Lyout Domposition for Qurupl Pttrning Lithogrphy n Byon Bi Yu ECE Dprtmnt Univrsity of Txs t Austin, Austin, TX i@r.utxs.u Dvi Z. Pn ECE Dprtmnt Univrsity of Txs t Austin, Austin, TX pn@.utxs.u ABSTRACT For nxt-gnrtion thnology nos, multipl pttrning lithogrphy (MPL) hs mrg s ky solution,.g., tripl pttrning lithogrphy (TPL) for 14/11nm, n qurupl pttrning lithogrphy (QPL) for su-10nm. In this ppr, w propos gnri n roust lyout omposition frmwork for QPL, whih n furthr xtn to hnl ny gnrl K-pttrning lithogrphy (K>4). Our frmwork is s on th smifinit progrmming (SDP) formultion with novl oloring noing. Mnwhil, w propos fst yt fftiv oloring ssignmnt n hiv signifint spup. To our st knowlg, this is th first work on th gnrl multipl pttrning lithogrphy lyout omposition. Ctgoris n Sujt Dsriptors B.7.2 [Hrwr, Intgrt Ciruit]: Dsign Ais Gnrl Trms Algorithms, Dsign, Prformn Kywors Multipl Pttrning Lithogrphy, Lyout Domposition 1. INTRODUCTION As th minimum ftur siz furthr rss, multipl pttrning lithogrphy (MPL) hs om on of th most vil solutions to su-14nm hlf-pith pttrning, long with xtrm ultr violt lithogrphy (EUVL), ltri m lithogrphy (EBL), n irt slf-ssmly (DSA) [1,2]. Lst fw yrs hv sn xtnsiv rsrhs on MPL thnology suh s oul pttrning [], n tripl pttrning [4]. Continuing growth of thnology no is xpt to shrink furthr own to 11nm or yon. Suh vn is, nonthlss, mking onvntionl pttrning prosss rly suffiint for th nxt gnrtion. Qurupl pttrning lithogrphy (QPL) is nturl xtnsion long th prigm of oul/tripl pttrning. In th QPL mnufturing, thr r four xposur/thing prosss, through whih th initil lyout n prou. Com- Prmission to mk igitl or hr opis of ll or prt of this work for prsonl or lssroom us is grnt without f provi tht opis r not m or istriut for profit or ommril vntg n tht opis r this noti n th full ittion on th first pg. To opy othrwis, to rpulish, to post on srvrs or to ristriut to lists, rquirs prior spifi prmission n/or f. DAC 14, Jun 01 05 2014, Sn Frniso, CA, USA Copyright 2014 ACM 978-1-450-270-5/14/06...$15.00. msk 1 msk 2 msk msk 4 () Figur 1: () A ommon ntiv onflit from tripl pttrning lithogrphy; () Th onflit n rsolv through qurupl pttrning lithogrphy. pr with tripl pttrning lithogrphy, QPL introus on mor msk. Although inrsing th numr of prossing stps y % ovr tripl pttrning, thr r svrl rsons/vntgs for QPL. Firstly, u to th ly or unrtinty of othr lithogrphy thniqus, suh s EUVL, smionutor inustry ns CAD tools to prpr n unrstn th omplxity/implition of QPL. Evn from thortil prsptiv, stuying th gnrl multipl pttrning is vlul. Sonly, it is osrv tht for tripl pttrning lithogrphy, vn with stith insrtion, thr r svrl ommon ntiv onflit pttrns. As shown in Fig. 1 (), ontt lyout within th stnr ll my gnrt som 4- liqu pttrns, whih r inomposl. This onflit n sily rsolv if four msks r vill (s Fig. 1 ()). Thirly, with on mor msk, som stiths my voi uring mnufturing. By this wy it is potntil to rsolv th ovrlpping n yil issus riv from th stiths. Th pross of QPL rings up svrl ritil yt opn sign hllngs, suh s lyout omposition, whr th originl lyout is ivi into four msks (olors). Doul/tripl pttrning lyout omposition with onflit n stith minimiztion hs n wll stui for full-hip lyout [ 12] n ll s sign [1 15]. Th prolm n optimlly solv through xpnsiv intgr linr progrmming (ILP) [ 5]. To ovrom th long runtim prolm of ILP solvr, for oul pttrning, prtitioning/mthing s mthos hv n propos [6,7]; whil for tripl pttrning, som spup thniqus,.g., smifinit progrmming (SDP) [4,10], n huristi oloring ssignmnt [8, 9] hv n propos. Howvr, how to fftivly solv th qurupl pttrning, or vn gnrl multipl pttrning prolms, is still n opn qustion. In this ppr, w l with th qurupl pttrning lyout omposition (QPLD) prolm. Our ontriutions r highlight s follows. (1) To our st knowlg, this is th first lyout omposition rsrh for QPLD prolm. W liv this work will invok mor futur rsrh into this fil thry promoting th sling of thnology no. (2) Our frmwork onsists of holisti lgorithmi prosss, suh s smifinit progrmming s lgorithm, linr olor s- ()
signmnt, n novl GH-tr s grph ivision. () W monstrt th viility of our lgorithm to suit with gnrl K-pttrning (K 4) lyout omposition, whih oul vn guilins for futur thnology. Th rst of th ppr is orgniz s follows. In Stion 2, w giv th prolm formultions n th ovrll omposition flow. In Stion n Stion 4 w propos th olor ssignmnt lgorithms n grph ivision thniqus, rsptivly. Stion 5 xtns our mthoologis to gnrl K-pttrning prolm. Stion 6 prsnts th xprimnt rsults, follow y onlusion in Stion 7. 2. PRELIMINARIES 2.1 Prolm Formultion Givn input lyout whih is spifi y fturs in polygonl shps, omposition grphs [4, 5] is onstrut y Dfinition 1. Dfinition 1 (Domposition Grph). A omposition grph is n unirt grph {V, CE, SE} with singl st of vrtis V, n two g sts CE n SE ontining th onflit gs (CE) n stith gs (SE), rsptivly. Eh vrtx v V rprsnts polygonl shp, n g CE xists iff th two polygonl shps r within minimum oloring istn min s, n n g SE iff thr is stith nit twn th two vrtis whih r ssoit with th sm polygonl shp. Now w giv th prolm formultion of qurupl pttrning lyout omposition (QPLD). Prolm 1 (QPLD). Givn n input lyout whih is spifi y fturs in polygonl shps n minimum oloring istn min s, th omposition grph is onstrut. Qurupl pttrning lyout omposition (QPLD) ssigns ll th vrtis into on of four olors (msks) to minimiz onflit numr n stith numr. Th QPLD prolm n xtn to gnrl K-pttrning lyout omposition prolm s follows. Prolm 2 (K-Pttrning Lyout Domposition). Givn n input lyout, th omposition grph is onstrut. Eh vrtx in grph woul ssign into on of K olors (msks) to minimiz onflit numr n stith numr. 2.2 Ovrviw of Lyout Domposition Flow min_s Input Lyout Figur 2: Domposition Grph Constrution Grph Division Color Assignmnt Output Msks SDP s Algorithm Linr Color Assignmnt Propos lyout omposition flow. Th ovrll flow of our lyout omposition is summriz in Fig. 2. W first onstrut omposition grph to trnsform th originl gomtri pttrns into grph mol. By this wy, th QPLD prolm n formult s 4 oloring on th omposition grph. To ru th prolm siz, grph ivision thniqus (s Stion 4) r ppli to prtition th grph into st of omponnts. Thn th olor ssignmnt prolm n solv inpnntly for h omponnt, through st of lgorithms isuss in Stion.. COLOR ASSIGNMENT IN QPLD Givn omposition grph G = {V, CE, SE}, olor ssignmnt woul rri out to ssign h vrtx into on of four olors (msks), to minimiz oth th onflit numr n th stith numr. In this stion, w propos two olor ssignmnt lgorithms, i.., smifinit progrmming (SDP) s lgorithm, n linr olor ssignmnt..1 SDP Bs Color Assignmnt ( Figur : p 6, p 2, 1 ) (0, 2p 2, 1 ) y z (0, 0, 1) ( p 6, p 2, 1 ) x Four vtors orrspon to four olors. Smifinit progrmming (SDP) hs n sussfully ppli to tripl pttrning lyout omposition [4, 10]. Hr w will show tht SDP formultion n xtn to solv QPLD prolm. To rprsnt four iffrnt olors (msks), s illustrt in Fig., four unit vtors r introu [16]: (0, 0, 1), (0, 2 2, 1 ), ( 6, 2, 1 ) n ( 6, 2, 1 ). W onstrut th vtors in suh wy tht innr prout for ny two vtors v i, v j stisfying: v i v j = 1 if v i = v j; v i v j = 1 if vi vj. Bs on th vtor finition, th QPLD prolm n formult s th following vtor progrmming: min ij CE 4 ( vi vj + 1 ) + α 4 ij SE (1 v i v j) (1) s.t. v i {(0, 0, 1), (0, 2 2, 1 6 2 ), (,, 1 ), 6 2 (,, 1 )} whr th ojtiv funtion is to minimiz th onflit numr n th stith numr. α is usr-fin prmtr, whih is st s 0.1 in this work. Aftr rlxing th isrt onstrints in (1) n rmoving th onstnt in ojtiv funtion, w rrw th following smifinit progrmming (SDP) formultion. min ij CE v i v j α s.t. v i v i = 1, i V v i v j 1, ij SE ij CE v i v j (2) Aftr solving th SDP, w gt st of ontinuous solutions in mtrix X, whr h vlu x ij in mtrix X orrspons to v i v j. If x ij is los to 1, vrtis v i, v j r tn to
in th sm msk (olor). A gry mpping lgorithm [4] n irtly ppli hr to gt olor ssignmnt solution. Howvr, th prformn of gry mtho my not goo. Algorithm 1 SDP + Bktrk Input: SDP solution x ij, thrshol vlu t th ; 1: for ll x ij t th o 2: Comin vrtis v i, v j into on lrgr vrtx; : n for 4: Construt mrg grph G = {V, CE, SE }; 5: BACKTRACK(0, G ); 6: rturn olor ssignmnt rsult in G ; 7: funtion BACKTRACK(t, G ) 8: if t siz[g ] thn 9: if Fin ttr olor ssignmnt thn 10: Stor urrnt olor ssignmnt; 11: n if 12: ls 1: for ll lgl olor o; 14: G [t] ; 15: BACKTRACK(t + 1, G ); 16: G [t] 1; 17: n for 18: n if 19: n funtion To ovrom th limittion of th gry mtho, in our frmwork ktrk s lgorithm (s Algorithm 1) is propos to onsir oth SDP rsults n grph informtion. Th ktrk s mtho pts two rgumnts of th SDP solution {x ij} n thrshol vlu t th. In our work t th is st s 0.9. As isuss ov, if x ij is los to 1, two vrtis v i n v j tn to in th sm olor (msk). Thrfor, w sn ll pirs, n omin som vrtis into on lrgr vrtx (lins 1 ). Aftr th omintion, th vrtx numr n ru, thus th grph hs simplifi (lin 4). Th simplifi grph is ll mrg grph [10]. On th mrg grph, BACKTRACK lgorithm is prsnt to srh n optiml olor ssignmnt (lins 7 19)..2 Linr Color Assignmnt Bktrk s mtho my still involv runtim ovrh, spilly for omplx s whr SDP solution nnot provi nough mrging nits. Thrfor, n ffiint olor ssignmnt is rquir. At first gln, th olor ssignmnt for qurupl pttrning n solv through four olor mp thorm [17] tht vry plnr grph is 4-olorl. Howvr, in mrging thnology no, th signs r so omplx tht w osrv mny K 5 or K, struturs, whr K 5 is th omplt grph on fiv vrtis, whil K, is th omplt iprtit grph on six vrtis. Du to Kurtowski s thorm [18], th omposition grph is not plnr, thus lssil four oloring thniqus [19] is hr to ppli. Hr w propos n ffiint olor ssignmnt lgorithm. Not tht our mtho is trgting gnrl grph, not just plnr grph. In ition, iffrnt from lssil four oloring mtho tht ns qurti runtim [19], our olor ssignmnt is linr runtim lgorithm. Th tils of linr olor ssignmnt is summriz in Algorithm 2, whih involvs thr stgs. Th first stg is itrtivly vrtx rmovl. For h vrtx v i, w not its onflit gr onf (v i) s numr of onflit gs inint to v i, Algorithm 2 Linr Color Assignmnt Input: Domposition grph G = {V, CE, SE}, Stk S; 1: whil v i V s.t. onf (v i) < 4 & stit(v i) < 2 o 2: S.push(v i); : G.lt(v i); 4: n whil 5: Construt vtor v; 6: C1 = SEQUENCE-COLORING(v); 7: C2 = DEGREE-COLORING(v); 8: C = ROUND-COLORING(v); 9: C = st oloring solution mong {C1, C2, C}; 10: POST-REFINEMENT(v); 11: whil!s.mpty() o 12: v i = S.pop(); 1: G.(v i); 14: (v i) lgl olor; 15: n whil Hlf Pith () () Figur 4: () Domposition grph; () Gry oloring with on onflit; () is tt s olor-frinly to ; () Coloring onsiring olor-frinly ruls. whil its stith gr stit(v i) s numr of stith gs. Th min i is tht th vrtis with onflit gr lss thn 4 n stith gr lss thn 2 r intifi s non-ritil, thus n tmporrily rmov n push into stk S (lins 1-4). Aftr oloring rmining vrtis, h vrtx in stk S woul pop up on y on n ssign on lgl olor (lins 11-15). This strtgy is sf in trms of onflit numr. In othr wors, whn vrtx is pop up from S, thr is lwys on olor vill without introuing nw onflit. In th son stg (lins 5-9), ll rmining vrtis woul ssign olors on y on. Howvr, olor ssignmnt through on spifi orr my stuk t lol optimum whih stms from th gry ntur. For xmpl, givn omposition grph in Fig. 4 (), if th oloring orr is ----, whn vrtx is grily slt gry olor, th following vrtx nnot fin ny olor without onflit (s Fig. 4 ()). In othr wors, vrtx orring signifintly impts th oloring rsult. To llvit th impt of vrtx orring, two strtgis r propos. Th first strtgy is ll olor-frinly ruls, s in Dfinition 2. In Fig. 4 (), ll onflit nighors of pttrn r ll insi gry ox. Sin th istn twn n is within th rng of (min s, min s + hp), is olor-frinly to. Intrstingly, w isovr rul tht for omplx/ns lyout, olor-frinly pttrns tn to with th sm olor. Bs on ths ruls, uring linr olor ssignmnt, to trmin on vrtx olor, inst of just ompring its on- () ()
flit/stith nighors, th olors of its olor-frinly vrtis woul lso onsir. Dtting olor-frinly vrtis is similr to th onflit nighor ttion, thus it n finish uring omposition grph onstrution without muh itionl fforts. Dfinition 2 (Color-Frinly). A pttrn is olor-frinly to pttrn, iff thir istn is lrgr thn min s, ut smllr thn min s + hp. Hr hp is th hlf pith. Our son strtgy is ll pr sltion, whr thr iffrnt vrtx orrs woul pross simultnously, n th st on woul slt s th finl oloring solution (lins 6-8). Although olor ssignmnt is solv thri, sin for h orr th oloring is in linr tim, th totl omputtionl tim is still linr. In th thir stg (lin 10), post-rfinmnt grily hks h vrtx to s whthr th solution n furthr improv. For omposition grph with olor-frinly informtion n n vrtis, in th first stg vrtx rmovl/pop up n finish in O(n). In th son stg, s mntion ov th oloring ns O(n). In post-rfinmnt stg, ll vrtis r trvl on, whih rquirs O(n) tim. Thrfor, th totl omplxity is O(n). 4. GRAPH DIVISION FOR QPLD Grph ivision is thniqu tht prtitions th whol omposition grph into st of omponnts, thn th olor ssignmnt on h omponnt n solv inpnntly. In our frmwork, th thniqus xtn from prvious work r summriz s follows, (1) Inpnnt Componnt Computtion [4 10, 1]; (2) Vrtx with Dgr Lss thn Rmovl [4,8 10] 1 ; () 2-Vrtx-Connt Componnt Computtion [8 10]. 4.1 GH-Tr s -Cut Rmovl Anothr thniqu, ut rmovl, hs n provn powrful in oul/pttrning lyout omposition [4, 7, 8]. A ut of grph is n g whos rmovl isonnts th grph into two omponnts. Th finition of ut n xtn to 2- ur (-ut), whih is oul (triplt) of gs whos rmovl woul isonnt th grph. Howvr, iffrnt from th 1- ut n 2-ut ttion tht n finish in linr tim [8], -ut ttion is muh mor omplit. In this sustion w propos n fftiv -ut ttion mtho. Bsis, our mtho n sily xtn to tt ny K-ut (K ). olor 0 olor 1 olor 2 olor omponnt 1 f omponnt 2 () omponnt 1 f omponnt 2 () rott y 1 omponnt 1 f omponnt 2 Figur 5: An xmpl of -ut ttion n rmovl. 1 In QPLD prolm, th vrtis with gr lss thn 4 woul tt n rmov tmporlly. () () 4 4 () Figur 6: () Domposition grph; () Corrsponing GH-tr; () Componnts ftr -ut rmovl. Fig. 5 () shows grph with -ut (,, f), n two omponnts n riv y rmoving this -ut. Aftr olor ssignmnt on two omponnts, for h ut g, if th olors of th two npoints r iffrnt, th two omponnts n mrg irtly. Othrwis, olor rottion oprtion is rquir to on omponnt. For vrtx v in grph, w not (v) s its olor, whr (v) {0, 1, 2, }. Vrtx v is si to rott y i, if (v) is hng to ((v) + i)%4. It is sy to s tht ll vrtis in on omponnt shoul rott y th sm vlu, so no itionl onflit is introu within th omponnt. An xmpl of suh olor rottion oprtion is illustrt in Fig. 5 ()-(), whr onflit twn vrtis, f woul rmov to intronnt two omponnts togthr. Hr ll th vrtis in omponnt 2 r rott y 1 (s Fig. 5 ()). W hv th following Lmm: Lmm 1. In QPLD prolm, olor rottion ftr intronnting -ut os not inrs th onflit numr. In ition, to tt ll -uts, w hv th following Lmm: Lmm 2. If th minimum ut twn two vrtis v i n v j is lss thn 4, thn v i, v j long to iffrnt omponnts tht ivi y -ut. Bs on Lmm 2, w n s tht if th ut twn vrtis v i, v j is lrgr or qul to 4 gs, v i, v j shoul long to th sm omponnt. On strightforwr -ut ttion mtho is to omput th minimum uts for ll th {s t} pirs. Howvr, for omposition grph with n vrtis, thr r n(n 1)/2 pirs of vrtis. Computing ll ths ut pirs my spn too long runtim, whih is imprtil for omplx lyout sign. Gomory n Hu [20] show tht th ut vlus twn ll th pirs of vrtis n omput y solving only n 1 ntwork flow prolms on grph G. Furthrmor, thy show tht th flow vlus n rprsnt y wight tr T on th n vrtis, whr for ny pir of vrtis (v i, v j), if is th minimum wight g on th pth from v i to v j in T, thn th minimum ut vlu from v i to v j in G is xtly th wight of. Suh wight tr T is ll Gomory-Hu tr (GH-tr). For xmpl, givn th omposition grph in Fig. 6 (), th orrsponing GH-tr is shown in Fig. 6 (), whr th vlu on g ij is th minimum ut numr twn vrtis v i n v j. Bus of Lmm 2, to ivi th grph through -ut rmovl, ll th gs with vlu lss thn 4 woul rmov. Th finl thr omponnts r in Fig. 6 (). Th prour of th -ut rmovl is shown in Algorithm. Firstly w onstrut GH-tr s on th lgorithm y [21] (lin 1). Dini s loking flow lgorithm [22] is ppli to hlp GH-tr onstrution. Thn ll gs in th GH-tr with wights lss thn four r rmov (lin 2). Aftr solving th onnt omponnt prolm (lin ), w n ssign olors to ()
Algorithm GH-tr s -Cut Rmovl Input: Domposition grph G = {V, CE, SE}; 1: Construt GH-tr s in [21]; 2: Rmov th gs with wight < 4; : Comput onnt omponnts on rmining GH-tr; 4: for h omponnt o 5: Color ssignmnt on this omponnt; 6: n for 7: Color rottion to intronnt ll omponnts; h omponnt sprtly (lins 4 5). At lst olor rottion is ppli to intronnt ll -uts k (lin 6). 5. GENERAL K-PATTERNING LAYOUT DE- COMPOSITION In this stion, w monstrt tht our lyout omposition frmwork is gnrlizl to K-pttrning lyout omposition, whr K > 4. Thorm 1: SDP formultion in () n provi v i v j pirs for K-pttrning olor ssignmnt prolm. min ( v i v j + 1 k 1 ) + α (1 v i v j) () ij CE s.t. v i v i = 1, i V v i v j 1 k 1, ij CE ij SE W n s tht if K = 4, formultion () quivlnts to (2). Rphrsing oth th SDP formultion in () n ktrk mtho in Algorithm 1, th olor ssignmnt prolm for K-pttrning n rsolv. In ition, th linr olor ssignmnt lgorithm in Stion.2 n xtn to gnrl K-pttrning prolm s wll. All th grph ivision thniqus in Stion 4 n xtn hr. Bsis, w rw th following Thorm: Thorm 2: For K-pttrning lyout omposition prolm, iviing grph through (K 1)-ut os not inrs th finl onflit numr. Th proof n provi y xtning Lmm 1. Bs on Thorm 2, GH-tr s ut rmovl in Stion 4 n ppli hr to srh ll (K 1)-uts. Tht is, ftr onstruting GH-tr, ll gs with wight lss thn K r rmov. 6. EXPERIMENTAL RESULTS W implmnt th propos lyout omposition lgorithms in C++, n tst on Linux mhin with 2.9GHz CPU. W hoos GUROBI [2] s th intgr linr progrmming (ILP) solvr, n CSDP [24] s th SDP solvr. Th nhmrks in [4,8] r us s our tst ss. W sl own th Mtl1 lyr to 20nm hlf pith. Both th minimum ftur with w m n th minimum sping twn fturs s m r 20nm. From Fig. 7 w n s tht whn minimum oloring istn min s = 2 s m + w m = 60nm, vn on imnsion rgulr pttrns n K 5 strutur, whih is not 4-olorl or plnr [18]. In our xprimnts, for qurupl pttrning min s is st s 2 s m + 2 w m = 80nm, whil for pntupl pttrning min s is st s s m + 2.5 w m = 110nm. Whn lrgr min s is ppli, thr r too mny ntiv onflits in lyouts, s th nhmrks r not multipl pttrning frinly. () w m s m () 2 s m + w m Figur 7: min s = 2 s m + w m my us K 5 strutur. 6.1 Qurupl Pttrning First w ompr iffrnt olor ssignmnt lgorithms for qurupl pttrning, n th rsults r list in Tl 1. ILP, SDP+Bktrk, SDP+Gry n Linr not ILP formultion, SDP follow y ktrk mpping (Stion.1), SDP follow y gry mpping, n linr olor ssignmnt (Stion.2), rsptivly. Hr w implmnt n ILP formultion xtn from th tripl pttrning work [4]. In SDP+Gry, gry mpping from [4] is ppli. All th grph ivision thniqus, inluing GH-tr s ivision, r ppli. Th olumns n# n st# not th onflit numr n th stith numr, rsptivly. Column CPU(s) is olor ssignmnt tim in sons. From Tl 1 w n s tht for smll ss th ILP formultion n hiv st prformn in trms of onflit numr n stith numr. Howvr, for lrg ss (S8417, S592, S8584, S15850) ILP suffrs from long runtim prolm tht non of thm n finish in on hour. Compr with ILP, SDP+Bktrk n hiv nr-optiml solutions, i.., in vry s th onflit numr is optiml, whil only in on s 2 mor stiths r introu. SDP+Gry mtho n hiv 2 spup ginst SDP+Bktrk. But th prformn of SDP+Gry is not goo tht for omplx signs hunrs of itionl onflits r rport. Th linr olor ssignmnt n hiv roun 200 spup ginst SDP+Bktrk, whil only 15% mor onflits n 8% mor stiths r rport. 6.2 Pntupl Pttrning W furthr ompr th lgorithms for pntupl pttrning, tht is, K = 5. To our st knowlg thr is no xt ILP formultion for pntupl pttrning in litrtur. Thrfor w onsir thr slins, i.., SDP+Bktrk, SDP+Gry, n Linr. All th grph ivision thniqus r ppli. Tl 2 vluts six most ns ss. W n s tht ompr with SDP+Bktrk, SDP+Gry n hiv roun 8 spup, ut 15% mor onflits r rport. In trms of runtim, linr olor ssignmnt n hiv 500 n 60 spup, ginst SDP+Bktrk n SDP+Gry, rsptivly. In trms of prformn, linr olor ssignmnt rports th st onflit numr minimiztion, ut mor stiths my introu. Intrstingly, w osrv tht whn lyout is multipl pttrning frinly, olor-frinly ruls n provi goo guilin, thus linr olor ssignmnt n hiv high prformn in trms of onflit numr. Howvr, whn lyout is vry omplx or involving mny ntiv onflits, linr olor ssignmnt rports mor onflits thn SDP+Bktrk. On possil rson is tht th olor-frinly ruls r not goo in moling glol onflit minimiztion, ut oth SDP n ktrk provi glol viw. 7. CONCLUSIONS In this ppr w hv propos th first lyout ompo-
Tl 1: Comprison for Qurupl Pttrning Ciruit ILP SDP+Bktrk SDP+Gry Linr n# st# CPU(s) n# st# CPU(s) n# st# CPU(s) n# st# CPU(s) C42 2 0 0.6 2 0 0.24 2 0 0.02 2 1 0.001 C499 1 4 0.7 1 4 0.16 1 4 0.05 1 4 0.001 C880 1 0 0. 1 0 0.02 1 0 0.02 1 2 0.001 C155 0 4 0.6 0 4 0.1 0 4 0.04 0 4 0.001 C1908 2 1.0 2 0.28 2 0.09 2 4 0.001 C2670 0 6 1.1 0 6 0.16 0 6 0.1 0 7 0.001 C540 1 1.1 1 0.09 2 2 0.05 1 0.001 C515 1 1 2.8 1 1 0.6 2 12 0.24 1 15 0.002 C6288 9 0 2. 9 0 0.6 9 0 0.17 9 1 0.001 C7552 2 1.4 2 1 0.6 12 0.22 2 18 0.00 S1488 0 6 0.7 0 6 0.05 4 2 0.01 0 6 0.001 S8417 20 549 1226.7 20 551 6.6 142 429 2.7 21 576 0.0 S592 N/A N/A >600 50 1745 28.7 460 18 16.4 64 1927 0.15 S8584 N/A N/A >600 41 165 21.1 470 1224 10.4 47 1744 0.12 S15850 N/A N/A >600 42 1462 18 420 1084 7.8 48 1571 0.11 vg. - - >802.7 11.5 64.0 5.14 101.2 274.7 2.56 1. 92.2 0.0 rtio - - >156. 1.0 1.0 1.0 8.8 0.75 0.49 1.15 1.08 0.005 Tl 2: Comprison for Pntupl Pttrning Ciruit SDP+Bktrk SDP+Gry Linr n# st# CPU(s) n# st# CPU(s) n# st# CPU(s) C6288 19 2 2.4 19 2 0.49 19 5 0.005 C7552 1 1 0. 1 1 0.05 1 4 0.001 S8417 0 4 1.45 0 4 0.21 0 4 0.001 S592 5 20 8.11 5 20 0.62 5 25 0.009 S8584 4 1.66 7 0. 6 0.008 S15850 6 5 2.7 7 5 0.4 5 15 0.007 vg. 5.7 6.0 2.77 6.5 5.8 0.5 5.5 9.8 0.005 rtio 1.0 1.0 1.0 1.15 0.97 0.12 0.97 1.64 0.002 sition frmwork for qurupl pttrning n yon. Exprimntl vlutions hv monstrt tht our lgorithm is fftiv n ffiint to otin high qulity solution. As ontinuing sling of thnology no to su-10nm, MPL my promising mnufturing solution. W liv this ppr will stimult mor futur rsrh into this fil, thry filitting th vnmnt of MPL thnology. Aknowlgmnt Th uthors woul lik to thnk Tsung-Wi Hung for hlpful isussions. This work is support in prt y NSF, NSFC, SRC, Orl, n Toshi. 8. REFERENCES [1] D. Z. Pn, B. Yu, n J.-R. Go, Dsign for mnufturing with mrging nnolithogrphy, IEEE Trnstions on Computr-Ai Dsign of Intgrt Ciruits n Systms (TCAD), vol. 2, no. 10, pp. 145 1472, 201. [2] B. Yu, J.-R. Go, D. Ding, Y. Bn, J.-S. Yng, K. Yun, M. Cho, n D. Z. Pn, Dling with IC mnufturility in xtrm sling, in Pro. ICCAD, 2012, pp. 240 242. [] A. B. Khng, C.-H. Prk, X. Xu, n H. Yo, Lyout omposition for oul pttrning lithogrphy, in Pro. ICCAD, 2008, pp. 465 472. [4] B. Yu, K. Yun, B. Zhng, D. Ding, n D. Z. Pn, Lyout omposition for tripl pttrning lithogrphy, in Pro. ICCAD, 2011, pp. 1 8. [5] K. Yun, J.-S. Yng, n D. Z. Pn, Doul pttrning lyout omposition for simultnous onflit n stith minimiztion, in Pro. ISPD, 2009, pp. 107 114. [6] Y. Xu n C. Chu, A mthing s omposr for oul pttrning lithogrphy, in Pro. ISPD, 2010, pp. 121 126. [7] X. Tng n M. Cho, Optiml lyout omposition for oul pttrning thnology, in Pro. ICCAD, 2011, pp. 9 1. [8] S.-Y. Fng, W.-Y. Chn, n Y.-W. Chng, A novl lyout omposition lgorithm for tripl pttrning lithogrphy, in Pro. DAC, 2012, pp. 1185 1190. [9] J. Kung n E. F. Young, An ffiint lyout omposition pproh for tripl pttrning lithogrphy, in Pro. DAC, 201, pp. 69:1 69:6. [10] B. Yu, Y.-H. Lin, G. Luk-Pt, D. Ding, K. Lus, n D. Z. Pn, A high-prformn tripl pttrning lyout omposr with ln nsity, in Pro. ICCAD, 201, pp. 16 169. [11] Y. Zhng, W.-S. Luk, H. Zhou, C. Yn, n X. Zng, Lyout omposition with pirwis oloring for multipl pttrning lithogrphy, in Pro. ICCAD, 201, pp. 170 177. [12] B. Yu, J.-R. Go, n D. Z. Pn, Tripl pttrning lithogrphy (TPL) lyout omposition using n-utting, in Pro. of SPIE, vol. 8684, 201. [1] H. Tin, H. Zhng, Q. M, Z. Xio, n M. Wong, A polynomil tim tripl pttrning lgorithm for ll s row-strutur lyout, in Pro. ICCAD, 2012, pp. 57 64. [14] B. Yu, X. Xu, J.-R. Go, n D. Z. Pn, Mthoology for stnr ll omplin n til plmnt for tripl pttrning lithogrphy, in Pro. ICCAD, 201, pp. 49 56. [15] H. Tin, Y. Du, H. Zhng, Z. Xio, n M. Wong, Constrin pttrn ssignmnt for stnr ll s tripl pttrning lithogrphy, in Pro. ICCAD, 201, pp. 57 64. [16] D. Krgr, R. Motwni, n M. Sun, Approximt grph oloring y smifinit progrmming, J. ACM, vol. 45, pp. 246 265, Mrh 1998. [17] K. Appl n W. Hkn, Evry plnr mp is four olorl. prt i: Dishrging, Illinois Journl of Mthmtis, vol. 21, no., pp. 429 490, 1977. [18] C. Kurtowski, Sur l prolm s ours guhs n topologi, Funmnt mthmti, vol. 15, no. 1, pp. 271 28, 190. [19] N. Rortson, D. P. Snrs, P. Symour, n R. Thoms, Effiintly four-oloring plnr grphs, in ACM Symposium on Thory of omputing, 1996, pp. 571 575. [20] R. E. Gomory n T. C. Hu, Multi-trminl ntwork flows, Journl of th Soity for Inustril & Appli Mthmtis, vol. 9, no. 4, pp. 551 570, 1961. [21] D. Gusfil, Vry simpl mthos for ll pirs ntwork flow nlysis, SIAM Journl on Computing, vol. 19, no. 1, pp. 14 155, 1990. [22] E. A. Dini, Algorithm for solution of prolm of mximum flow in ntworks with powr stimtion, in Sovit Mth. Dokl, vol. 11, no. 5, 1970, pp. 1277 1280. [2] GUROBI, http://www.guroi.om/html/mi.html. [24] B. Borhrs, CSDP, C lirry for smifinit progrmming, Optimiztion Mthos n Softwr, vol. 11, pp. 61 62, 1999.