VIA2 VIA1 MET2 VIA1. VIA2 w3 VIA2 MET1 MET3 MET3 MET2

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1 VIA DESIGN RULE CONSIDERATION IN MULTI-LAYER MAZE ROUTING ALGORITHMS Jon Cong, Jie Fng nd Kei-Yong Khoo Comuer Siene Dermen, UCLA Lo Angele, CA el , fx fong, jfng, ABSTRACT Mze rouing lgorihm re widely ued for nding n oiml h in deiled rouing for VLSI, PCB nd MCM. In hi er, we how h nding n oiml roue of wo-in ne in muli-lyer rouing environmen under ril vi deign rule n e urriely diul. Furhermore, righforwrd exenion o he mze rouing lgorihm h dillow vi-rule inorre roue my eiher ue uoiml roue o e found, or more eriouly, ue he filure o nd ny roue even if one exi. We reen rened heurii o hi rolem y emedding he dine o he mo reenly led vi in n exended onneion grh o h he mze rouing lgorihm h higher hne of nding vi-rule orre oimum h in he exended onneion grh. We furher reen eien d-ruure o imlemen he mze rouing lgorihm wihou he need o reonru he exended onneion grh. Exerimenl reul onrmed he uefulne of our lgorihm nd i liiliy o wide rnge of CMOS ehnologie. 1 INTRODUCTION Finding n oiml oin-o-oin h i he fundmenl oerion in re-ed deiled rouing for VLSI, PCB nd MCM. The mo ommon roh i o rereen he rouing re wih rouing grid nd erform rouing over he grid. The grid-oin in he rouing grid rereen he ermiile loion h he ener-line of h n hrough, nd he edge eween he grid-oin deermine he ermiile rouing ern. In generl, he gridoin nd grid-edge n e rereened e of node nd edge, reeively, in n undireed grh G =(V; E) lled he onneion grh. The edge re uully weighed o ree he rouing o, uh he ul lengh of he Thi work i rilly uored y DARPA/ETO under Conr DAAL01-96-K-3600 mnged y he US Army Reerh Lorory. 1 w3 w1 3 4 VIA2 e1 2 VIA2 w2 5 VIA2 w3 w2 1 Figure 1. Illurion of he 0.5 m CMOS deign rule. grid-edge, nd he o of h F () i he um of edgeweigh long he h. In hi er, we will ume h he edge weigh re uniform in eh direion for eh lyer. For exmle, ll horizonl h on he r rouing lyer hve he me o er uni lengh. We lo dene F ( e)=1if e n invlid h due o deign rule violion Pril Vi Deign Rule The lyou deign rule eify e of ing nd widh onrin on lyou geomerie o enure oh he yield nd he eleril erformne of he mnufured deign. For inne, minimum wire ing nd widh rimrily reven eleril hor nd oen, reeively. Minimum ing in vi enure good yield well good onneion eween he onneing mel lyer. Tle 1.1. how ome deign rule, (lo illured in Fig. 1), reled o he mel nd u lyer for hree-level-mel 0:5 m CMOS roe. While u i lerly dened he onneion eween wo djen onduing lyer, \vi" i le well dened nd ommonly men he onneing oje eween mel lyer. In hi er, he diinion of vi onneing only mel lyer i no neery nd we will ue \u" nd \vi" inerhngely when referring o he onneion eween o rouing lyer. There re hree roerie regrding he deign rule h re generlly rue in rie: Proery 1The minimum ing for u e only he me or djen u lyer. For exmle, here i no minimum ing reuiremen eween he CONTACT nd VIA2 lyer ine hey re no djen u lyer.

2 Tle 1. Deign Rule for 0:5 m CMOS Proe Rule Dimenion (m) w1 Minimum nd widh 0:6 w2 Minimum widh 1:2 w3 CONTACT, nd VIA2 ize 0:8 0:8 e1 Minimum mel enloure of CONTACT, nd VIA2 0:2 1 Minimum nd. 0:8 2 Minimum ing 1:2 3 Minimum CONTACT o CONTACT ing 0:6 4 Minimum CONTACT o ing 0:3 5 Minimum o ing 0:6 6 Minimum o VIA2 ing 0:3 7 Minimum VIA2 o VIA2 ing 0:6 The r, eond nd hird mel lyer re, nd, reeively. The u lyer re CONTACT, nd VIA2. CONTACT onne he olyilon lyer wih lyer. onne lyer wih lyer. VIA2 onne lyer wih lyer. Lyer 3 Lyer 2 Lyer 1 Figure 2. An exmle where rdiionl mze rouer nno nd h from o due o he vi ing rule h reuire minimum vi-ovi ing of wo grid-oin. The vlid h i,,,, d,. Proery 2The minimum ing for u on djen lyer i mller hn or eul o he minimum ing for u on he me lyer. For exmle, he minimum ing eween nd VIA2 i 0.3 m, where he minimum ing eween nd or eween VIA2 nd VIA2 i 0.6 m. Proery 3The minimum ing for wo u, on eiher he me or djen u lyer, i mller hn he minimum wire widh lu wo ime he wire ing (W2S) of eiher of i onneing mel lyer. For exmle, he minimum ing eween VIA2 nd VIA2 i 0.6-m, u he W2S for i (0:6 2 0:8) = 1:4 m, nd he W2S for i (1:2 21:2) = 3:6 m The Clil Mze Rouing Algorihm nd i Limiion Algorihm: Mze Rouing Algorihm (G; ; ) 1 Q ; 2 while Q 6= 3 Po(Q); 4 if ( = ) hen h found; 5 for-eh 2 Neighor() do 6 Exnd(; ; Q); 7 end for-eh 8 end while end Figure 3. The mze rouing lgorihm nd minimum-o h o in he grh G. Given onneion grh G =(V; E) nd oure node nd deinion (where ; 2 V ), he minimum-o h rolem i o nd h in G uh h F ( )i miniml mong ll feile h from o in G. I i ler h he h orreond o deiled rouing oluion in he rouing region rereened y G. The minimum-o h rolem n e olved uing he mze rouing lgorihm [1, 2], hown in Fig. 1.2., whih nd he minimumo h uing oin-y-oin exnion regy ed on he dynmi rogrmming rinile. I minin rioriy ueue Q of ndide node for exnion, ordered ording o heir rioriy. The rioriy deermine he exnion regy of he lgorihm. For exmle, uing he ul o o he node in Q he rioriie will reul in reh-r erh. Uing he ul o lu he eimed o o he deinion will reul in n A erh. A eh ierion, he highe rioriy node i rerieved from Q nd exnded ino eh of i feile neighor. Exnd(; ; Q) ude node (nd dd o Q if neery) if he h!! i eer hn he h o (if here i one). The oimliy of he mze lgorihm i redied on o funion F () h i monoone nd ie he rinile of oimliy [3] in dynmi rogrmming dened follow: A monoone o funion F () imlie h F () F ( 0 ) for ll uh 0, for ll h in G. Thi i eily ied y hving only oiive weigh for he edge in G. Inuiively, monoone o funion llow he h erhing roe o lwy rogre wy from he oure. Therefore, eh node in Gi exnded mo one in he mze rouing lgorihm. The rinile of oimliy in dynmi rogrmming [3] e h: Prinile of Oimliy: An oiml oliy h he roery h whever he iniil e nd iniil deiion re, he remining deiion mu oniue n oiml oliy wih regrd o he e reuling from he r deiion. In he mze lgorihm, he e h found o ll viied node o fr oniue e, nd how o ude he e h o viied node oniue deiion. The rinile of oimliy imlie h ny given oin in he mze exnion (line 6 in Fig. 1.2.) roe, ril rouing owrd he deinion i indeenden of he ril rouing h hve een lredy een found. However, hi

3 *, β, β u * β, u i βnode Figure 4. Illurion of node. i uully no rue in ril lyou deign ine ril rouing h h een omleed immediely imoe oile deign rule reriion round i viiniy. A oluion o hi rolem i o enure h he grid ing i greer hn or eul o he lrge lile ing rule. However, in \gridle" rouing, he rouing grid i mller hn he wor-e vi-o-vi ing nd n e ne he \mnufuring" grid (The ul grid ize i deermined y he reoluion of he ehnology nd/or he deign de). In hi e, he lemen of vi will reri where he nex vi n e led. The rolem of vi-rule on he mze rouing lgorihm n now e illured wih imlied exmle for lriy. Le' uoe h he rouing grid h uniform ing nd uniform edge o, nd he minimum vi ing i wo grid-ing. Fig. 2 how n exmle wih roeionl (i.e., veril wo-dimenionl lne) rouing region. Wih he mze rouing lgorihm, he oure node will e exnded r ino node nd. If node i exnded nex (ine he h, nd, hve he me o), hen will e exnded ino node nd d. Now node dn e exnded o u he oluion,, d, will e deign rule inorre! If invlid h re dillowed during mze exnion, hen node d will e dirded nd node will e exnded ino node. Bu ine node n e exnded mo one in he mze rouing lgorihm, node nno e exnded ino node eue node h een exnded efore. Therefore, he feile h,,,,d, will no e found. Noie h even if node i exnded efore (y y weighing he vi edge wih higher o), he feile h ill nno e found ine node will lwy e exnded efore node Prolem Formulion In he reviou exmle, he filure o nd h i due o node, lled he -node nd i dened follow: Deniion 1 Anode u in G i -node if (i) here exi n oiml vi-rule-orre h ; from o h onin u, nd (ii) here exi minimum-o h (lled he - h) 0 ; from o, h i vi-rule orre u o u, nd h mller o hn he uh from o u in ;. The -node i illured in Fig. 4. For he exmle hown in Fig. 2, node i -node ine (i) here i n oiml vi-rule-orre h,,,, d,, nd (ii) here i h, whoe o of 1 i mller hn he o of 3 of he uh,,,. The exiene of Figure 5. Vi ing violion in wo lyer rouing rolem n e eily orreed eue Proery 3 imlie h here nno e ny ole eween he wo vi. Therefore, he vi (eiher one or oh deending on oher onneion o he vi) n e reled wih wire egmen he mel lyer. -node reven n oiml h from eing diovered y he mze rouing lgorihm: Proery 4If here i n oiml vi-rule-orre h ; from o whih h -node long he h, hen he h will no e found y he mze rouing lgorihm. Proof: The mze rouing lgorihm will lwy exnd he -node ed on he -h eue i i he mlle o h mong ll he h from o. Sine noden only e exnded one in he mze rouing lgorihm, he oiml h ; h e hrough will no e found. Proery 5The mze rouing lgorihm my no nd he oiml vi-rule-orre h. Proof: Thi follow direly from Proery 4 if ll he oiml h hve -node long heir h. Noie h Proery 5 doe no reven he mze rouing lgorihm from reurning u-oiml h. Thi i in f uie ele in rie. However, he ide h n oiml vi-rule-orre h h uh h i longer (higher o) hn lolly oiml h ugge h he iuion deied in Proery 4 involve ome kind of rouing deour. For exmle, he oiml vi-rule-orre h in Fig. 2 involve he deour,,,. Thi imlie h he vi-rule will more likely e rouing in ongeed re. Thi n hve more eriou oneuene. Proery 6If every minimum-o vi-rule-orre h from o h -node long he h, hen he mze rouing lgorihm will no nd vi-rule-orre h from o even if uh h exi. Proof: The mze rouing lgorihm nd only he minimumo h. If ll uh h hve -node long hem, hen y Proery 4, none of he h will e found y he mze rouing lgorihm. Fig. 2 i uh n exmle. We wn o re h in rie, he vi-rule re no eriou rolem in he erly he of rouing where ri-und-reroue, nd lol modiion n eeively hndle mny of he iue wih vi-rule. I i in he ler he of rouing when he rouing region i exremely ongeed or omed, nd he free e re nrrow nd irregulr, h reful oniderion of he vi rule eome riil. The rooed rouing lgorihm i men o funion n uxiliry u more ure rouer h eek ril h when he rdiionl mze rouing lgorihm fil or eing uoiml. Thi will e deried in more deil in Seion 3.

4 Previou work in deiled rouing hve onidered he inerion of vi wih oher oje [4, 5] u no eween vi wihin he me roue. The rolem of deign rule inerion wihin he me roue h een knowledged in [4] u no olved exe for ome eily idenile eil e. Noie h he rouer in [4,5] re ully gridle rouer uing re exnion. Thi i o eue erly gridded rouer hve ued lrge enough grid ing nd evded he vi-ing rolem. I i lo inereing o noe h Proery 3 imlie h he vi ing i no rolem for wo-lyer rouing illured in Fig. 5. Therefore, he vi ing rolem i limied o hree or more lyer rouing whih doe no ly o mny erly work in deiled rouing. In induril rouer, heurii re ofen ued o mke he mze rouing lgorihm more rou o h i i unlikely o fil o nd roue u he roue my e u-oiml. The generl oimliy of h erhing in grh h doe no ify he rinile of oimliy of dynmi rogrmming i lo diued in [6]. However, he heme rooed in [6] i oo generl nd nno exloi mny roerie in ril deign. In hi er, we will reen heurii o hi rolem in Seion 2 uing n exended onneion grh h emed he dine o he mo reenly led vi in h. While righforwrd imlemenion of he mze rouing lgorihm on he exended onneion grh n olve he rolem, we reen in Seion 3 eien d-ruure o imlemen he mze rouing lgorihm wihou he need o reonru he exended onneion grh. Seion 4 how n ul rouing exmle where our lgorihm n nd oluion, where rdiionl mze rouing lgorihm nno. We lo how he liiliy of our lgorihm o vriey ofcmos ehnologie. We onlude our er in Seion 5. 2 EXTENDED CONNECTION GRAPH There re wo i rolem h re ued y he vi rule. One i h grid oiion my need o e exnded more hn one o nd he oimum h. The oher i h we need o minin he dine o he mo reenly led vi long he h o deermine when he nex vi n e led. Our oluion o hi rolem i o oneully ree n exended onneion grh h emed he vi dine well o rovide mulile node eh grid-oiion o h eh grid oiion n eeively e exnded more hn one during he mze rouing. We le K e he minimum numer of uni griding eween he vi on djen lyer (e.g., K=2 in Fig. 2). Our ide i o rnform eh node v in he originl onneion grh G ino 2K exended node v,(k,1) ; ;v,1;v 0;v 1;;v K,1;v K in n exended direed onneion grh G 0 = (E 0 ;V 0 ). The node in G he o nd oom lyer re he exeion, nd hey re dded o G 0 wihou eing rnformed. Eh exended node v i (jij < K) ure he e h h i jij griding wy from he mo reenly led vi h i o he lef (if K>i>0) or o he righ (if i<0) of he urren node. The exended node v K ure he e h h i K or more grid-ing wy from he mo reenly led vi. The edge in G 0 re dded follow: (i) if v Lyer 3 Lyer 2 Lyer uri Figure 6. The exended onneion grh G 0 for he exmle hown on he lef. n rvere o i neighor u in G, hen v K n rvere o u K, nd v ij jij<k n rvere o u i1 if u i o righ ofv or u i,1 if u i o he lef of v, nd (ii) he exended node h n e onneed from vi re hoe wih uri 0 nd he exended node h n onne o vi re hoe wih uri K. Bed on he wo rule, we n onru he exended onneion grh for he exmle in Fig. 2 hown in Fig. 6. Noie h he oiml vi-rule-orre h i emedded in G 0,, 0, 1,d 2,. Noie lo h he minimum-o u vi-rule-inorre h i no in G 0 ; i.e., here exi no indie i; j uh h, i, d j, i in G 0. The exended onneion grh hown in Fig. 6 i vlid only for ro-eionl (wo-dimenionl) rouing region. For muli-lyer generl re rouing, he exended node mu enode he dine from he mo reenly led vi in oh he x-direion nd he y-direion. Therefore, eh node v in G, h i no in he o or oom lyer, i rnformed o (2K, 1) 2 1 exended node v K;K nd v i;j for i; j =0;1;2;;(K,1). The r nd eond indie (uri) rereen he (eiher oiive or negive) dine in he x-direion nd he y-direion from he mo reenly led vi, reeively. For eh edge e = (u; v) in he originl grh G h rereen wire egmen (i.e., u nd v rereen grid-oin in he me rouing lyer) in he x-direion, we dd he following edge o G 0 : (i) n undireed edge (u K;K;v K;K); (ii) direed edge in he oiive x-direion: (u i;j;v i1;j) for ll oile j, nd i =0;;K,2 nd he direed edge (u K,1;j;v K;K) for ll oile j; nd (iii) direed edge edge in he negive x-direion: (v i;j;u i,1;j) for ll oile j, nd i =0;,1;;,(K,2), nd (v,(k,1);j ;v K;K) for ll oile j. Similrly for edge in he y-direion. Therefore, n exended node v i;j ure he e h h i i nd j grid-oin wy from he mo reenly led vi in he x nd y direion, reeively. For eh edge e =(u; v) 2 E rereening vi, (i.e., u nd v rereen grid-oin on djen rouing lyer), we dd he edge (u 0;0;v K;K) nd (v 0;0;u K;K), o G 0. If u i on he oom or o lyer, hen he edge dded o G 0 re (u; v 0;0) nd (v K;K;u). Finlly, noie h K n e dieren for differen djen lyer. For exmle, in four-level rouing region, K 1 nd K 2 n orreond o he ing of o VIA2 nd VIA2 o VIA3, nd lied o he rnformion of grh node on lyer 2 nd lyer 3, reeively. The numer of node nd edge in G 0 re jv 0 j =

5 O(K 2 jv j) nd je 0 j = O(K 2 jej), reeively. However, he vi-ing K need no e very lrge in rie. For exmle, he 0.5- CMOS ehnology hown in Tle 1.1. h o VIA2 minimum ing of 0.3-m. Therefore, uing mnufuring grid-ing of 0.1-m give u reonly mll K vlue of only 3. Furhermore, we will reen eien d-ruure o imlemen he mze rouing lgorihm on G 0 wihou he need o onru G 0 efore rouing egin in he nex eion. The exended onneion grh G 0 lone doe no reven nding vi-rule inorre h. For exmle, he h,, 0, 1, d 2, 2, 2, in Fig. 6 i feile h in G 0 u no vi-rule-orre h ine he vi, 0 i no le wo grid-oin wy from he vi 2,. Therefore, we need o ly he following reriion when exnding node o i neighor in G 0. Reriion 1 Given he node v K;K, u K;K, nd 0;0 in G 0 where (i) v K;K nd u K;K re djen node, nd (ii) 0;0 i he oiion of he mo reenly led vi long he erh h o v K;K, hen v K;K n only e exnded ino u K;K if u K;K i le K grid wy from 0;0. For exmle, hi reriion will reven exloring he h,, 0, 1, d 2, 2, in Fig. 6 ine d 2 will no e llowed o exnd ino 2 due o he reene of 0. Thi reriion n e eily imlemened when he erh h i ored exliily h (deried in he nex Seion) rher \k link" in rdiionl mze rouer. We n now ly he mze rouing lgorihm in Fig o G 0 o nd he orreonding minimum-o vi-rule-orre h in G. Wih he reriion, G 0 will lwy reurn vi-ruleorre h if one i found. However, he reriion le ondiionl rule on G 0 nd in ee viole he Prinile of Oimliy. Thi will reven vlid vi-rule-orre h o e found in G 0 even if one exi. For exmle, Fig. 7 how ro-eionl onneion grh G nd i orreonding exended grh G 0 (imilr o Fig. 6) h onin he vi-rule-orre h,, 0, r,1, d 2, 2, 2,. If he edge re uniformed weighed, hen node d 2 will e viied y he eul o h,, 0, 1, d 2 nd,, 0, r,1, d 2. If he h,, 0, 1, d 2 i hoen, hen no oluion will e found in G 0. Therefore, node d 2 i -node nd he vi-rule-orre h my no e found in G 0. In f, he vi-rule-orre h will no e found if he vi, i more hn 2K grid-oin wy from he vi,. On he oher hnd, if he vi, i wihin 2K grid-oin from he vi,, hen he vi-rule-orre h will e found. Therefore, our lgorihm i eeive in reolving he rolem of vi deign rule inerion for vi h re loe y u i doe no olve he rolem of vi deign rule inerion dine. 3 SEARCHING ON THE EXTENDED CONNECTION GRAPH A righforwrd roh o nd he oiml vi-rule orre h i o onru G 0 nd ly he mze rouing lgorihm on G 0. Thi i ineien eue of he overhed in onruing G 0, nd mny exended node will never e viied. Furhermore, erhing on G 0 i only ueful round ongeed re. The liion of he rooed rouer i G r G r uri Figure 7. Exmle howing he filure of nding he minimum-o vi-rule-orre h in G 0. Se Ex- Prioriy ueue Q Hh Diionry M nd ho; nodei ho; hi Sr h0; i h; i 1 h1; 0 ih1;i h 0 ;ih; i 2 0 h1;ih2;,1 ih2;d 1 i h 0 ;ih; i h,1 ; ihd 1 ; di 3 h2;,1 ih2;d 1 ih2; 0 i h,1 ; ihd 1 ; di h 0 ; i 4,1 h2;d 1 ih2; 0 i hd 1 ; dih 0 ; i 5 d 1 h2; 0 i h 0 ; i 6 0 h3; 1 i h 1 ; i 7 1 h4;d 2 i hd 1 ; di 8 d 2 h5;i h; di Done Figure 8. On-he-y exnion of G 0 for he exmle in Fig. 2. We how he oiion for enrie in H ymolilly wih he node nme, nd hve omied he o nd viiion informion for reviy. Noie h he exended node re reed only hey re viied. For inne, e 2, he exended node 0 i reed when i exnded. Oviouly, only mll ue of node in G 0 i reed. o ke over he k of nding hor ril h (i.e., o ueeze hrough ongeed re) when rdiionl mze rouer fil o nd h. Therefore, i i no deirle o onru G 0 hed of ime. We will now how mze exnion lgorihm h doe no reuire G 0 o e onrued hed of ime. The mze exnion roe rvere G 0 o nd he oiml h. The d needed eh nodev2v 0 (we hve droed he uri here for reviy) during he exnion re: (i) he o of he e h found o v, nd (ii) he re k ode o genere h h. Relizing h only very mll ue of he node in he exnding wvefron re ively involved in he mze exnion oerion, Souku rooed in [7] o ue ere d-ruure o minin he mze exnion informion. Thi wy, he ize of onneion grh node n e igninly redued. Thi i riulrly ueful eue he exended onneion grh node n e omued on-he-y ed on he originl onneion grh nd need no e relized ll. Only he informion needed during exnion need o e re-

6 ed nd ored in emorry node lled he mze node. The d-ruure for imlemening he mze exnion informion re follow. We le m e mze node for n exended node v nd m oni of o (m:o), he oiion of v (m:o), g o mrk viiion y neighor (m:vii) nd he e h (m:h) from o m:o. The h i euene of egmen nd eh egmen i rereened uing 2-ule hd; li where d i direion nd l he lengh, reeively. The overll h i onrued y ring from uing he direion nd lengh in eh egmen. Le M e he e of mze node orgnized hh diionry uing m:o he erh key. An enry in he rioriy ueue Q for mze node m oni of m:o nd he omued rioriy ed on m:o. During he mze exnion, n enry i oed from Q h rovide he oiion of he mze-node n h i eing exnded o n n e rerieved from M. The oiion o 0 of feile neighor in G 0 h h no een viied, found uing n:vii, i omued on-he-y ed on he originl onneion grh G. If o 0 i no erh key in M, hen new mze node m wih m:o = o 0 i dded o M; oherwie m where m:o = o 0 i rerieved from M. If he h o m hrough n i eer, hen m:o nd m:h, nd he rioriy ueue re uded ordingly. The viiion g in m i lo mrked roriely. Finlly, when n i fully exnded y viiing ll i feile neighor, i i deleed from M. Thi i illured in Fig. 8 for he exmle hown in Fig. 2. Our heme dier from h deried in [7] in h M i no orgnized in [7] nd h we ore he oiml h direly eh mze node ined of genering k-re ode for every viied node. Our lgorihm eeively lloe memory only for he wvefron node in he mze exnion. Therefore, he memory reuiremen i deenden on he mximum numer of wvefron node nd he ize of eh h in hee node. The numer of wvefron node i inuened y oh he mze exnion regy nd he reene of ole in he rouing region. The ize of h in he mze node i he numer of egmen. In rie, in ongeed lyou, he numer of wvefron node re firly onn due o he very limied erh e. 4 EXPERIMENTAL RESULTS We hve uefully imlemened our mze rouing lgorihm in omrion wih wo oher rouer. One i Iroue [4] in he Mgi IC lyou edior nd he oher i rdiionl mze rouer h erform on-he-y vi-rule hek nd dird ll inorre h. In Fig. 9, we how hree lyer exmle imilr o he ro-eionl digrm in Fig. 2. The rdiionl mze rouer filed o nd ny h eue i exnded r. Iroue reurned vi-ruleinorre h,, d, while our lgorihm found he oiml vi-rule-orre h,,,d,. Fig. 10 how four lyer exmle uing MOSIS SCMOS deign rule. The h,,,, found y Iroue i vi-rule inorre eue he vi-o-vi ing eween nd i oo mll. By deeing hi inorre h, rdiionl mze rouer found deoured h,, 1, 1,. Wih he iliy o exnd node more hn one, our lgorihm found he oiml h, 1, 2,,. () Rouing rolem d Inorre Vi Sing () Iroue reul Conneion No Found () Trdiionl mze lgorihm reul d (d) Our reul Figure 9. A e uing he 0.5-m CMOS ehnology hown in Tle 2 where our rooed rouing lgorihm found he vi-rule orre oiml h. The ring oin i olyilon-o- on nd he rge i omewhere o he righ of he gure. The lyou re drwn o le. The vlidiy of our lgorihm deend on Proerie 1, 2 nd 3 (deried in Seion 1) eing rue. A urvey of few IC ehnologie how h hey re in f rue hown in Tle 2. The ize of K i lo given in he le. I i very likely h in dvned ehnology, he minimum ing rule eween djen u lyer will remin mll, o h he ize of he exended onneion remin reonle. Noie h wih he on-he-y exnion heme deried in Seion 3, i i he numer of viied exended node h im he memory nd runime erformne of our lgorihm. A reul, he im of K i le hn udri in rie. We imlemened our lgorihm nd eed i wih hree exmle on Sun Ulr 1 workion, hown in Tle 3. The erh window for eh exmle rnge from o Our exerimenl reul how h he run ime inree i u-udri wih ree o K. 5 CONCLUSION We hve hown h oluion of he rdiionl mze rouing lgorihm n viole ril vi-rule in muli-lyer rouing environmen. Furhermore, righforwrd exenion o he mze rouing lgorihm h dillow virule inorre roue my eiher ue uoiml roue o e found, or more eriouly, ue he filure o nd ny roue even if one exi. We reen heurii o hi rolem y emedding he dine o he mo reenly led vi in n exended onneion grh o h he mze rouing lgorihm h higher hne of nding vi-rule orre oimum h in he exended onneion grh. We furher reen eien d-ruure o imlemen he mze rouing lgorihm wihou he need o reonru he exended onneion grh.

7 MET4 MET4 () Rouing rolem Inorre Vi Sing 1 2 Su-Oiml Ph 1 () Trdiionl mze lgorihm reul Tle 2. Exmle of CMOS Deign Rule Feure Miron rule (m) rule 0:5m 0:8m SCMOS w1 Min widh 0:6 1:2 3 w2 Min widh 0:6 1:4 3 w3 Min widh 1:2 2:1 6 w4 Min MET4 widh N/A N/A 6 e1 Min enl. of 0:2 0:5 1 e2 Min enl. of VIA2 0:2 0:6 1 e3 Min enl. of VIA3 N/A N/A 1 1,2 Min,. 0:8 1:2 3 3 Min ing 1:2 1:6 4 4 Min MET4 ing N/A N/A 6 5 Min CONTACT o ing 0:6 0:8 3 6 Min o. 0:6 1:2 3 7 Min o VIA2. 0:3 1:2 2 8 Min VIA2 o VIA2. 0:6 1:5 3 9 Min VIA2 o VIA3. N/A N/A 3 10 Min VIA3 o VIA3. N/A N/A 4 Uni grid ing 0:1 0:1 1 K ; 3 1 () Iroue reul (d) Our reul Figure 10. A e uing he four lyer SCMOS ehnology hown in Tle 2 where our rooed rouing lgorihm found he vi-rule orre oiml h. The ring oin i olyilon-o- on nd he rge oin i on MET4. The lyou re drwn o le. REFERENCES [1] E. W. Dijkr, \A noe on wo rolem in onnexion wih grh," Numerihe Mhemik, vol. 1,. 269{ 271, [2] C. Lee, \An lgorihm for h onneion nd i liion," IRE Trn Eleroni Comuer, vol. EC- 10, no. 1,. 346{65, [3] R. Bellmn, Dynmi Progrmming. Prineon Univeriy Pre, [4] M. Arnold nd W. So, \An inerive mze rouer wih hin," in Pro. 25h ACM/IEEE Deign Auomion Conf., no. 1,. 672{76, Tle 3. Run Time on Differen K' Ex. Serh Window # Ne Run Time (e) (X/Y grid) K=3 K=5 K=12 T T T [7] J. Souku, \Mze rouer wihou grid m," in Pro. IEEE/ACM In. Conf. Comuer-Aided Deign Dig. Teh. Per,. 382{385, 8-12 Nov [5] W. Shiele, T. Kruger, K. Ju, nd F. Kirh, \A gridle rouer for induril deign rule," in Pro. 27h ACM/IEEE Deign Auomion Conf., June [6] A. Teelum, \Generlized oimum h erh," IEEE Trn. Comuer-Aided Deign, vol. 14, no. 12,. 1586{1590, De