A High-Performance Triple Patterning Layout Decomposer with Balanced Density

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1 A High-Prormn Tripl Pttrning Lyout Domposr with Blnd Dnsity Bi Yu, Yn-Hung Lin, Grrd Luk-Pt, Duo Ding, Kvin Lus, Dvid Z. Pn ECE Dpt., Univrsity o Txs t Austin, Austin, USA Synopsys In., Austin, USA CS Dpt., Ntionl Chio Tung Univrsity, Tiwn Orl Corp., Austin, USA Astrt Tripl pttrning lithogrphy (TPL) hs rivd mor nd mor ttntions rom industry s on o th lding ndidt or 4nm/nm nods. In this ppr, w propos high prormn lyout domposr or TPL. Dnsity lning is smlssly intgrtd into ll ky stps in our TPL lyout domposition, inluding dnsity-lnd smi-dinit progrmming (SDP), dnsity-sd mpping, nd dnsitylnd grph simpliition. Our nw TPL domposr n otin high prormn vn omprd to prvious stt-o-th-rt lyout domposrs whih r not lnd-dnsity wr,.g., y Yu t l. (ICCAD ), Fng t l. (DAC ), nd Kung t l. (DAC 3). Furthrmor, th lnd-dnsity vrsion o our domposr n provid mor lnd dnsity whih lds to lss dg plmnt rror (EPE), whil th onlit nd stith numrs r still vry omprl to our non-lnd-dnsity slin. I. INTRODUCTION As th minimum tur siz urthr drss, th smiondutor industry s grt hllng in pttrning su-nm hl-pith du to th dly o vil nxt gnrtion lithogrphy, suh s xtrm ultr violt (EUV) nd ltri m lithogrphy (EBL). Tripl pttrning lithogrphy (TPL), long with sl-lignd doul pttrning (SADP), r solution ndidts or th 4nm logi nod []. Both TPL nd SADP r similr to doul pttrning lithogrphy (DPL), ut with dirnt or mor xposur/thing prosss []. SADP my signiintly rstritiv on dsign, i.., nnot hndl irrgulr rrngmnts o ontts nd dos not llow stithing. Thror, TPL gn to riv mor ttntion rom industry, spilly or mtl lyr pttrns. For xmpl, industry hs lrdy xplord tst-hip pttrns with tripl pttrning nd vn qudrupl pttrning [3]. Similr to DPL, th ky hllng o TPL lis in th domposition pross whr th originl lyout is dividd into thr msks. During domposition, whn th distn twn ny two turs is lss thn th minimum oloring distn dis m, thy nd to ssignd into dirnt msks to void onlit. Somtims, onlit n rsolvd y splitting pttrn into two touhing prts, lld stiths. Atr th TPL lyout domposition, th turs r ssignd into thr msks (olors) to rmov ll onlits. Th dvntg o TPL is tht th tiv pith n tripld whih n urthr improv lithogrphy rsolution. Bsids, som ntiv onlits in DPL n rsolvd. In lyout domposition, spilly or TPL, dnsity ln should lso onsidrd, long with th onlit nd stith minimiztion. A good pttrn dnsity ln is lso xptd to onsidrtion in msk CD nd rgistrtion ontrol [4], whil unlnd dnsity would us lithogrphy hotspots s wll s lowrd CD uniormity du to irrgulr piths [5]. Howvr, rom th lgorithmi prsptiv, hiving lnd dnsity in TPL ould hrdr thn tht in DPL. () In DPL, two olors n mor impliitly lnd; whil in TPL, otn tims xisting/prvious strtgis my try to do DPL irst, nd thn do som pth with th third msk, whih uss ig hllng to xpliitly onsidr th dnsity ln. () Du to th on mor olor, th solution sp is muh lrgr [6]. (3) Instd o glol dnsity ln, lol dnsity ln should onsidrd to rdu th potntil hotspots, sin nighoring pttrns r on o th min sours o hotspots. As shown in Fig. ()(), whn 3 () () 4 Fig.. Domposd lyout with () () glol lnd dnsity. () lol lnd dnsity in ll ins. only glol dnsity ln is onsidrd, tur is ssignd whit olor. Sin two lk turs r los to h othr, hotspot my introdud. To onsidr th lol dnsity ln, th lyout is prtitiond into our ins {,, 3, 4} (s Fig. ()). Ftur is ovrd y ins nd, thror it is olord s lu to mintin th lol dnsity lns or oth ins (s Fig. ). Thr r invstigtions on TPL lyout domposition [6] [] or TPL wr dsign [] [4]. [6] providd thr oloring lgorithm, whih dopts SAT Solvr. Yu t l. [7] proposd systmti study or th TPL lyout domposition, whr thy showd tht this prolm is NP-hrd. Fng t l. [8] prsntd svrl grph simpliition thniqus to rdu th prolm siz, nd mximum indpndnt st (MIS) sd huristi or th lyout domposition. [9] proposd lyout domposr or row strutur lyout. Howvr, ths xisting studis sur rom on or mor o th ollowing issus: () nnot intgrt th stith minimiztion or th gnrl lyout, or n only dl with stith minimiztion s post-pross; () dirtly xtnd th mthodologis rom DPL, whih loss th glol viw or TPL; (3) ssigning olors on y on prohiits th ility or dnsity ln. In this ppr, w propos high prormn lyout domposr or TPL. Comprd with prvious works, our domposr provids not only lss onlit nd stith numr, ut lso mor lnd dnsity. In this work, w ous on th oloring lgorithms nd lv othr lyout rltd optimiztions to post-oloring stgs, suh s ompnstion or vrious msk ovrly rrors introdud y snnr nd msk writ ontrol prosss. Howvr, w do xpliitly onsidr lning dnsity during oloring, sin it is known tht msk writ ovrly ontrol gnrlly nits rom improvd dnsity ln. Our ky ontriutions inlud th ollowing. () Aurtly intgrt dnsity ln into th mthmtil ormultion; () Dvlop thr-wy prtition sd mpping, whih not only hivs lss onlits, ut lso mor lnd dnsity; (3) Propos svrl thniqus to spdup th lyout domposition; (4) Our xprimnts show th st rsults in solution qulity whil mintining ttr 3 () /3/$ IEEE 63

2 lnd dnsity (i.., lss EPE). Th rst o th ppr is orgnizd s ollows. Stion II prsnts th si onpts nd th prolm ormultion. Stion III givs th ovrll domposition low. Stion IV prsnts th dtils to improv ln dnsity nd domposition prormn, nd Stion V shows how w urthr spdup our domposr. Stion VI prsnts our xprimntl rsults, ollowd y onlusion in Stion VII. II. PROBLEM FORMULATION Givn input lyout whih is spiid y turs in polygonl shps, w prtition th lyout into n ins B = {,..., n}. Not tht nighoring ins my shr som ovrlpping. For h polygonl tur r i, w dnot its r s dn i, nd its r ovrd y in k s dn ki. Clrly dn i dn ki or ny in k. During lyout domposition, ll polygonl turs r dividd into thr msks. For h in k, w din thr dnsitis (d k, d k, d k3 ), whr d k = dn ki, or ny tur r i ssignd to olor. Thror, w n din th lol dnsity uniormity s ollows: Dinition (Lol Dnsity Uniormity) For th in k S, th lol dnsity uniormity is mx{d k }/min{d k } givn thr dnsitis d k, d k nd d k3 or thr msks nd is usd to msur th rtio dirn o th dnsitis. A lowr vlu mns ttr lol dnsity ln. Th lol dnsity uniormity is dnotd y DU k. For onvnin, w us th trm dnsity uniormity to rr to lol dnsity uniormity in th rst o this ppr. It is sy to s tht DU k is lwys lrgr thn or qul to. To kp mor lnd dnsity in in k, w xpt DU k s smll s possil, i.., los to. Prolm (Dnsity Blnd Lyout Domposition) Givn lyout whih is spiid y turs in polygonl shps, th lyout grphs nd th domposition grphs r onstrutd. Our gol is to ssign ll vrtis in th domposition grph into thr olors (msks) to minimiz th stith numr nd th onlit numr, whil kping ll dnsity uniormitis DU k s smll s possil. III. Fst Color Assignmnt Tril Ys 0 onlit, 0 stith? No [Dnsity Blnd] SDP Formultion [Dnsity Blnd] Prtition sd Mpping Fig.. OVERALL DECOMPOSITION FLOW Input Lyout Grphs Constrution nd Simpliition Color Assignmnt on h Domposition Grph Color Assignmnt on h Domposition Grph Output Msks Ovrll low o proposd dnsity lnd domposr. Lyout Grph Constrution [Dnsity Blnd] Lyout Grph Simpliition Stith Cndidt Gnrtion Domposition Grph Constrution nd Simpliition Th ovrll low o our TPL domposr is illustrtd in Fig.. It onsists o two stgs: grph onstrution / simpliition, nd olor ssignmnt. Givn input lyout, lyout grphs nd domposition grphs r onstrutd, thn grph simpliitions [7] [8] r pplid to rdu th prolm siz. Two dditionl grph simpliition thniqus r introdud in S. V-A nd V-B. During stith ndidt gnrtion, th mthods dsrid in [] r pplid to srh ll stith ndidts or TPL. In sond stg, or h domposition grph, olor ssignmnt is proposd to ssign h vrtx on 64 Fig. 3. () (g) d d 3 () () stith (h) An xmpl o th lyout domposition low. 4 () () stith olor. Bor lling SDP ormultion, st olor ssignmnt tril is proposd to hiv ttr spdup (s Stion V-C). Fig. 3 illustrts n xmpl to show th domposition pross stp y stp. Givn th input lyout s in Fig. 3(), w prtition it into st o ins {,, 3, 4} (s Fig. 3()). Thn th lyout grph is onstrutd (s Fig. 3()), whr th tn vrtis rprsnting th tn turs in th input lyout, nd h vrtx rprsnts polygonl tur (shp) whr thr is n dg (onlit dg) twn two vrtis i nd only i thos two vrtis r within th minimum oloring distn min s. During th lyout grph simpliition, th vrtis whos dgr qul or smllr thn two r itrtivly rmovd rom th grph. Th simpliid lyout grph, shown in Fig. 3, only ontins vrtis,, nd d. Fig. 3 shows th projtion rsults. Followd y stith ndidt gnrtion [], thr r two stith ndidts or TPL (s Fig. 3()). Bsd on th two stith ndidts, vrtis nd d r dividd into two vrtis, rsptivly. Th onstrutd domposition grph is givn in Fig. 3(). It mintins ll th inormtion out onlit dgs nd stith ndidts, whr th solid dgs r th onlit dgs whil th dshd dgs r th stith dgs nd untion s stith ndidts. In h domposition grph, olor ssignmnt, whih ontins smidinit progrmming (SDP) ormultion nd prtition sd mpping, is rrid out. During olor ssignmnt, th six vrtis in th domposition grph r ssignd into thr groups: {, }, {} nd {, d, d } (s Fig. 3(g) nd Fig. 3(h)). Hr on stith on tur is introdud. Atr itrtivly rovr th rmovd vrtis, th inl domposd lyout is shown in Fig. 3(i). Our lst pross should domposition grphs mrging, whih omins th rsults on ll domposition grphs. Sin this xmpl hs only on domposition grph, this pross is skippd. (i) d d

3 TABLE I. IV. NOTATIONS USED IN COLOR ASSIGNMENT CE SE V B th st o onlit dgs th st o stith dgs th st o turs th st o lol ins DENSITY BALANCED DECOMPOSITION Dnsity ln, spilly lol dnsity ln, is smlssly intgrtd into h stp o our domposition low. In this stion, w irst lort how to intgrt th dnsity ln into th mthmtil ormultion nd orrsponding SDP ormultion. Followd y som disussion or dnsity ln in ll othr stps. A. Dnsity Blnd SDP Algorithm For h domposition grph, dnsity lnd olor ssignmnt is rrid out. Som nottions usd r listd in Tl I. S Appndix or som prliminry o smidinit progrmming (SDP) sd lgorithm. ) Dnsity Blnd Mthmtil Formultion: Th mthmtil ormultion or th gnrl dnsity lnd lyout domposition is shown in (), whr th ojtiv is to simultnously minimiz th onlit numr, th stith numr nd th dnsity uniormity o ll ins. Hr α nd β r usr-dind prmtrs or ssigning th rltiv wights mong th thr vlus. min ij + α s ij + β DU k () ij CE ij SE k B s.t. ij = (x i == x j) ij CE () s ij = x i x j ij SE () x i {,, 3} r i V () d k = x i = dn ki r i V, k B DU k = mx{d k }/min{d k } k B () Hr x i is vril rprsnting th olor (msk) o tur r i, ij is inry vril or th onlit dg ij CE, nd s ij is inry vril or th stith dg ij SE. Th onstrints () nd () r usd to vlut th onlit numr nd stith numr, rsptivly. Th onstrint () is nonlinr, whih mks th progrm () hrd to ormultd into intgr linr progrmming (ILP) s in [7]. Similr nonlinr onstrints our in th loorplnning prolm [5], whr Tyor xpnsion is usd to linriz th onstrint into ILP. Howvr, Tyor xpnsion will introdu th pnlty o ury. Comprd with th trditionl tim onsuming ILP, smidinit progrmming (SDP) hs n shown to ttr pproh in trms o runtim nd solution qulity trdos [7]. Howvr, how to intgrt th dnsity ln into th SDP ormultion is still n opn qustion. In th ollowing w will show tht instd o using th pinul Tyor xpnsion, this nonlinr onstrint n intgrtd into SDP without losing ny ury. ) Dnsity Blnd SDP Formultion: In SDP ormultion, th ojtiv untion is th rprsnttion o vtor innr produts, i.., v i v j. At th irst gln, th onstrint () nnot ormultd into n innr produt ormt. Howvr, w will show tht dnsity uniormity DU k n optimizd through onsidring nothr orm DU k = d k d k +d k d k3 +d k d k3. This is sd on th ollowing osrvtion: mximizing DU k is quivlnt to minimizing DU k. Lmm DU k = /3 i,j V dn ki dn kj ( v i v j), whr dn ki is th dnsity o tur r i in in k. Proo: First o ll, lt us lult d d. For ll vtors v i = (, 0) nd ll vtors v j = (, 3 ), w n s tht ln i ln j ( v i v j) = ln i ln j 3/ i j i j =3/ ln i ln j = 3/ d d i j So d d = /3 i j lni lnj ( vi vj), whr vi = (, 0) nd v j = (, 3 ). W n lso lult d d3 nd d3 using similr mthods. Thror, DU = d d + d d 3 + d d 3 = /3 ln i ln j ( v i v j) i,j V Bus o Lmm, th DUk n rprsntd s vtor innr produt, thn w hv hivd th ollowing thorm. Thorm Mximizing DUk n hiv ttr dnsity ln in in k. Not tht w n rmov th onstnt i,j V dn ki dn kj in DUk xprssion. Similrly, w n limint th onstnts in th lultion o th onlit nd stith numrs. Th simpliid vtor progrm is s ollows: min ij CE ( v i v j) α ij SE ( v i v j) β DUk () k B s.t. DUk = dn ki dn kj ( v i v j) k B () i,j V v i {(, 0), ( 3, ), ( 3, )} () Formultion () is quivlnt to th mthmtil ormultion (), nd it is still NP-hrd to solvd xtly. Constrint () rquirs th solutions to disrt. To hiv good trdo twn runtim nd ury, w n rlx () into SDP ormultion, s shown in Thorm. Thorm Rlxing vtor progrm () n gt th SDP ormultion (3). SDP: min A X (3) X ii =, i V (3) X ij, ij CE (3) X 0 (3) whr A ij is th ntry tht lis in th i-th row nd th j-th olumn o mtrix A: + β k dn ki dn kj, k B, ij CE A ij = α + β k dn ki dn kj, k B, ij SE β k dn ki dn kj, othrwis Du to sp limit, th dtild proo is omittd. Th solution o (3) is ontinuous instd o disrt, nd provids lowr ound o vtor progrm (). In othr words, (3) provids n pproximtd solution to (). B. Dnsity Blnd Mpping Eh X ij in solution o (3) orrsponds to tur pir (r i, r j). Th vlu o X ij provids guidlin, i.., whthr two turs r i nd r j should in sm olor. I X ij is los to, turs r i nd r j tnd to in th sm olor (msk); whil i it is los to 0.5, r i 65

4 nd r j tnd to in dirnt olors (msks). With ths guidlins mpping produr is doptd to inlly ssign ll input turs into thr olors (msks). ) Limittions o Grdy Mpping: In [7], grdy pproh ws pplid or th inl olor ssignmnt. Th id is strightorwrd: ll X ij vlus r sortd, nd vrtis r i nd r j with lrgr X ij vlu tnd to in th sm olor. Th X ij n lssiid into two typs: lr nd vgu. I most o th X ijs in mtrix X r lr (los to or -0.5), this grdy mthod my hiv good rsult. Howvr, i th domposition grph is not 3-olorl, som vlus in mtrix X r vgu. For th vgu X ij,.g., 0.5, th grdy mthod my not so tiv. ) Dnsity Blnd Prtition sd Mpping: Contrry to th prvious grdy pproh, w propos prtition sd mpping, whih n solv th ssignmnt prolm or th vgu X ijs in mor tiv wy. Th nw mpping is sd on thr-wy mximumut prtitioning. Th min ids r s ollows. I X ij is vgu, instd o only rlying on th SDP solution, w lso tk dvntg o th inormtion in domposition grph. Th inormtion is pturd through onstruting grph, dnotd y G M. Through ormulting th mpping s thr-wy prtitioning on th grph G M, our mpping n provid glol viw to srh ttr solutions. Algorithm Prtition sd Mpping Rquir: Solution mtrix X o th progrm (3). : Ll h non-zro ntry X i,j s triplt (X ij, i, j); : Sort ll (X ij, i, j) y X ij; 3: or ll tripls with X ij > th unn do 4: Union(i, j); 5: nd or 6: or ll tripls with X ij < th sp do 7: Sprt(i, j); 8: nd or 9: Construt grph G M ; 0: i grph siz 3 thn : rturn; : ls i grph siz 7 thn 3: Bktrking sd thr-wy prtitioning; 4: ls 5: FM sd thr-wy prtitioning; 6: nd i Algorithm shows our prtition sd mpping produr. Givn th solutions rom progrm (3), som triplts r onstrutd nd sortd to mintin ll non-zro X ij vlus (lins ). Th mpping inorports two stgs to dl with th two dirnt typs. Th irst stg (lins 3 8) is similr to tht in [7]. I X ij is lr thn th rltionship twn vrtis r i nd r j n dirtly dtrmind. Hr th unn nd th sp r usr-dind thrshold vlus. For xmpl, i X ij > th unn, whih mns tht r i nd r j should in th sm olor, thn untion Union(i, j) is pplid to mrg thm into lrg vrtx. Similrly, i X ij < th sp, thn untion Sprt(i, j) is usd to ll r i nd r j s inomptil. In th sond stg (lins 9 6) w dl with th vgu X ij vlus. During th prvious stg som vrtis hv n mrgd, thror th totl vrtx numr is not lrg. Hr w onstrut grph G M to rprsnt th rltionships mong ll th rmnnt vrtis (lin 9). Eh dg ij in this grph hs wight rprsnting th ost i vrtis i nd j r ssignd into sm olor. Thror, th olor ssignmnt prolm n ormultd s mximum-ut prtitioning prolm on G M (lin 0 6). Through ssigning wight to h vrtx rprsnting its dnsity, grph G M is l to ln dnsity mong dirnt ins. Bsd (0) (00) () d(5) () d (5) (5) (0) (0) A (00) + (00) d(5) d(5) Fig. 4. Dnsity Blnd Mpping. () Domposition grph. () Construt grph G M. () Mpping rsult with ut vlu 8. nd dnsity uniormitis 4. A ttr mpping with ut 8. nd dnsity uniormitis 3. on th G M, prtitioning is prormd to simultnously hiv mximum-ut nd lnd wight mong dirnt prts. Not tht w nd to modiy th gin untion, thn in h mov, w try to hiv mor lnd nd lrgr ut prtitions. An xmpl o th dnsity lnd mpping is shown in Fig. 4. Bsd on th domposition grph (s Fig. 4 ()), SDP is ormultd. Givn th solutions o SDP, tr th irst stg o mpping, vrtis nd d r mrgd in to lrg vrtx. As shown in Fig. 4(), th grph G M is onstrutd, whr h vrtx is ssoitd with wight. Thr r two prtition rsults with th sm ut vlu 8. (s Fig. 4 () nd Fig. 4 ). Howvr, thir dnsity uniormitis r 4 nd 3, rsptivly. To kp mor lnd dnsity rsult, th sond prtitioning in Fig. 4 () is doptd s olor ssignmnt rsult. It is wll known tht th mximum-ut prolm, vn or - wy prtition, is NP-hrd. Howvr, w osrv tht in mny ss, tr th glol SDP optimiztion, th grph siz o G M ould quit smll, i.., lss thn 7. For ths smll ss, w dvlop ktrking sd mthod to srh th ntir solution sp. Not tht hr ktrking n quikly ind th optiml solution vn through thr-wy prtitioning is NP-hrd. I th grph siz is lrgr, w propos huristi mthod, motivtd y th lssi FM prtitioning lgorithm [6] [7]. Dirnt rom th lssi FM lgorithm, w mk th ollowing modiitions. () In th irst stg o mpping, som vrtis r lld s inomprl, thror or moving vrtx rom on prtition to nothr, w should hk whthr it is lgl. () Clssil FM lgorithm is or minut prolm, w nd to modiy th gin untion o h mov to hiv mximum ut. Th runtim omplxity o grph onstrution is O(m), whr m is th vrtx numr in G M. Th runtim o thr-wy mximumut prtitioning lgorithm is O(mlogm). Bsids, th irst stg o mpping nds O(n logn) [7]. Sin m is muh smllr thn n, th omplxity o dnsity lnd mpping is O(n logn) () C. Dnsity Blnd Lyout Grph Simpliition Hr w show tht th lyout grph simpliition, whih ws proposd in [7], n onsidr th lol dnsity ln s wll. During lyout grph simpliition, w itrtivly rmov nd push ll vrtis with dgr lss thn or qul to two. Atr th olor ssignmnt on th rmind vrtis, w itrtivly rovr ll th rmovd (5) (5) (5) (5) 66

5 vrtis nd ssign lgl olors. Instd o rndomly piking on, w srh lgl olor whih is good or th dnsity uniormitis. V. SPEEDUP TECHNIQUES d +d Our lyout domposr pplis st o grph simpliition thniqus proposd y rnt works: Indpndnt Componnt Computtion [7] [8] []; Vrtx with Dgr Lss thn 3 Rmovl [7] [8] []; -Edg-Conntd Componnt Computtion [7] [8] []; -Vrtx-Conntd Componnt Computtion [8] []. Aprt rom th ov grph simpliitions, our domposr proposs st o novl spdup thniqus, whih would introdud in this stion. A. LG Cut Vrtx Stith Foriddn Fig. 5. d g d g () () g d d Lyout grph ut vrtx stith oriddn. A vrtx o grph is lld ut vrtx i its rmovl domposs th grph into two or mor onntd omponnts. Cut vrtis n idntiid through th pross o ridg omputtion [7]. During stith ndidt gnrtion, oridding ny stith ndidt on ut vrtis n hlpul or ltr domposition grph simpliition. Fig. 5 () shows lyout grph, whr tur is ut vrtx, sin its rmovl n prtition th lyout grph into two prts: {,, d} nd {,, g}. I stith ndidts r introdud within, th orrsponding domposition grph is illustrtd in Fig. 5 (), whih is hrd to urthr simpliid. I w orid th stith ndidt on, th orrsponding domposition grph is shown in Fig. 5 (), whr is still ut vrtx in domposition grph. Thror w n pply -onntd omponnt omputtion [8] to simpliy th prolm siz, nd pply olor ssignmnt sprtly (s Fig. 5 ). B. Domposition Grph Vrtx Clustring Domposition grph vrtx lustring is spdup thniqu to urthr rdu th domposition grph siz. As shown in Fig. 6 (), vrtis nd d shr th sm onlit rltionships ginst nd. Bsids, thr is no onlit dgs twn nd d. I no onlit is introdud, vrtis nd d should ssignd th sm olor, thror w n lustr thm togthr, s shown in Fig. 6 (). Not tht th stith nd onlit rltionships r lso mrgd. Applying vrtx lustring in domposition grph n urthr rdu th prolm siz. g g () ' DG DG g g Fig. 6. () DG vrtx lustring to rdu th domposition grph siz. C. Fst Color Assignmnt Tril Although th SDP nd th prtition sd mpping n provid high prormn or olor ssignmnt, it is still xpnsiv to pplid to ll th domposition grphs. W driv st olor ssignmnt tril or lling SDP sd mthod. I no onlit or stith is introdud, our tril solvs th olor ssignmnt prolm in linr tim. Not tht SDP mthod is skippd only whn domposition grph n olord without stith or onlit, our st tril dos not los ny solution qulity. Bsids, our prliminry rsults show tht mor thn hl o th domposition grphs n domposd using this st mthod. Thror, th runtim n drmtilly rdud. Algorithm Fst Color Assignmnt Tril Rquir: Domposition grph G, stk S. : whil n G s.t. d on (n) < 3 & d stit(n) < do : S.push(n); G.dlt(n); 3: nd whil 4: i G is not mpty thn 5: Rovr ll vrtis in S; 6: rturn ls; 7: ls 8: whil!s.mpty() do 9: n = S.pop(); G.dd(n); 0: Assign n lgl olor; : nd whil : rturn tru; 3: nd i Th st olor ssignmnt tril is shown in Algorithm. First, w itrtivly rmov th vrtx with onlit dgr (d on ) lss thn 3 nd stith dgr (d stit) lss thn (lins 3). I som vrtis nnot rmovd, w rovr ll th vrtis in stk S, thn rturn ls; Othrwis, th vrtis in S r itrtivly poppd (rovrd) (lins 8 ). For h vrtx n poppd, sin it is onntd with t most on stith dg, w n lwys ssign on olor without introduing onlit or stith. VI. () EXPERIMENTAL RESULTS W implmnt our domposr in C++ nd tst it on n Intl Xon 3.0GHz Linux mhin with 3G RAM. ISCAS 85&89 nhmrks rom [7] r usd, whr th minimum oloring sping dis m ws st th sm with prvious studis [7] [8]. Bsids, to prorm omprhnsiv omprison, w lso tst on othr two nhmrk suits. Th irst suit is with six dns nhmrks ( 9 totl - s5 totl ), whil th sond suit is two synthsizd OpnSPARC T dsigns mul top nd xu with Nngt 45nm stndrd ll lirry [8]. Whn prossing ths two nhmrk suits w st th minimum oloring distn dis m = w min + 3 s min, whr w min nd s min dnot th minimum wir width nd th minimum sping, rsptivly. Th prmtr α is st s 0.. Th siz o h in is st s 0 dis m 0 dis m. W us CSDP [9] s th solvr or th smidinit progrmming (SDP). 67

6 TABLE II. COMPARISON OF RUNTIME AND PERFORMANCE. Ciruit ICCAD [7] DAC [8] DAC 3 [] SDP+PM n# st# ost CPU(s) n# st# ost CPU(s) n# st# ost CPU(s) n# st# ost CPU(s) C C C C C C C C C C S S S S S vg rtio TABLE III. COMPARISON ON VERY DENSE LAYOUTS Ciruit ICCAD 0 [7] DAC 0 [8] SDP+PM n# st# ost CPU(s) n# st# ost CPU(s) n# st# ost CPU(s) mul top xu totl totl s totl s3 totl s4 totl NA NA NA > s5 totl NA NA NA > vg. NA NA NA > rtio - > A. Comprison with othr domposrs In th irst xprimnt, w ompr our domposr with th stto-th-rt lyout domposrs whih r not lnd dnsity wr [7] [8] []. W otin th inry ils rom [7] nd [8]. Sin urrntly w nnot otin th inry or domposr in [], w dirtly us th rsults listd in []. Hr our domposr is dnotd s SDP+PM, whr PM mns th prtition sd mpping. Th β is st s 0. In othr words, SDP+PM only optimizs or stith nd onlit numr. Tl III shows th omprison in trms o runtim nd prormn. For h domposr w list its stith numr, onlit numr, ost nd runtim. Th olumns n# nd st# dnot th onlit numr nd th stith numr, rsptivly. ost is th ost untion, whih is st s n# +0. st#. CPU(s) is omputtionl tim in sonds. First, w ompr SDP+PM with th domposr in [7], whih is sd on SDP ormultion s wll. From Tl III w n s tht th nw stith ndidt gnrtion (s [] or mor dtils) nd prtition-sd mpping n hiv ttr prormn (rduing th ost y round 55%). Bsids, SDP+PM n gt nrly 4 spd-up. Th rson is tht, omprd with [7], st o spdup thniqus, i.., -vrtx-onntd omponnt omputtion, lyout grph ut vrtx stith oriddn (S. V-A), domposition grph vrtx lustring (S. V-B), nd st olor ssignmnt tril (S. V-C), r proposd. Sond, w ompr SDP+PM with th domposr in [8], whih pplis svrl grph sd simpliitions nd mximum indpndnt st (MIS) sd huristi. From Tl III w n s tht lthough th domposr in [8] is str, MIS sd huristi hs wors solution qulitis (round 33% ost pnlty omprd to SDP+PM). Comprd with th domposr in [], lthough SDP+PM is slowr, it n rdu th ost y round 6%. In ddition, w ompr SDP-PM with othr two domposrs [7] [8] or som vry dns lyouts, s shown in Tl IV. W n s tht or som ss th domposr in [7] nnot inish in 000 sonds. Comprd with [8] work, SDP+PM n rdu ost y 65%. It is osrvd tht omprd with othr domposrs, SDP+PM Th rsults o DAC 3 domposition r rom []. TABLE IV. BALANCED DENSITY IMPACT ON EPE Ciruit SDP+PM SDP+PM+DB ost CPU(s) EPE# ost CPU(s) EPE# C C C C C C C C C C S S S S S vg rtio dmonstrts muh ttr prormn whn th input lyout is dns. Th rson my tht whn th input lyout is dns, through grph simpliition, h indpndnt prolm siz my still quit lrg, thn SDP sd pproximtion n hiv ttr rsults thn huristi. It n osrvd tht or th lst thr ss our domposr ould rdu thousnds o onlits. Eh onlit my rquir mnul lyout modiition or high ECO orts, whih r vry tim onsuming. Thror, vn our runtim is mor thn [8], it is still ptl (lss thn 6 minuts or th lrgst nhmrk). B. Comprison or Dnsity Bln In th sond xprimnt, w tst our domposr or th dnsity lning. W nlyz dg plmnt rror (EPE) using Clir- Worknh [0] nd industry-strngth stup. For nlyzing th EPE in our tst ss, w us systmti lithogrphy pross vrition, suh s ous ±50nm nd dos ±5%. In Tl IV, w ompr SDP+PM with SDP+PM+DB, whih is our dnsity lnd domposr. Hr β is st s 0.04 (w hv tstd dirnt β vlus, w ound tht iggr β dos not hlp muh ny mor; mnwhil, w still wnt to giv onlit nd stith highr wights). Column ost lso lists 68

7 TABLE V. ADDITIONAL COMPARISON FOR DENSITY BALANCE Ciruit SDP+PM SDP+PM+DB ost CPU(s) EPE# ost CPU(s) EPE# mul top xu totl totl s totl s3 totl s4 totl s5 totl vg rtio th wightd ost o onlit nd stith, i.., ost = n#+0. st#. From Tl IV w n s tht y intgrting dnsity ln into our domposition low, our domposr (SDP+PM+DB) n rdu EPE hotspot numr y 4%. Bsids, dnsity lnd SDP sd lgorithm n mintin similr prormn to th slin SDP implmnttion: only 7% mor ost o onlit nd stith, nd only 8% mor runtim. In othr words, our domposr n hiv good dnsity ln whil kping omprl onlits/stiths. W urthr ompr th dnsity ln, spilly EPE distriutions or vry dns lyouts. As shown in Tl V, our dnsity lnd domposr (SDP+PM+DB) n rdu EPE distriution numr y 7%. Bsids, or vry dns lyouts, dnsity lnd SDP pproximtion n mintin similr prormn with plin SDP implmnttion: only 4% mor runtim. C. Slility o SDP Formultion Runtim (s) Fig Runtim omplxity o SDP Slility o SDP Formultion. Numr o nods runtim o SDP O(x^.) O(x^.4) In ddition, w dmonstrt th slility o our domposr, spilly th SDP ormultion. Pnros nhmrks rom [6] r usd to xplor th slility o SDP runtim. No grph simpliition is pplid, thror ll runtim is onsumd y solving SDP ormultion. Fig. 7 illustrts th rltionship twn grph (prolm) siz ginst SDP runtim. Hr th X xis dnots th numr o nods (.g., th prolm siz), nd th Y xis shows th runtim. W n s tht th runtim omplxity o SDP is lss thn O(n. ). VII. CONCLUSION In this ppr, w propos high prormn TPL lyout domposr with lnd dnsity. Dnsity lning is intgrtd into ll th ky stps o our domposition low. In ddition, w propos st o spdup thniqus, suh s lyout grph ut vrtx stith oriddn, domposition grph vrtx lustring, nd st olor ssignmnt tril. Comprd with stt-o-th-rt rmworks, our domposr dmonstrts th st prormn in minimizing th ost o onlits nd stiths. Furthrmor, our lnd domposr n otin lss EPE whil mintining vry omprl onlit nd stith rsults. As TPL my doptd y industry or 4nm/nm nods, w liv mor rsrh will ndd to nl TPL-rindly dsign nd msk synthsis. ACKNOWLEDGMENT This work is supportd in prt y NSF grnts CCF nd CCF-8906, SRC tsk 44.00, NSFC grnt 6800, nd IBM Sholrship. REFERENCES [] ITRS. [Onlin]. Avill: [] B. Yu, J.-R. Go, D. Ding, Y. Bn, J.-S. Yng, K. Yun, M. Cho, nd D. Z. Pn, Dling with IC mnuturility in xtrm sling, in IEEE/ACM Intrntionl Conrn on Computr-Aidd Dsign (ICCAD), 0, pp [3] Y. Borodovsky, Lithogrphy 009 ovrviw o opportunitis, in Smion Wst, 009. [4] K. Lus, C. Cork, B. Yu, G. Luk-Pt, B. Pintr, nd D. Z. Pn, Implitions o tripl pttrning or 4 nm nod dsign nd pttrning, in Pro. o SPIE, vol. 837, 0. [5] J.-S. Yng, K. Lu, M. Cho, K. Yun, nd D. Z. Pn, A nw grphthorti, multi-ojtiv lyout domposition rmwork or doul pttrning lithogrphy, in IEEE/ACM Asi nd South Pii Dsign Automtion Conrn (ASPDAC), 00. [6] C. Cork, J.-C. Mdr, nd L. Brns, Comprison o tripl-pttrning domposition lgorithms using priodi tiling pttrns, in Pro. o SPIE, vol. 708, 008. [7] B. Yu, K. Yun, B. Zhng, D. Ding, nd D. Z. Pn, Lyout domposition or tripl pttrning lithogrphy, in IEEE/ACM Intrntionl Conrn on Computr-Aidd Dsign (ICCAD), 0, pp. 8. [8] S.-Y. Fng, W.-Y. Chn, nd Y.-W. Chng, A novl lyout domposition lgorithm or tripl pttrning lithogrphy, in IEEE/ACM Dsign Automtion Conrn (DAC), 0. [9] H. Tin, H. Zhng, Q. M, Z. Xio, nd M. Wong, A polynomil tim tripl pttrning lgorithm or ll sd row-strutur lyout, in IEEE/ACM Intrntionl Conrn on Computr-Aidd Dsign (ICCAD), 0. [0] B. Yu, J.-R. Go, nd D. Z. Pn, Tripl pttrning lithogrphy (TPL) lyout domposition using nd-utting, in Pro. o SPIE, vol. 8684, 03. [] J. Kung nd E. F. Young, An iint lyout domposition pproh or tripl pttrning lithogrphy, in IEEE/ACM Dsign Automtion Conrn (DAC), 03. [] Q. M, H. Zhng, nd M. D. F. Wong, Tripl pttrning wr routing nd its omprison with doul pttrning wr routing in 4nm thnology, in IEEE/ACM Dsign Automtion Conrn (DAC), 0, pp [3] Y.-H. Lin, B. Yu, D. Z. Pn, nd Y.-L. Li, TRIAD: A tripl pttrning lithogrphy wr dtild routr, in IEEE/ACM Intrntionl Conrn on Computr-Aidd Dsign (ICCAD), 0. [4] B. Yu, X. Xu, J.-R. Go, nd D. Z. Pn, Mthodology or stndrd ll omplin nd dtild plmnt or tripl pttrning lithogrphy, in IEEE/ACM Intrntionl Conrn on Computr-Aidd Dsign (ICCAD), 03. [5] P. Chn nd E. S. Kuh, Floorpln sizing y linr progrmming pproximtion, in IEEE/ACM Dsign Automtion Conrn (DAC), 000. [6] C. M. Fidui nd R. M. Mtthyss, A linr-tim huristi or improving ntwork prtitions, in ACM/IEEE Dsign Automtion Conrn (DAC), 98, pp [7] L. A. Snhis, Multipl-wy ntwork prtitioning, IEEE Trns. Comput., vol. 38, pp. 6 8, Jnury 989. [8] NnGt FrPDK45 Gnri Opn Cll Lirry, [9] B. Borhrs, CSDP, C lirry or smidinit progrmming, Optimiztion Mthods nd Sotwr, vol., pp , 999. [0] Mntor Clir, 69