Floorplanning. Obtained by subdividing a given rectangle into smaller rectangles. Each smaller...

Floorplanning

Obtained by subdividing a given rectangle into smaller rectangles.

Each smaller rectangle corresponds to a module.

Slicing Floorplan

Slicing Floorplan

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

Slicing Floorplan and Slicing Graph

Slicing Floorplan

Non-slicing Floorplan

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

Graph representation of Floorplan

Polar graphsAdjacency graphsChannel Intersection GraphsChannel position graphs

Polar Graphs

Vertical polar graph

Horizontal polar graph

sourcesink

source

sink

Polar Graphs

Vertical polar graph

Horizontal polar graph

sourcesink

source

sink

Used to determine the area of the smallest floorplan.

Adjacency Graphs

A D

B

C

E

F

G

H

A

B

C

D

E

FH

G

source

sink

Horizontal adjacency graphsource

sink

A

B

C

D

EF

G

H Vertical adjacency graph

Channel Intersection Graphs

A

C

B

D

Channel Position Graphs

A

C

B

D

A B

C D

A B

C D

Horizontal channel Position graph

Vertical channel position graph

Given circuit modules (or cells) and their connections, determine the approximate location of circuit elements

Cost functions for floorplanning

Minimize areaMinimize wirelengthMaximize routabilityMinimize delays

EA

B

C

F

G

D

AB

EC

F

G

D

VLSI Floorplanning with adjacency requrementsVLSI Floorplanning with adjacency requrements

Interconnection graph VLSI floorplan

EA

B

C

F

G

D

AB

EC

F

G

D

VLSI FloorplanningVLSI Floorplanning


EA

B

C

F

G

D

AB

EC

F

G

D


AB

E

C

F

G

D


Dual-like graph

EA

B

C

F

G

D

AB

EC

F

G

D


AB

E

C

F

G

D

AB

E

C

F

G

D


Dual-like graph Add four corners

EA

B

C

F

G

D

AB

EC

F

G

D


AB

E

C

F

G

D

AB

E

C

F

G

D


Dual-like graph Add four corners

Rectangular drawing

EA

B

C

F

G

D

AB

EC

F

G

D



Rectangular drawing

EA

B

C

F

G

D

AB

EC

F

G

D



Rectangular drawing

Unwanted adjacency

Not desirable for MCM floorplanning andfor some architectural floorplanning.

A

B

E C

F

G

D

MCM FloorplanningMCM Floorplanning SherwaniSherwani

Architectural FloorplanningArchitectural Floorplanning Munemoto, Katoh, ImamuraMunemoto, Katoh, Imamura

EA

B

C

F

G

D

Interconnection graph

A

B

E C

F

G

D

MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning

EA

B

C

F

G

D



A

E

B

FG

CD

EA

B

C

F

G

D


Dual-like graph

A

B

E C

F

G

D


A

E

B

FG

CD

EA

B

C

F

G

D

A

E

B

FG

CD


Dual-like graph

A

B

E C

F

G

D


A

E

B

FG

CD

EA

B

C

F

G

D

A

E

B

FG

CD

Box-Rectangular drawing


Dual-like graph

A

B

E C

F

G

D

dead space

Floorplanning with Area Requirements

Octagonal Drawing

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.

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 Tree

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.

Floorplanning. Obtained by subdividing a given rectangle into smaller rectangles. Each smaller...

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