツディン　メンジディ　ピンメントウディ　バージョシン　ミョーワォ

Sendai city

Hsinchu city

Tokyo

Hsinchu city and Sendai city

Only 2400 km

新竹仙台

仙台

新竹

東京

Tohoku University （東北大学） Tohoku University was established in 1907.

春夏

秋冬

Lu Xian and Prof. FujinoBook Cover 1st page

GSIS, Tohoku University Graduate School of Information Sciences (GSIS), Tohoku University, was established in 1993.

150 Faculties

Math.

Robotics

Economics

Human Social Sciences

Interdisciplinary School

情報科学研究科

Transportation

Computer Science

450 students

Octagonal Drawings of Plane Graphs with Prescribed Face Areas

Takao Nishizeki (Tohoku Univ.)

指定面積的平面図的八角形描画

西関　隆夫　（東北大学）

ツディン　メンジディ　ピンメントウディ　バージョシン　ミョーワォ

Prescribed-Area Octagonal Drawing

InputPlane graph

K 27

G 19A 46

C 11

J 14

I 12

B 65

D 56

E 23

F 8

H 37


InputPlane graph

A real number for each inner face

K 27

G 19A 46

C 11

J 14

I 12

B 65

D 56

E 23

F 8

H 37

A 46

B 65

C 11D 56

E

23

F 8

H 37

G 19

I 12 J 14

K 27


Input Output

Plane graph Prescribed-area octagonal drawingA real number for each inner face

K 27

G 19A 46

C 11

J 14

I 12

B 65

D 56

E 23

F 8

H 37

A 46

B 65

C 11D 56

E

23

F 8

H 37

G 19

I 12 J 14

K 27


Input Output

Each inner face is drawn as a rectilinear polygon of at most eight corners.The outer face is drawn as a rectangle.

Each face has its prescribed area.

ApplicationsVLSI Floorplanning

Each module has area requirements.

Obtained by subdividing a given rectangle into smaller rectangles.

Each smaller rectangle corresponds to a module.

2

1

1

2

Area requirements cannot be satisfied if each module is allowed to be only a rectangle.

2

1

1

2

Area requirements can be satisfied if each module is allowed to be a simple rectilinear polygon.

1

1

2

2

ApplicationsVLSI Floorplanning

It is desirable to keep the shape of each rectilinear polygon as simple as possible.

Our Results

G: a good slicing graph

O(n) time algorithm.

Slicing Floorplan

Slicing Floorplan

A slicing floorplan can be obtained by repeatedly subdividing rectangles horizontally or vertically.

Slicing Floorplan and Slicing Graph

Slicing Floorplan Slicing Graph

corner

Not a Slicing Floorplan Not a Slicing Graph

Slicing Tree

Slicing Tree

P1

Slicing Tree

P1

P : V1

root

Slicing Tree

P1

P : V1

Right subgraph becomes right subtree

Left subgraph becomes left subtree

root

Slicing Tree

P2

P : V1

root

Slicing Tree

P2

P : V1

P : V2

root

Slicing Tree

P3

P : V1

P : V2

root

Slicing Tree

P3

P : V1

P : V2

P : H3

root

Slicing Tree

P3

P : V1

P : V2

P : H3

Upper subgraph becomes right subtree

Lower subgraph becomes left subtree

root

Slicing Tree

f

f

ff

f

ff

ff

1

2

34

5

6

7

8

9

10f

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

root

Good Slicing Graph

A slicing tree is good if each horizontal slice is a face path.

P

E

P P P

PP

P

P

P

P

f

f

ff

f

ff

ff

1 23

5

4

67

8

9

1

2

34

5

6

7

8

9

10f

P

P

N

S

W

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

P

Three face paths

On a boundary of a single face

Not a face path

Good Slicing Graph


P P P

PP

P

P

P

P

f

f

ff

f

ff

ff

1 23

5

4

67

8

9

1

2

34

5

6

7

8

9

10f

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

Good Slicing Graph


P P P

PP

P

P

P

P

f

f

ff

f

ff

ff

1 23

5

4

67

8

9

1

2

34

5

6

7

8

9

10f

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

We call a graph a good slicing graphif it has a good slicing tree.

Good Slicing Graph

For a horizontal slice, at least one of the upper subgraphand the lower subgraph cannot be vertically sliced.

Cannot be vertically sliced

Can be vertically sliced

Not every slicing graph is a good slicing graph.

Input

P P P

PP

P

P

P

P

f

f

ff

f

ff

ff

1 23

5

4

67

8

9

1

2

34

5

6

7

8

9

10f

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

A Good Slicing Graph

A Good Slicing Tree

Output

Prescribed-areaOctagonal drawing

Each inner face is drawn as a rectilinear polygon with at most eight corners.

Particularly, each inner face is drawn as a rectilinear polygon of the following nine shapes.

A 46

B 65

C 11D 56

E

23

F 8

H 37

G 19

I 12 J 14

K 27

Octagons8 corners

6 corners 4 corners

OctagonsA 46

B 65

C 11D 56

E

23

F 8

H 37

G 19

I 12 J 14

K 27

Do not appear

Feasible Octagons

Octagons of nine shapes must satisfy some conditions on size

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

First traverse root

Algorithm

Then traverse the right subtree

Finally, traverse the left subtree

Depth-first search

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

Algorithm

f

f

ff

f

ff

ff

1

2

34

5

6

7

8

9

10f

P1

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

Algorithm

f

f

ff

f

ff

ff

1

2

34

5

6

7

8

9

10f

P2

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

Algorithm

f

f

ff

f

ff

ff

1

2

34

5

6

7

8

9

10f

P3

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

Algorithm

f

f

ff

f

ff

ff

1

2

34

5

6

7

8

9

10f

P4

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

Algorithm

f

f

ff

f

ff

ff

1

2

34

5

6

7

8

9

10f

Algorithm

Initialization at root

Algorithm

Initialization at rootDraw the outer cycle as an arbitrary rectangle of area A(G).

A(G): sum of the prescribed areas of all inner faces in G.

Algorithm



59

15

8

46081595)( GA

Algorithm



59

15

8

W = 23

H = 20

46081595)( GA 2023

Algorithm



Fix arbitrarily the position of vertices on the right side of the rectangle preserving their relative positions.

Root : vertical sliceOperation at root

Algorithm

A(G)

Pr : V

Pv : VPw : V

r

vw

A(Gv)A(Gw)

Pr

A(Gw) A(Gv)

Root : vertical sliceOperation at root

Algorithm Pr : V

Pv : VPw : V

r

vw

Root : horizontal sliceOperation at root

AlgorithmPr : H

Pv : VPw : V

r

vw

RA(Gv)

A(Gw)


AlgorithmPr : H

Pv : VPw : V

r

vw

A(Gw)

A(Gv) R

Case 1: A(Gv) = A(R)

A(Gv)

A(Gw)


Algorithm

A(Gw)

A(Gv)

Case 2: A(Gv) > A(R)

RA(Gv)

A(Gw)


Algorithm Case 3: A(Gv) < A(R)


A(Gw)

A(Gv)A(Gv)

A(Gw)


Algorithm

6 corners A(Gw)

A(Gv)

Case 3: A(Gv) < A(R)


A(Gw)

A(Gv)A(Gv)

A(Gw)

Polygon of exactly 8 corners may appearwhen a horizontal slice is embedded insidea polygon of 6 corners.

6 corners

8 corners

Pu : V

Pv : VPw : V

u

vw

General computation at an internal node

Feasible Octagon

Ru

Vertical slice

Pr : V

Pv : VPw : V

u

vw


Feasible Octagon

Ru

Vertical slice Fixed

Pr : V

Pv : VPw : V

u

vw


Feasible Octagon

Ru

Vertical slice FixedFace

Pr : V

Pv : VPw : V

u

vw


Feasible Octagon

Ru

Embed the slicing path in Ru

Vertical slice

Pr : V

Pv : VPw : V

u

vw


Feasible Octagon

Ru

Vertical slice

Feasible Octagon

Rv

Feasible Octaon

Rw

A(Gv )

A(Gw)

A(Gv) A(Gv)

A(Gw)

Large

10 corners

Horizontal slice

very small foot-lenght

Ru

Pu

fA(Gv)

10 corners


Ru

Pu

f

Hf

Aminf number of inner faces in G


Ru

Pu

f

Hf


K 27

G 19A 46

C 11

J

14

I 12

B 65

D 56

E 23

F 8H 37

Input

Amin

Amin = 8


Ru

Pu

f

Hf


H height of initial rectangle Rr

Input at root Rr

A(R ) = A(G)r

H=20


Ru

Pu

f

Hf


K 27

G 19A 46

C 11

J

14

I 12

B 65

D 56

E 23

F 8H 37

Input

Amin

Amin = 82015

8

A(R ) = A(G)r

H=20


Ru

Pu

f

Hf



Ru f

large enough

East side


Ru f

large enoughEf

East side

Ef The number of inner faces each of which has an edge on the east side.

Computation at a leaf node

PE

PW

PS

NP

PE

PW

PS

NP

Time Complexity

Overall time complexity is linear.

Conclusion

We have presented a linear algorithm for prescribed area octagonal drawings of good slicing graphs.

Obtaining such an algorithm for larger classes of graphs is our future work.

We also give a sufficient condition for a graph of maximum degree 3 to be a good slicing graph and give a linear-time algorithm to find a good slicing tree of such graphs.

A sufficient condition for a good slicing graph

A good slicing tree can be found in linear time for a cyclically 5-edge connected plane cubic graph.

Cyclically 5-edge Connected Cubic Plane Graphs

Removal of any set of less than 5-edges leaves a graph such that exactly one of the connected components has a cycle.

Ｘ

ＸＸ

Ｘ

Ｘ

ＸＸ

Ｘ

ＸＸ

Ｘ

Ｘ

Not cyclically 5-edge connected

Two connected componentshaving cycles.

No connected component has a cycle.

A sufficient condition for a good slicing graph

Any graph G obtained from a cyclically 5-edge connected plane cubic graph by inserting four vertices of degree 2 on outer face is a good slicing graph.

Augmentation

Algorithm

Initialization at root

Ru

PE

If a large number of inner faces have edges on PE then foot-lengh should be large enough.

Dimensions of Ru play crucial roles.

large

Dimensions of a Feasible Octagon ?

lt

lb

lt

lb

flfl bEt

f number of inner faces in G


a positive constant.

Hf

Amin0

lt

lb




Hf

Amin0

flfl bEt

K 27

G 19A 46

C 11

J

14

I 12

B 65

D 56

E 23

F 8H 37

Input

Amin

Amin = 8

lt

lb




Hf

Amin0

flfl bEt H height of initial rectangle Rr

Input at root Rr

A(R ) = A(G)r

H=20

lt

lb



flfl bEt


Hf

Amin0

K 27

G 19A 46

C 11

J

14

I 12

B 65

D 56

E 23

F 8H 37

Input

Amin

Amin = 82015

8

A(R ) = A(G)r

H=20

K 27

G 19A 46

C 11

J

14

I 12

B 65

D 56

E 23

F 8H 37

Input

Amin

Amin = 82015

8

A(R ) = A(G)r

H=20


Input at root Rr

A(R ) = A(G)r

H=20

a

b

c

d

lt

lb

},max{ bt lll

l < f δ

a

b

c

d

ltu

lbu},max{ butuu lll

l < f δ

Red area < lu H < f δH < Amin

A(Gv )

A(Gw)

A(Gv) A(Gv)

A(Gw)

Large

This situation does not occur.

Eb fl

lblb

Eb fl

lblb

for a facial octagon whose xS1 is convex.bul

Pr : V

Pv : VPw : V

u

vw


Feasible Octagon Ru

Feasible Octagon

Rv

Feasible Octagon

Rw

Time Complexity

Using a bottom-up computation on slicing tree, areaof subgraphs for all internal nodes can be computed inlinear time.

With an O(n) time preprocessing, embedding of theslicing path at each internal node takes constant time.

Overall time complexity is linear.

Computation time at a leaf node is proportional tothe number of non-corner vertices on the west side of the face.

Conclusion

We have presented a linear algorithm for prescribed area octagonal drawings of good slicing graphs.

Obtaining such an algorithm for larger classes of graphs is our future works.

P

P

P

P

N

E

S

W uR

a

b

c

d

ltu

lbu

xN2

xN1

xS1

xS1

},max{ butuu lll

l < f δ

ltu = 0

lbu = 0

lu = 0

If the octagon is a rectangle

Ru is a feasible octagon if Ru satisfies the following eight conditions

(ii) lu < fδ(i) A(Ru) =A(Gu)

(iii) if xN 2 is a convex corner then u

Etu fl

(iv) if xS1 is a convex corner then u

Ebu fl

(v) if both xN2 and xS then u

Etubu fll

Feasible Octagon

uEbu fl

(vi) if both xN1 and xS1 are concave corners then u

Ebutu fll

(vii) if xN 2 is a concave corner then )( u

Etu ffl

(viii) if xS1 is a concave corner then )( u

Ebu ffl

P

E

P P P

PP

P

P

P

P

f

f

ff

f

ff

ff

1 23

5

4

67

8

9

1

2

34

5

6

7

8

9

10f

P

P

P

N

S

W

P : V1

P : V2

P : H4

P : H3P : H7P : H9

P : H8

P : V5

P : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

u

(i) A(Ru) =A(Gu)

Ru

(iii) if xN 2 is a convex corner then u

Etu fl

xN1

xN2

ltu

for a facial octagon whose xN2 is convex.tul

xN1

xN2

ltu

(vi) if xS1 is a convex corner then u

Ebu fl

xs1

xs2

xs1

xs2lbulbu

for a facial octagon whose xS1 is convex.bul

(v) if both xN2 and xS2 are concave corners then u

Etubu fll

xs1

xs2

xN1

xN2

lbu - ltu


Etubu fll

xs1

xs2

xN1

xN2 lbu - ltu

tubu ll

for a facial octagon whose xN2 and xS2 are concave

(vi) if both xN1 and xS1 are concave corners then u

Ebutu fll

lbu - ltu

butu ll

for a facial octagon whose xN1 and xS1 are concave

Pr : V

Pv : VPw : V

u

vw


Feasible Octagon Ru

Feasible Octagon

Rv

Feasible Octagon

Rw

Embedding of a vertical slicing path

P

P

P

N

E

S

WuR

a

b

c

d

xN2

xN1

xS1

xS1

P

P

P

P

N

E

S

W uR

a

b

c

d

ltu

lbu

xN2

xN1

xS1

xS1

},max{ butuu lll

(ii) lu < f δ

Red area < lu H < f δH < Amin

The vertical slicing path is always embedded as a vertical line segment.

Embedding of a vertical slicing path

P

P

P

N

E

S

W

a

b

c

d

xN2

xN1

xS1

xS1

Rw

Rv

Rw is a rectangewhich is a feasible octagon.

Rv is a feasible octagonsince Ru is a feasible octagon.

Embedding of a horizontal slicing path

xs1

xs2

xN1

xN2


Etubu fll

Rv

Rw

Ru

vEtubv fll

Both Rv and Rw are feasible octagons.

We can prove for other cases.

Computation at a leaf node x

PE

PW

PS

NP

PE

PW

PS

NP

Algorithm Octagonal-DrawS : V1

S : V2

S : H4

S : H3S : H7S : H9

S : H8

S : V5

S : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

Reverse preorder traversal from root

very small

Intuitively

Ru

Pu A(Gv )

Pu can be embedded successfully although A(Gv) is very large.

Dimensions of Ru play crucial roles.

We call Ru a feasible octagon if Pu can embedded successfully irrespective of size of A(Gv).

P

P

P

P

N

E

S

W xG

AlgorithmS : V1

S : V2

S : H4

S : H3S : H7S : H9

S : H8

S : V5

S : H6

f9f10

f8

f6f7

f4

f3 f2

f5

f1

P

EP

P

P

N

S

WG

Input at root

(a) (b) (c) (d) (e)

Px Px PxPx

Px

A face in Rz may need to be drawn with more than eight corners.

Rz

Ry

Rz

Ry

Rz

Ry

Rz

Ry

Rz

Ry

Computation at a V-node x

Let y be the right child of x and z be left child of x.

Foot Length lx of an Octagon Rx

P

P

P

P

N

E

S

W xG tx

bxlx = max {tx, bx}

P

P

P

P

N

E

S

W uR

a

b

c

d

ltu

lbu

fEf The number of inner faces in Gu

each of which has an edge on the east side.


fH

Amin0 f number of inner faces in G


xN2

xN1

xS1

xS1

If the footlength of Rx is resonably small then Rz willalways be drawn as a rectngle.

Px Px PxPx

Px

Rz

Ry

Rz

Ry

Rz

Ry

Rz

Ry

Rz

Ry

Neck Length d of a Facial Octagon

dd d

d

Maximum foot length

P

P

P

N

S

W

G

P

Efdl max

lmax

f : number of faces in G

P

P

N

W P

PE

S

tx

bxlx = max {tx, bx}

H

minAfdH

minAfdHHfdHlx

Estimating d

We fix d such that holds.

Px Px PxPx

Px

Rz

Ry

Rz

Ry

Rz

Ry

Rz

Ry

Rz

Ry

Rz is always a rectangle.

Computation at a H-node x

Let y be the right child of x and z be left child of x.

PW

PE

PS

yG tx

bx

NP

zG

xG

PE

PW

PS

tx

bx

NP

yG

zG

xG

PE

PW

PS

tx

bx

NP

yG

zG

xG

According to Areas of Gy and Gz

Satisfying some Invariants

dfbt Exxx |

Exf number of faces in Gx having an edge on

xEP

xEP

xEP

Invariants

tx

tx

bx

bx

dft Exx

dfb Exx

dft Exx

dfb Exx

Invariants

tx

tx

bx

bx


PE

PW

PS

tx

bx

NP

yG

zG

xG

PE

PW

PS

tx

bx

NP

yG

zG

xG

Time Complexity


K 27

G 19

F 8

C 11

J 14

I 12

A 46

B 65

D 56

E 23

H 37

G’11 65

s1

1sEG1s

WG

1sWG 1s

EGs1

76

s2

C B

s2

46

318

242

196

56

140

23

117

4572

37 8 1953

27

12

26

14

s4

s5

E

s5

Ds6

s6

s10

FH G

s7

s8

K s9

JI

s3As4

s3

Time Complexity



Exf number of faces in Gx having an edge on

xEP x

EP

xEP

txtx

bx bx

PE

PW

PS

tx

bx

NP

yG

zG

xG

PE

PS

yG tx

bx

NP

zG

xG

Time Complexity



Computation time at a leaf node is proportional tothe number of non-corner vertices on the west side of the face.


PE

PW

PS

tx

bx

NP

yG

zG

xG

PE

PW

PS

tx

bx

NP

yG

zG

xG

Foot Length lx of an Octagon Rx

P

P

P

P

N

E

S

W xG

tx

bx

lx = max {tx, bx}

According to Areas of Gy and Gz

Satisfying Invariant

A Linear Algorithm for Prescribed-Area Octagonal Drawings

of Plane Graphs

Md. Saidur RahmanKazuyuki MiuraTakao Nishizeki

TOHOKU UNIVERSITY

Previous works on prescribed-area drawing

Thomassen, 1992

Obtained from cyclically 5-edge connected plane cubic graphs by inserting four vertices of degree 2 on outer face.

G:

Cyclically 5-edge Connected Cubic Plane Graphs

Removal of any set of less than 5-edges leaves a graph such that exactly one of the connected components has a cycle.

Ｘ

ＸＸ

Ｘ

Ｘ

ＸＸ

Ｘ

ＸＸ

Ｘ

Ｘ

Not cyclically 5-edge connected

Two connected componentshaving cycles.

No connected component has a cycle.

Previous works on prescribed area drawing

Thomassen, 1992

Obtained from cyclically 5-edge connected plane cubic graphs by inserting four vertices of degree 2 on outer face.

G:

G has a prescribed area straight-line drawing.

An inner face is not always drawn as a rectilinear polygon.

Outer face is a rectangleInner faces are arbitrary polygons

O(n3) time algorithm.

Theorem

Let G be a 2-3 plane graph obtained from a cyclically 5-edgeconnected plane cubic graph by inserting four vertices ofdegree 2 on four distinct edges. Then G is a good slicing graph.

That is, Thomassen’s graph is a good slicing graph.

Cyclically 5-edge Connected cubic graph with four vertices of

degree 2 inserted on outer face

Good Slicing Graphs

Slicing Graphs

Slicing Graph

P

P

P

P

N

E

S

W NS-path

P

P

P

P

N

E

S

WWE-path

G is a slicing graph if either it has exactly one inner face or it has an NS-path or a WE-path P such that both the subgraphs corresponding to P are slicing graphs.

PP

P is called a slicing path.

G: 2-3 plane graph

GG

G

G

PP

P

PW

EN

S

octagonal drawings of plane graphs with prescribed face areas takao nishizeki (tohoku univ.)...

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