Electronics and Communications i n Japan, Vol. 65-A, No, 2, 1982

An Analysis of Wiring Performances of a Routing System for

High Density Printed Wiring Boards

Shigeo Asahara, Regular Member

Corporate Engineering Division, Matsushita E l e c t r i c Indus t r i a l Co., Ltd. , Moriguchi, Japan 570

Masayasu Odani, Regular Member

Semiconductor Division, Toshiba Corporation, Kawasaki, Japan 210

Yasunori Ogura, Regular Member

CAD Center, Sharp Corporation, Tenri , Japan 632

Isao Shirakawa and Hiroshi Ozaki, Regular Members

Faculty of Engineering, Osaka Universi ty , Sui ta , Japan 565

SUMMARY

Recent advances i n the technology of microelectronics have changed the design r u l e f o r pr inted wiring boards, allowing the num- ber of wiring t racks between consecutive p ins of an ordinary dual i n l i n e package (DIP) t o be two o r more. When the wiring densi ty augments t o tha t ex ten t , conventional rou ters a r e confronted with var ious d i f f i c u l t i e s .

To cope with t h i s s i t ua t ion , the s ingle- row router which has topological f l u i d i t y can be employed i n conjunction with the l ine- search router . new rout ing system constructed of a l i ne - search router combined with a single-row router and of a maze-running router . ever i n each routing process there are many parameters, the combination of whose values have much influence on the wiring perform- ance of t he whole system.

We have already developed a

How-

In t h i s paper we ou t l i ne t h i s rout ing system, descr ibe some experimental r e s u l t s i n order t o determine values of these parameters,

and show tha t t he ana lys i s of implemented re- s u l t s suggests high performance f o r high dens- i t y pr in ted c i r c u i t boards (PWB's) .

1. Introduct ion

Most ex i s t ing rout ing systems are con- s t ruc t ed of severa l d i s t i n c t rou ters , such as maze-running routers , l ine-search rou te r s , o r channel routers . Thus the merits of one may compensate f o r t he defec ts of another. These rout ing systems have been ab le t o cont r ibu te much t o reducing the t i m e and cos t incurred i n laying out w i r e pa t te rns on PWB's.

However, recent advances i n the tech- nology of microelectronics have changed the design r u l e f o r PWB's i n such a way t h a t t he between-pins capaci ty of wiring t racks can be ra i sed t o two o r more.

When the spec i f i ca t ions fo r a PWB are t o undergo such a change, conventional rou ters are confronted with var ious d i f f i c u l t i e s . The reasons are as follows:

32

Wire segment on the 1st layer Cqll

Pi

v i

track

Fig. 1. Channels and t r a c k s on two-layer p r i n t e d wir ing boards.

(1) When t h e between-pins capac i ty increases two o r more, t h e number of cel ls ( t h e smallest wir ing u n i t s ) i nc reases rapidly. This causes a decrease i n processing speed and an inc rease of memory space, and t h e r e f o r e l e a d s t o t h e decrease of w i r ing performances.

(2 ) For high dens i ty boards, t h e f ixed v i a s t r a t e g y [ l ] i s usua l ly t o be taken, t h a t is , v i a s a r e located a t predetermined r e g u l a r g r ids . I n t h i s case, conventional r o u t e r s cannot be app l i ed without modif icat ions.

To cope with t h i s s i t u a t i o n , we have already developed a new rou t ing system con- s t r u c t e d of t h e l ine-search r o u t e r combined with a single-row r o u t e r which has "topol- og ica l f l u i d i t y " [ 21, and of a maze-running rou te r based on Hadlock's method [3 ] . How- ever , i n each rou t ing process t h e r e a r e many parameters, t h e choice of whose va lues has much in f luence on t h e wir ing performance of t h e whole system.

I n t h i s paper, we o u t l i n e t h i s rou t ing system [ 4 ] , desc r ibe some experimental re- s u l t s i n o rde r t o determine t h e va lues of t hese parameters, and show t h a t t h e a n a l y s i s of experimental r e s u l t s suggests s t rong ly t h a t t h i s rou t ing system may a t t a i n high p o t e n t i a l i t i e s i n t h e p r a c t i c e of layout of high dens i ty PWB's.

2. Overview of t h e Routing System

2 .1 P r e l i m i n a r i e s

When a rou t ing system i s t o be con- s t r u c t e d so as t o treat PWB's of d i f f e r e n t s p e c i f i c a t i o n s such t h a t :

( i ) t h e between-pins capac i ty of wir ing t r a c k s is t o be two o r more, and

( i i ) t h e fixed-via s t r a t e g y i s t o be taken, t h a t i s , t h e p o s i t i o n s of vias are t o be f i x e d ,

it is of primary importance how t o descr ibe w i r e p a t t e r n s on such PWB's. We f i r s t touch on a scheme t o desc r ibe t h e in t e rconnec t ions on a PWB.

The smallest u n i t t o d e s c r i b e w i r e pat- t e r n s on each l a y e r of a FWB i s a t r a c k on which wire segments are a l l o c a t e d . Given a PWB, l e t u s provide h o r i z o n t a l and v e r t i c a l t r a c k s on t h e working a r e a of each l aye r ac- cording t o s p e c i f i c a t i o n s of t h e dens i ty i m - posed on t h e board, and denote by a square r eg ion a t t h e i n t e r s e c t i o n of a hori- z o n t a l t r a c k and a v e r t i c a l one, as shown i n Fig. 1. In o r d e r t h a t ' a l ine-search r o u t e r can be combined with a single-row r o u t e r , and t h a t t h e fixed-via s t r a t e g y can be taken, l e t

a

33

Line - sea rch r o u t e r

S i n g I c - r o 1v r o u t e r

hla z e - r u 11 n i ng r o u t e r

I via elimination I

6 s Lo [I

F i g . 2 . A g e n e r a l f l o w c h a r t of r o u t i n g pro- cess.

u s i n t r o d u c e a n o t h e r u n i t , c a l l e d a c h a n n e l , which c o n s i s t s of ku + kw + 1 t r a c k s such t h a t

( i ) each c h a n n e l c o n t a i n s t h e u p p e r and lower s t reets composed of ku and kw t r a c k s , r e s p e c t i v e l y , as i l l u s t r a t e d i n F i g . 1 (where t h e c a s e of ku = 2 and k l = 1 i s shown), on which p o r t i o n s of w i r e segments e x c e p t v i a s are p e r m i t t e d t o b e g e n e r a t e d , and

( i i ) each c h a n n e l c o n t a i n s e x a c t l y one t r a c k between t h e upper and lower s t r e e t s , on which v i a s and p i n s are p e r m i t t e d t o b e p l a c e d . Denote by a b&k a s q u a r e r e g i o n a t the i n t e r s e c t i o n of a h o r i z o n t a l c h a n n e l and a v e r t i c a l c h a n n e l . Thus, i t f o l l o w s t h a t t h e r e i s one and o n l y one c e l l i n a b l o c k a t which a v i a o r p i n c a n be l o c a t e d , a s c a n b e s e e n from F i g . 1 .

Assume t h a t t h e PWB's t o b e c o n s i d e r e d h e n c e f o r t h a r e of two l a y e r s ; t h e f i r s t l a y e r is f o r u s e of h o r i z o n t a l w i r e segment and t h e second i s f o r v e r t i c a l w i r e segment i n p r i n - c i p l e .

2 . 2 Rout ing p r o c e d u r e s

A g e n e r a l f l o w c h a r t of the r o u t i n g pro- c e s s i s shown i n F i g . 2 . I n o r d e r t h a t t h e sys tem can d e a l w i t h PWB's of v a r i o u s d e n s i t y s p e c i f i c a t i o n s , w e se t up several v a r i a t i o n s i n a p p l y i n g t h e l i n e - s e a r c h r o u t e r combined w i t h t h e s ing le- row r o u t e r and t h e maze- r u n n i n g r o u t e r as f o l l o w s :

[LINE-SEARCH ROUTER]

A l i n e - s e a r c h r o u t e r and a s i n g l e - r o w r o u t e r i s i n c o r p o r a t e d i n s u c h a manner t h a t t h e l i n e - s e a r c h r o u t e r f i r s t a s s i g n s i n t e r - c o n n e c t i o n r e q u i r e m e n t s o r n e t s t o e a c h

c h a n n e l , and t h e n t h e s i n g l e - r o w r o u t e r real- i z e s a l l t h e n e t s a s s i g n e d t o e a c h c h a n n e l .

Mode 0 assumes t h a t a c h a n n e l c o n s i s t s o n l y of a s i n g l e t r a c k , and t h a t p o s i t i o n s o f v i a s a re f l o a t i n g , a l t h o u g h some con- s t r a i n t s c a n b e imposed on t h e i r r e l a t i v e l o c a t i o n s a c c o r d i n g t o s p e c i f i c a t i o n s . Thus t h i s mode can b e a p p l i e d t o PWB's f o r which t h e f l o a t i n g - v i a s t r a t e g y i s a v a i l a b l e and t h e between-pins c a p a c i t y i s one .

I n t h e o t h e r modes, the l i n e - s e a r c h r o u t e r f i r s t s e e k s a s e t of f e a s i b l e c h a n n e l segments t o i n t e r c o n n e c t two b l o c k s , and t h e n c h e c k s whether o r n o t each n e t t o b e a s s i g n e d to such a c h a n n e l segment t o g e t h e r w i t h a l l t h o s e so f a r a s s i g n e d t o t h e c h a n n e l s a t i s f y t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s [ 5 ] f o r s i n g l e - r o w r o u t i n g w i t h p r e s c r i b e d upper and lower s t reet c a p a c i t i e s . I f a l l such n e t s prove t o s a t i s f y t h e c o n d i t i o n s , t h e n t h e r o u t e r a l l o c a t e s t h e s e n e t s t o t h e f e a s i b l e c h a n n e l segments .

Mode 1 assumes t h a t ku = 1 and kw = 1; Mode 2 assumes t h a t k, = kw = 1; Mode 3 assumes t h a t k u = 2 and kw = 1 ;

Mode 4 assumes t h a t kLl = kw = 2.

C l e a r l y , t h e s e f o u r modes also assume t h a t t h e f i x e d - v i a s t r a t e g y i s t o b e t a k e n , and t h a t t h e be tween-p ins c a p a c i t i e s a re 1 , 2 , 3 , and 4 , r e s p e c t i v e l y ,

and

[SINGLE-ROW ROUTER]

A f t e r t h e l i n e - s e a r c h r o u t e r a s s i g n s i n t e r c o n n e c t i o n r e q u i r e m e n t t o e a c h c h a n n e l , i t i s a lways g u a r a n t e e d t h a t a l l the n e t s can b e r e a l i z e d w i t h i n e a c h c h a n n e l . T h e r e f o r e , t h e s i n g l e - r o w r o u t e r can b e a p p l i e d t o chan- n e l s independent ] -y , and w i t h i n e a c h c h a n n e l i t r e a l i z e s a l l the a s s i g n e d n e t s u s i n g t h e a l g o r i t h m i n ( 5 ) .

[MAZE-RUNNING ROUTER]

The maze-running r o u t e r employed i n t h i s r o u t i n g s y s t e m is b a s e d on H a d l o c k ' s a l g o r i t h m [ 31 which i n t r o d u c e s a depth- f i r s t - t e c h n i q u e so as t o a v o i d u n n e c e s s a r y s c a n of c e l l s . T h i s r o u t e r i s m o d i f i e d t o b e a p p l i e d f o r two- l a y e r PIJB's, and f i n d s t h e r o u t e s between two unconnected c e l l s .

Mode A. T h i s assumes t h a t a l l h o r i z o n t a l and v e r t i c a l wire segments are l i m i t e d t o be g e n e r a t e d o n l y i n h o r i z o n t a l and v e r t i c a l c h a n n e l s on t h e f i r s t and second l a y e r s , re- s p e c t i v e l y , and the f l o a t i n g - v i a s t r a t e g y i s t o b e t a k e n .

Mode B. T h i s a l s o assumes t h a t a l l h o r i - z o n t a l and v e r t i c a l w i r e segments a re l i m i t e d

34

Table 1. PWB s t a t i s t i c s

PWB’S

279 X 203 137X 188 178X 159 183 X 274

t o be generated only i n h o r i z o n t a l and v e r t i - cal channels on t h e f i r s t and second l a y e r s , r e spec t ive ly , but t he fixed-via s t r a t e g y is t o be taken.

Mode C . This assumes t h a t any such r e s t r i c t i o n a s i n Modes A and B is no t i m - posed on generating w i r e segments on t h e f i r s t o r second l a y e r , and t h a t t h e f loa t ing - v i a s t r a t e g y i s t o be taken.

Mode D. This a l s o assumes t h a t no re- s t r i c t i o n is imposed on generat ing w i r e seg- ments on t h e f i r s t o r second l a y e r , but t h a t t he fixed-via s t r a t e g y i s t o be taken.

3. An Experimental Analysis of t he Perform- ance of t he Line Search Router

A l ine-search r o u t e r i s b a s i c a l l y heur- i s t i c , which can f ind wir ing rou te s much f a s t e r than a maze-running rou te r , bu t rou te s thus generated do not always c o n s t i t u t e short- est paths.

The l ine-search r o u t e r employed h e r e is based on ( 6 ) , whose c h a r a c t e r i s t i c i s t h a t t h i s method can p red ic t t h e nonexistence of a path which c o n s i s t s of s i x o r less w i r e seg- ments and the re fo re does no t waste searches. The o u t l i n e of t h i s a lgori thm is shown i n Fig. 3, where n is t h e upper l i m i t of t h e number of l i n e segments.

3.1 Preliminary

In the following, some experimental re- s u l t s a r e shown t o r evea l what kinds of fac- t o r s have much inf luence on t h e wir ing per- formance of t h e l ine-search rou te r . Table 1 shows t h e l i s t of t h e nine boards used i n these experiments.

To estimate t h e f i g u r e of merit of w i r e p a t t e r n s o r t o i n v e s t i g a t e the wir ing

No. of from- tos 398 437 60 1 686

1,142 50 1 303 91 1 277

Fig.

~~

3etween-pins capac i ty

1

( s t a r t ) J

I Y e s

no Are there unscanned from-tos ? )--c=3 s t o p

Select an unscanned from-to t

Seek a wiring route composed of three or less wire segments for the from-to

i t 4 I no

.-<n~i ?>- I Yes

Seek a wiring route composed of i wire segments for the from-to

3 . A gene ra l f lowchart of t h e l ine-search r o u t e r .

performance of t h e r o u t i n g system, we f i r s t introduce f i v e c r i t e r i a , which are:

(1) wi r ing performance rate:

t h e number of i n t e r -

x 100 (%), connected from-tos

t h e number of from-tos w = t o be interconnected

(2) t o t a l w i r e l eng th :

L = t h e t o t a l sum of t h e number of A ce l l s o r w i r ing routes ,

( 3 ) number of vias:

V t h e t o t a l number of v i a s ,

35

(4) c e l l occupation r a t i o (or wiring density) :

the number of unused c e l l s

1 a f t e r implementation the number of unused cells D e ( 1 -

before implementation

x 100 ( X ) ,

(5) processing t i m e :

T e CPU t i m e spent on searching wiring routes (sec) .

A t any s tage o r routing process, a set of c e l l s which cons t i tu te a maximal in te r - connected pa r t of a ne t N i is re fer red t o as a subnet of N i , and each p a i r of subnets of N i i s designated a s a from-to of N i , with i t s dis tance defined t o be the shor tes t one be- tween two c e l l s , one taken from a subnet and the other from another one.

3.2 From-to generation and ordering schemes

The l ine-search router employed here interconnects unconnected from-tos succes- sively; therefore , i t s wiring performance depends much not only on the method of de- composing a multi-terminal net i n t o from-tos, but on the ordering scheme of from-tos.

F i r s t , we consider two methods of de- composing a multi-terminal net i n t o from-tos,

( i ) P r io r t o rout ing process, construct a complete graph f o r each ne t k such tha t each ver tex corresponds t o a terminal and each edge t o a pa i r of terminals. each edge a weight defined as the r e c t i l i n e a r dis tance between the corresponding terminals , and seek a minimum spanning tree f o r each ne t .

At any s tage of routing process,

Assign t o

( i i ) fo r each net k s e l e c t the from-to which has the minimum r e c t i l i n e a r dis tance, and then search a wiring route f o r t h i s from-to.

In what follows, th ree conventional methods a re s t a t ed as t o the problem i n what order the selected from-tos should be in t e r - connected:

8 Among a l l from-tos generated, s e l ec t one randomly and then search f o r a wiring route f o r i t .

6) smallest area, seek wiring routes f o r from- tos i n an ascending order .of dis tance.

For the ne t k whose MDR* has the

*MDR of a net denotes the smallest rec- tangle which contains a l l terminals of t he net.

100

90

n x B L-l

80

70

LDnta 1 Data 2 1

/Data 4

Data 8

Data 5 Data 6

I I I I I I 1-a 2:a 1-6 2-6 1-c 2-12

Fig. 4. Wiring performance vs. combinations of s e l ec t ion and ordering schemes of from-tos.

.. (Xl, Yl) Search area

MDR

Fig. 5. Res t r ic t ion of search area.

@ A t any s tage of rout ing process, select the from-to whose d is tance i s the mini- mum among a l l generated from-tos, and then seek a wiring route f o r it.

PWB's of Table 1 a r e interconnected by the l ine-search router using severa l combina- t i ons of s e l ec t ion and ordering schemes of from-tos s t a t ed above, and r e s u l t s as shown i n Fig. 4 are obtained.

From t h i s f i gu re i t can be seen t h a t (ii) - @ i s the bes t combination f o r a l l the boards. Consequently, i n the following, ex- periments t he combination ( i i ) - @, is adopted.

36

3.3 R e s t r i c t i o n of s ea rch area

h :tion M Z 92.7 614 810

25,824 93.7 399 788

22,350 91.4 336 724

87.9 1,352 1,242

62,031

31,214

Thus, t h e number of v i a s i n c r e a s e s , and t h e t o t a l w i r ing performance may decrease.

Without r e s t r i c t i o n

L S M Z 89.7 92.5 323 776 722 769

23,167 25,553 90.6 93.6 270 594 773 811

20,670 23,003 81.1 89.8 95 354 577 669

84.8 88.4 464 1,437

1,059 1,250 56,518 63,547

24,908 29,886

In t h e l ine-search r o u t i n g p rocess , t h e search a rea of rou t ing between two blocks, A = (xa, ya) and B = (Xb, yb) is r e s t r i c t e d i n t h e r e c t a n g l e determined by f o u r param- eters a s i n Fig. 5 , where only a case i s shown i n which t h e c e n t e r of t h e board is i n t h e r igh t - and upper-hand s i d e of t h e c e n t e r of MDR con ta in ing A and B.

Now t h a t t h e optimal values of t h e s e parameters depend on PWB's, they cannot be determined uniquely. However, t h e experi- mental r e s u l t s suggest t h a t t h e bes t combina- t i o n is (a, B , y, 6 ) = (0.7, 0.2, 0.3, 5). Therefore , a l l t h e experiments i n t h e follow- i n g are t o be performed us ing t h i s combina- t i o n . The experimental r e s u l t s due t o t h e re- s t r i c t i o n of t h e search area are shown i n Table 2 , where LS denotes t h e wir ing perform- ance be fo re t h e maze-running r o u t e r is ap- p l i e d , and MZ denotes t h e f i n a l w i r ing per- formance a f t e r t h e maze-running r o u t e r . A s can be seen from t h i s t a b l e , t h e e f f e c t s of t h e r e s t r i c t i o n of t h e sea rch area are sum- marized as follows:

(1) searching e f f i c i e n c y (= W/T) i s i m - proved;

( 2 ) long de tou r s are p roh ib i t ed , s o t h a t t h e f i n a l wir ing performance a f t e r t h e maze- running r o u t e r tends t o increase; and

(3) t h e r e is a tendency t h a t long and simple wir ing p a t t e r n s decrease, whereas s h o r t and complicated w i r e p a t t e r n s inc rease .

3 . 4 R e s t r i c t i o n on t h e number of l i n e segments i n t h e l i ne - sea rch r o u t e r

I n t h i s rou t ing system, t h e l i ne - sea rch r o u t e r f i r s t gene ra t e s w i r e p a t t e r n s , and then t h e maze-running r o u t e r i n t e rconnec t s from-tos remaining incomplete. One of t h e important f a c t o r s t o improve e f f i c i e n c y is t o what ex- t e n t w i r e p a t t e r n s should be s i m p l i f i e d i n t h e l i ne - sea rch r o u t e r , i n o t h e r words, what is t h e upper l i m i t of t h e number of l ine seg- ments f o r each from-to,

Figure 6 shows t h e r a t i o of t h e number of from-tos r e a l i z e d w i t h n segments t o a l l t hose r e a l i z e d by t h e l i ne - sea rch r o u t e r , where t h e s ix boards shown in Table 1 are in- terconnected under t h e cond i t ion t h a t t h e upper l i m i t of l i n e segments f o r each from-to is s i x . It is obvious from t h i s f i g u r e t h a t most of w i r e p a t t e r n s are r e a l i z e d wi th f o u r o r less l i n e segments, and t h a t t h e upper l i m i t on t h e number of l i n e segments f o r each from-to has l i t t l e i n f l u e n c e on t h e f i n a l w i r i n g performance. Thus, i t may be concluded t h a t , i n terms of s ea rch ing e f f i c i e n c y t h e opt imal va lue of t h e upper l i m i t can be set t o fou r .

4 , The Analysis of Wiring Performance of t h e Maze-Running Router

The maze-running r o u t e r , f i r s t proposed by Lee [7], has been improved by many

Table 2. The r e s u l t s due t o t h e r e s t r i c t i o n on t h e sea rch area

PWB'S

DATA 3

DATA 4

DATA 6

DATA 8

22,603

174 767

T (sec) 87 V I 585

T (sec) 1,040 54,502

37

X

Fig. 6. The ratio of the number of from-tos realized with n segments to all those real-

ized by the line-search router.

researchers. first-technique in such a manner that the closest cell to the target cell should be first searched.

Hadlock has introduced a depth-

As far as the interconnection for a from-to is possible, the number of cells to be searched by his router is generally less than that by Lee's router. the interconnection for a net is impossible, it takes more processing time in Hadlock's router than in Lee's one, due to stack opera- t ions.

However, when

Thus, at the final stage when it occurs frequently that the search for incomplete from-tos fails because of the nonexistence of wiring route, Hadlock's router is not neces- sarily efficient. Therefore, we have inveeti- gated the difference between the conventional Lee-type breadth-first search (BFS) router and the Hadlock-type depth-first search (DFS) router.

Table 3 shows the implementation re- sults of these two routers for PWB's 1 and 3 of Table 1. As can be seen from this table, when the interconnection is successful, the DFS router is much faster than the BFS router, and when the interconnection is unsuccessful,

DFS router is not so fast as BFS router, Consequently, it is concluded that DFS is superior to BFS when the required intercon- nections are mostly successful (for example PWB 1). However, BFS is superior to DFS when the interconnections are unsuccessful (for example PWB 3) .

In our routing system both DFS and BFS are incorporated such that DFS is applied so long as the rate of successful interconnec- tions does not exceed 30%, and BFS is applied otherwise.

5. Evaluation of the Wiring Performance

After all the parameter values in each router are determined, the wiring performance of the whole system has been evaluated through some experimental results. The routers de- scribed in this paper have been programmed in FORTRAN and run on an ACOS 771900 computer.

Table 4 shows the results obtained for three input data. As can be seen from this table, the line-search router performs most of the interconnections in a fairly short time, and the maze-running router raises the final wiring performance in a comparatively short time.

It should be noted that the wiring per- formance of Mode 0 is slightly better than that of Mode 1, and therefore one might have an impression that the line-search router by itself is superior to such a combination of the line-search and single-row routers as de- scribed here. However, this tendency is natur- ally to be expected, considering that Mode 1 is implemented under the fixed-via constraint, whereas Mode 0 is run in an environment free from such a severe constraint. Moreover, it should be noted that application of Mode 0 to PWB's of between-pins capacity two or more may have not only to require a considerable modification to cope with the fixed-via con- straint, but also to suffer the loss of "topological fluidity , "

It is observed from the implementation results that wirabllity of this system tends to rise rapidly as the between-pins capacity increases, and that there is not much differ- ence among processing times spent in Modes 1 to 4 , although the total numbers of cells on a PWB in Modes 2, 3, and 4 are 2.25, 4 , and 6.25 times as many as that in Mode 1, respec- tively.

Now, let us investigate the characteris- tic of the obtained wire patterns. The aver- age number of vias per from-to is 1.5, and hence most of the wiring routes are of simple pattern. Moreover, this system has a

38

Table 3. The implementation results of DFS-type and BFS-type routers

Average CPU time spent on an unsuccessful search (sec)

scanned cells per f rom-to

Average number of

Average number of scanned cells in a successful search

scanned cells in an unsuccessful search

Average number of

No. of intercon- nected from-tos No.of from-tos to be intercon-

nected

Average CPU time per from-to (sec) 1 0.70 1 1.24 1 0.70 1 0.52

0.27 0.21 0.65 0.39

2,917 6,882 2,737 2,847

3,624 9,176 3,322 5,208

1,149 1,146 2,549 2,094

7 1 71 7 1 7 1

Average CPU time spent on a successful search (sec) 1 0.88 1 1.6 l 1 0.84 1 0.94

Fig. 7. Final wire patterns obtained by Modes 3 , B and D for the third board of Table 4.

39

Table 4. Implementation r e s u l t s

’ LINE-SEARCH ROUTER + WE-RUNNING-

ROUTER V rom- t o

Example S 1NGI.E-ROW BOUT W(%)

- 100

99.5

100

100

100 - 92.9

90.5

98.0

100

100

D (%I

- 36.2

39.1

22.7

16.4

12.9

42.6

43.3

27.9

21.7

16.9

-

- u1 (ZZ) - 99.6

95.2

100

100

100

[OD€

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

I

-

-

-

-

~-

88.012.0

76.2113.0

40.8112.3

40.7112.7

36.6/10.3

Data 1

s i ze 241x191 (mmrm)

#net 222

#from-to 437

# pin 903

1.21

1.39

1.02

0.94

0.85 _ _ ~

Data 2

s i z e 318x188

#net 30 7

#from-to 601

# Pin 1508

(m-1

268r2.5

194119.6

217119.6

98/20.0

71119 .O

289

287

314 - -

1.46

1.65

1.30

1.16

1.08

89.2

85.7

97.2

100

100 - 82.2

17.5

94.3

99.5

100 - 72.2

66.4

85.2

94.6

100

~-

643r4.5

363133.5

408/41.0

261143.7

251/41.2

299

298

544

196 -

85.6

83.3

96.8

99.5

100 - 77.4

72.9

88.6

97.8

100 -

1.24

1.35

1.33

1.04

0.96

50.7

51.3

43.9

34.1

26.6

50.6

50.1

45.5

42.6

33.6

-

-

Data 3 s i ze 27 X198 ?mimml ine t 517

#f rom-to 1142 # p in 3668

295

4 84

1176

599 -

50513.3

334/2l.l

436129 .O

447133.6

1611348

1.47

1.51

1.64

1.55

1.19

Data 4

size 183x274

#net 481 l f r o n r t o 911

# pin 2179

(mmKmm1

Size: (Maximum length i n the X-direction) (Maximum length i n the Y-direction); Ti, T2, T3:

V/from-to: W1, W:

CPU time spent i n the l ine-search router , t he single-row router , and the maze-running router , respect ively;

t he average number of v i a s per from-to; and the wiring performance before and a f t e r t h e maze-running router , respec-

t ive ly .

40

d i s t i n c t i v e high dens i ty r o u t i n g such t h a t t h e c e l l u t i l i z a t i o n r a t i o may exceed 40%. F ina l ly Fig. 7 shows t h e f i n a l wire p a t t e r n s obtained by Modes 3, B and D f o r t h e t h i r d d a t a of Table 4 , where t h e s c a l e denotes t h e number of t r a c k s .

6. Conclusion

F i r s t we have descr ibed system f o r h igh dens i ty PWB's

a new rou t ing and then d e t e r -

mined t h e parameter va lues which have much in f luence on t h e wir ing performance.

It can be seen from samples of implemen- t a t i o n r e s u l t s t h a t t h i s system h a s h igh per- formance i n t h e p r a c t i c e of l ayou t f o r m's of high dens i ty .

Thus development i s cont inuing on t h e l a y e r i n g problem [8] f o r high dens i ty multi- l a y e r PWB's.

Acknowledgement. The au tho r s would l i k e t o express t h e i r a p p r e c i a t i o n t o T. Chiba, Sharp Corp., N. Yoshida, K. Kawakita, H. Kawanishi, Nippon E l e c t r i c Co., L t d . , and H. Hamamura, F u j i t s u L td . , f o r va luab le cooperat ion i n providing PWB examples.

This work was supported i n p a r t by t h e Grant i n Aid f o r S c i e n t i f i c Research of t h e Minis t ry of Education, Science, and Cu l tu re of Japan under Grant: Cooperative Research (A) 435013 (1980).

1.

2.

3.

4.

5.

6.

7.

8.

REFERENCES

H.C. So. Some t h e o r e t i c a l r e s u l t s on t h e r o u t i n g of m u l t i l a y e r p r i n t e d wir ing boards, Proc. IEEE I n t . Symp. on C i r c u i t s and Systems, pp. 296-303 (1974). M.T. Doreau and L.C. Abel. A topologic- a l l y based non-minimum d i s t a n c e r o u t i n g algori thm, Proc. 15 th Design Automation Conf, , pp. 92-99 (1978). F.O. Hadlock. A s h o r t e s t path a lgo r i thm f o r g r i d graphs, Networks, 7, 4 , pp. 323- 334 (1977). S. Asahara, M. Odani. Y. Ogura, I. Shira- kawa and H. Ozaki. A rou t ing system based on single-row r o u t i n g f o r high d e n s i t y p r i n t e d w i r i n g boards, Proc. IEEE I n t . Conf. on C i r c u i t s and Com- p u t e r s , pp. 290-294 (Oct. 1980). S. Tsukiyama, E.S. Kuh and I. Shirakawa. An a lgo r i thm f o r single-row r o u t i n g with p re sc r ibed street conges t ions , IEEE Trans. C i r c u i t s and Systems, CAS-27, pp. 756-771 (1980). H. Yamamura, I. Shirakawa and H. Ozaki. A l ine-search method f o r t h e r o u t e con- n e c t i n g problem on 2-layer p r i n t e d cir- c u i t board, Trans. I .E.C.E., E, pp. 671-678 (1974) ( i n Japanese) . C.Y. Lee. An a lgo r i thm f o r pa th connec- t i o n s and its a p p l i c a t i o n s , IRE Trans. EC-10, pp. 346-365 (Sept. 1961). S. Tsukiyama, E.S. Kuh and I. Shirakawa. On t h e l a y e r i n g problem of m u l t i l a y e r PWB wi r ing , Proc. 18 th Design Automation Conf, t o appear.

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