Lec18 Steiner

8/3/2019 Lec18 Steiner

1/12

j

A l g o r i t h m s f o r V L S I P h y s i c a l D e s i g n A u t o m a t i o n

c S h e r w a n i 9 2

3 . 6

D a t a S t r u c t u r e s a n d B a s i c A l g o r i t h m s

S p a n n i n g T r e e

~

P r o b l e m F o r m u l a t i o n :

G i v e n a g r a p h G = ( V E ) , s e l e c t a s u b s e t V

0

V ,

s u c h t h a t V

0

h a s p r o p e r t y P .

~

M i n i m u m S p a n n i n g T r e e

~

P r o b l e m F o r m u l a t i o n :

G i v e n a n e d g e - w e i g h t e d g r a p h G = ( V E ) , s e l e c t a s u b s e t

o f e d g e s E

0

E s u c h t h a t E

0

i n d u c e s a t r e e a n d t h e

t o t a l c o s t o f e d g e s

P

e

i

2 E

0

w t ( e

i

) , i s m i n i m u m o v e r

a l l s u c h t r e e s , w h e r e w t ( e

i

) i s t h e c o s t o r w e i g h t o f

t h e e d g e e

i

.

{ U s e d i n r o u t i n g a p p l i c a t i o n s .


2/12

j



3 . 1 3


S t e i n e r T r e e s

~

1 . P r o b l e m f o r m u l a t i o n :

G i v e n a n e d g e w e i g h t e d g r a p h G = ( V E ) a n d a s u b s e t D V ,

s e l e c t a s u b s e t V

0

V , s u c h t h a t D V

0

a n d V

0

i n d u c e s a t r e e o f m i n i m u m c o s t o v e r a l l s u c h t r e e s .

T h e s e t D i s r e f e r r e d t o a s t h e s e t o f d e m a n d p o i n t s a n d

t h e s e t V

0

; D i s r e f e r r e d t o a s S t e i n e r p o i n t s .

U s e d i n t h e g l o b a l r o u t i n g o f m u l t i - t e r m i n a l n e t s .

Demand Point

B

CD

FJ

H

7

7

5

4

9

5 6

12

8

6 5 5

2 3

5

6

6

6

A

I

E

G

(a) (b)


3/12

j



3 . 1 4


U n d e r l y i n g G r i d G r a p h

~

T h e u n d e r l y i n g g r i d g r a p h i s d e n e d b y t h e i n t e r s e c t i o n s o f t h e

h o r i z o n t a l a n d v e r t i c a l l i n e s d r a w n t h r o u g h t h e d e m a n d p o i n t s .

Hanan's Thm (69'):

There exists an optimal

RST with all Steiner

points (set S) chosen

from the intersection points of horizontal

and vertical lines

drawn from points of D.


4/12

j



3 . 1 5


D i e r e n t S t e i n e r t r e e s c o n s t r u c t e d f r o m a M S T

(a) (b)

(c) (d)

(e)

Hwang's Thm (76'):The ratio of the cost of

a rectilinear MST to

that of an optimal RST

is no greater than 3/2.


5/12

Rectilinear Steiner TreesGiven a set of points on the planeDetermine a Steiner tree using only horizontal and vertical

wires( lines)

Manhattan distance:

cost(v1,v2) =|x1-x2|+|y1-y2|

v1=(x1,y1), v2=(x2,y2)

Steiner pointsDraw a horizontal and a vertical line through each point.

Need to consider only grid points as steiner points

Prim-based algorithm:

Grow a connected subtree by iteratively adding the closest

points

It gives 3/2-approximation, i.e. cost(T)3/2cost(Topt)

v1=(x1,y1)

v2=(x2,y2)


6/12

Steiner Tree Heuristics Observation: MST approximation can be easily improved

Difficulty: where to add Steiner points??

cost(T)=6 cost(T)=4


7/12

L-Shaped MST approach Ho, Vijayan and Wong, A new approach to the rectilinear steiner tree problem, DAC89, pp161-166

Basic Idea: Each on-degenerated edge in MST has two possible

L-shaped layouts, choose one for each edge in MST to maximizeoverlap.

degenerate edges non-degenerate edge two L-shaped layouts

MST one L-shaped mapping another L-shaped

mapping

Problem: Compute the best L-shaped mapping


8/12

Key Ideas in L-RST ApproachSeparable MST: bounding boxes of every two non-adjacent edges dont intersect or overlap

Theorem: Every point set has a separable MSTTheorem: Each node is adjacent to at most 8 edges

(6 non-degenerate edges) in a rectilinear MST

Theorem: We can compute an optimal L-shaped

implementation of an MST in O(2dn) time.

( Dynamic Programming Approach).

Note that d6

non-separable MST separable MST


9/12

Compute a Separable MSTWeight (i,j) of each edge is a 4-tuplew(i,j)= (dij, -|yi-yj|, -max(yi,yj), -max(xi,xj))

Weights are compared underlexicographic ordering

Use Prims algorithm to compute a MST based on

the weight function

we obtain a rectilinear MST since the 1st

component of w(i,j) is dij

This MST is separable since the next threecomponents in w(i,j) help break ties

Run time O(n2)

dij

i (xi,yi)

j (xj,yj)


10/12

Compute an Optimal L-shaped Mapping

Choose an arbitrary node as root

Using dynamic programming

Compute at each node v the following

l(vi): min cost of Ti with (v,vi) using lower L-shapeu(vi): min cost of Ti with (v, vi) using upper L-shape

Ti Ti


11/12

Compute an Optimal L-shaped Mapping(Contd)

l(v) (or u(v) ) can be computed by examining 2dcombinations ofl(vi) and u(vi) (1id) with (pv,v) takingupper (or lower) L

T1 T2 Td

v

pv

T1 T2 Td

v

pv


12/12

Applying Spanning/Steiner Tree Algorithms General cell design: channel intersection graphs

Standard cell/Gate array/Sea-of-gate designrectilinear steiner/spanning trees or grid graphs

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Lec18 Steiner