Layout Decomposition for Triple Patterning Lithography
|
|
- Alfred Turner
- 6 years ago
- Views:
Transcription
1 Lyout Domposition or Tripl Pttrning Lithogrphy Bi Yu, Kun Yun, Boyng Zhng, Duo Ding, Dvi Z. Pn ECE Dpt. Univrsity o Txs t Austin, Austin, TX USA 7871 Cn Dsign Systms, In., Sn Jos, CA USA 9514 Emil: {i, pn}@r.utxs.u Astrt As imum tur siz n pith sping urthr rs, tripl pttrning lithogrphy (TPL) is possil 19nm xtnsion long th prigm o oul pttrning lithogrphy (DPL). Howvr, thr is vry littl stuy on TPL lyout omposition. In this ppr, w show tht TPL lyout omposition is mor iiult prolm thn tht or DPL. W thn propos gnrl intgr linr progrmg ormultion or TPL lyout omposition whih n simultnously imiz onlit n stith numrs. Sin ILP hs vry poor slility, w propos thr lrtion thniqus without sriiing solution qulity: inpnnt omponnt omputtion, lyout grph simpliition, n rig omputtion. For vry ns lyouts, vn with ths spup thniqus, ILP ormultion my still too slow. Thror, w propos novl vtor progrmg ormultion or TPL omposition, n solv it through tiv smiinit progrmg (SDP) pproximtion. Exprimntl rsults show tht th ILP with lrtion thniqus n ru 8% runtim ompr to th slin ILP. Using SDP s lgorithm, th runtim n urthr ru y 4% with som tro in th stith numr (ru y 7%) n th onlit (9% mor). Howvr, or vry ns lyouts, SDP s lgorithm n hiv 140 sp-up vn ompr with lrt ILP. 1. INTRODUCTION As imum tur siz urthr sls, th smionutor inustry is grtly hllng o pttrning su-nm hlpith u to th ly o vil nxt gnrtion lithogrphy suh s Extrm Ultr Violt (EUV). Doul pttrning lithogrphy (DPL) is wily rogniz s promising solution or nm, nm, n possily 16nm volum hip proution. As shown in Fig. 1, th ky hllng o DPL lis in th omposition pross y whih th originl lyout is ivi into two msks. Thn, thr r two xposur/thing stps, through whih th lyout n prou. Th vntg o this pproh is tht th tiv pith n oul, whih improvs th lithogrphy rsolution. During th omposition, whn th istn twn th two pttrns is lss thn imum olorl istn s, thy n to ssign to irnt msks to voi onlit. Somtims onlit n solv y splitting pttrn into two touhing prts, ll stiths. In Fig. 1(), polygon is split into two polygons 1 n in orr to rsolv th omposition onlits. Howvr th introu stiths l to yil loss u to ovrly rror [1]. Thror, two o th hllngs in lyout omposition r onlit n stith imiztion. Th prigm o oul pttrning my urthr xtn to tripl pttrning lithogrphy (TPL). Inustry hs lry xplor th tst-hip pttrns with tripl pttrning or vn () 1 () 1 msk1 msk Fig. 1: In DPL, singl lyr is ompos into two msks n th pith n inrs tivly. () Conlits Fig. : () In DPL, vn stith insrtion n not voi ntiv onlit. () Th ntiv onlits n rsolv y TPL. qurupl pttrning []. By using TPL, w n hiv urthr tur-siz sling through pith-tripling. It shll not tht in DPL, vn with stith insrtion, thr my ntiv onlits []. Fig. () shows thrwy onlit yl twn turs, n, whr ny two o thm r within th s. As onsqun, thr is no hn to prou onlit-r solution using DPL omposition. Howvr, w n sily rsolv this prolm i th lyout is ompos into thr msks s shown in Fig. (). Yt this os not mn TPL lyout omposition prolm oms sir. Atully sin th turs n pk losr, th prolm turns out to mor iiult, s to shown in Stion. Muh prvious rsrh ouss on th oul pttrning lyout omposition prolm, whih is gnrlly rgr s two-oloring prolm on onlit grph. Intgr Linr Progrmg (ILP) is opt in [4][5] to imiz th stith numr n/or th onlit numr. Xu t l. [6] propos n iint grph rution-s lgorithm or stith imiztion, n Yng t l. [7] propos st -ut s pproh. A mthing s omposr is propos to imiz oth th onlit n th stith numrs [8]. To rsolv th ntiv () () /11/$ IEEE 1
2 onlit, svrl works introu lyout moiition to urthr imiz th onlit numr [9][10][11]. Howvr, lyout moiition my us nw prolms, i.., tig losur, hotspot. Until now, thr r vry w invstigtions on TPL lyout omposition. [1] proposs tripl pttrning oloring lgorithm, whih opts SAT Solvr. Howvr, thir work only ls with ontt rrys, not gnrl lyout struturs with wirs, ontts, n so on. Bsis, it os not involv ny stith imiztion. [1][14] propos sl-lign tripl pttrning (SATP) pross to xtn 19nm immrsion lithogrphy to hl-pith 15nm pttrning. But th SATP pross nnot insrt ny stith, whih woul grtly onstrin th possil lyout pttrns tht r omposl [15]. To our st knowlg, thr is no stuy so r on lyout omposition on TPL or gnrl lyout styls. In this ppr, w propos th irst systmti stuy on lyout omposition or tripl pttrning lithogrphy. W irst ormult gnrl ILP ormultion or TPL lyout omposition to simultnously imiz onlit n stith. To improv slility, w urthr propos thr lrtion thniqus without loss o solution qulity: lyout grph simpliition, inpnnt omponnt omputtion n rigs omputtion. A smiinit progrmg s pproximtion lgorithm is urthr propos to improv slility. Smiinit progrmg is n xtnsion o linr progrmg to pproximtly solv NP-hr prolms n it hs n sussully ppli to mny otoril prolms [16][17]. Our ontriutions o this ppr inlu: Gnrl ILP ormultion to simultnously imiz onlit n stith or TPL lyout omposition; Thr lrtion thniqus to improv ILP slility; A novl vtor progrmg ormultion or TPL omposition n its smiinit progrmg s pproximtion lgorithm whih n urthr l with vry ns lyouts whr vn lrt ILP oms too slow; Our xprimntl rsults r vry promising in trms o qulity o rsults n runtim tro. Th rst o th ppr is orgniz s ollows: in Stion, w isuss th prolm ormultion n thn nlyz prolm omplxity. Th si lgorithm n som lrtion thniqus r sri in stion. Stion 4 proposs smiinit progrmg s lgorithm to urthr lrt th si lgorithm. Stion 5 prsnts th xprimnt rsults, ollow y onlusion in Stion 6.. PROBLEM FORMULATION AND COMPLEXITY Som prliris on TPL r provi in this stion, inluing som initions n th prolm ormultion. W lso monstrt th omplxity o th prolm..1. Prolm Formultion Givn lyout whih is spii y turs in polygonl shps, lyout grph [4] n omposition grph [9] r onstrut. () () Fig. : Lyout grph onstrution n omposition grph onstrution () Input lyout rprsnting s irrgulr polygons. ()Corrsponing lyout grph, whr ll gs r onlit gs. ()Th no projtion. ()Corrsponing omposition grph, whr sh gs r stith gs. Dinition 1 (Lyout Grph): Th lyout grph (LG) is n unirt grph whos nos r th givn lyout s polygonl shps n whr n g xists i n only i th two polygonl shps r within imum oloring istn s o h othr. Fig. () givs n xmpl o n input lyout; th orrsponing lyout grph is shown in Fig. (). All th gs in lyout grph r ll Conlit Egs (CE). A onlit xists i n only i two nos r onnt y CE n r in th sm msk. In othr wors, h onlit g is onlit nit. Dinition (Domposition Grph): Givn lyout rprsnt y st o polygonl shps, th omposition grph (DG) is n unirt grph with singl st o nos V, n two sts o gs, CE n SE, whih ontin th onlit gs n stith gs, rsptivly. V hs on or mor nos or h polygonl shp n h no is ssoit with polygonl shp. An g is in CE i th two polygonl shps r within imum oloring istn s o h othr. An g is in SE i thr is stith twn th two nos whih r ssoit with th sm polygonl shp. On th lyout grph, th no projtion is irst prorm, whr projt sgmnts r highlight y ol lins in Fig. (). Bs on th projtion rsult, ll th lgl splitting lotions r omput. Thn th omposition grph is onstrut, s shown in Fig. (). Not tht th onlit gs r mrk s lk gs, whil stith gs r mrk s sh gs. Prolm 1 (TPL lyout omposition): Givn lyout whih is spii y turs in polygonl shps, th lyout grph n th omposition grph r onstrut. Our gol () () 1 1
3 CE SE V r i x i ij s ij x i1, x i ij1, ij s ij1, s ij TABLE I: Nottion Nottion us in Mthmtil progrmg st o onlit gs st o stith gs. th st o polygns. th i th lyout polygons vril noting th oloring o r i 0-1 vril, ij = 1 whn onlit twn r i n r j 0-1 vril, s ij = 1 whn stith twn r i n r j Nottion us in ILP ormultion two 1-it 0-1 vrils to rprsnts olors two 1-it 0-1 vrils to tr ij two 1-it 0-1 vrils to tr s ij is to ssign ll th nos in th omposition grph to thr msks to imiz th stith numr n th onlit numr... Prolm Complxity At irst gln, th lyout omposition is similr to grph oloring prolm. Howvr, sin stith gs r introu, th prolm to imiz onlit n stith is mor omplit. For oul pttrning s, iing whthr grph is -olorl is sy y tring i thr xists o yls. For onlit n stith imiztion, i omposition grph is plnr DPL lyout omposition n solv in polynomil tim [8]. In orr to solv th tripl pttrning issu, th prolm oms mor omplit. Lmm 1: Diing whthr plnr grph is -olorl is NP-omplt [18]. Lmm 1 n nturlly xtn to gnrl grph. Bs on Lmm 1, th mthoology in [1] is not suitl or TPL omposition: SAT solvr n only work or -olorl lyout grph, whih nnot hk in polynomil tim. Lmm : Coloring -olorl grph with 4 olors is NP-omplt [19]. -oloring prolm is to ssign th nos in on -olorl grph to olors. Sin oloring th grph with 4 olors nnot inish in polynomil tim, it n shown - oloring prolm is NP-hr. Bs on ov lmms, vn th omposition grph is plnr, w rh th ollowing thorm: Thorm 1: TPL lyout omposition prolm is NPhr. W n prov this thorm y ruing -Coloring prolm to th TPL omposition prolm. Du to pg limit, th til proo is skipp hr.. BASIC ALGORITHM In this stion, w will prsnt our si lgorithms, whih r s on th Intgr Linr Progrmg (ILP). Sin th tig omplxity or ILP is vry high, w propos thr lrtion thniqus to ivi th whol prolm into svrl smllr ons. Th ntir low is shown in Fig Mthmtil Formultion or TPL Domposition Th mthmtil ormultion or TPL lyout omposition is shown in (1). For onvnin, som nottions in mthmtil progrmg n ILP ormultion r list in Tl I. Th ojtiv is to simultnously imiz oth th onlit numr n th stith numr. Th prmtr α is usr-in prmtr or ssigning rltiv importn twn th onlit numr n th stith numr. ij CE ij + α ij SE s ij (1) s.t. ij = (x i == x j ) ij CE (1) s ij = x i x j ij SE (1) x i {0, 1, } i V (1) whr x i is vril or th thr olors o rtngls r i, ij is inry vril or onlit g ij CE n s ij is inry vril or stith g ij SE. Constrint (1) is us to vlut th onlit numr whn touh nos r i n r j r ssign irnt olors (msks). Constrint (1) is us to lult th stith numr. I no r i n no r j r ssign th sm olor (msk), stith s ij is introu... ILP Formultion or TPL Lyout Domposition W will now show how to ormult (1) with Intgr Linr Progrmg. Not tht qs. (1) n (1) n linriz only whn x i is 0-1 vril [4], whih nnot rprsnt thr irnt olors. To hnl this prolm, w rprsnt th olor o h no using two 1-it 0-1 vrils x i1 n x i. In orr to limit th numr o olors or h no to, or h pir (x i1, x i ) th vlu (1, 1) is not prmitt. In othr wors, only vlus (0, 0), (0, 1) n (1, 0) r llow. Thus, (1) n ormult s ollows: ij + α s ij () ij CE s.t. x i1 + x i 1 ij SE () x i1 + x j1 1 + ij1 ij CE () (1 x i1 ) + (1 x j1 ) 1 + ij1 ij CE () x i + x j 1 + ij ij CE () (1 x i ) + (1 x j ) 1 + ij ij CE () ij1 + ij 1 + ij ij CE () x i1 x j1 s ij1 ij SE (g) x j1 x i1 s ij1 ij SE (h) x i x j s ij ij SE (i) x j x i s ij ij SE (j) s ij s ij1, s ij s ij ij SE (k) Th ojtiv untion is th sm s tht in (1), whih imizs th wight summtion o th onlit numr n th stith numr. Constrint () is us to limit th numr o olors or h no to.
4 Input Lyout Lyout Grph Constrution..1 Inpnnt Componnt Computtion.. Lyout Grph Simpliition Fig. 4: Bsi Algorithms Flow Domposition Grph Constrution.. Brig Computtion ILP Formultion Output Msks Constrints () to () r quivlnt to onstrint (1), whr 0-1 vril ij1 monstrts whthr x i1 quls to x j1, n ij monstrts whthr x i quls to x j. 0-1 vril ij is tru only i two nos onnt y onlit g ij r in th sm olor,.g. oth ij1 n ij r tru. Similrly, onstrints (g) to (k) r quivlnt to onstrint (1). 0-1 vril s ij1 monstrts whthr x i1 is irnt rom x j1, n s ij monstrts whthr x i is irnt rom x j. Stith s ij is tru i ithr s ij1 or s ij is tru... Alrtion Thniqus Sin ILP is n NP-hr prolm, its runtim inrss rmtilly with th siz o omposition grph. W propos thr lrtion thniqus to simpliy th lyout grph n th omposition grph in orr to ru th tim omplxity o ILP. As shown in Fig.4, our lrtion low onsists o thr stps: Inpnnt Componnt Computtion, Lyout Grph Simpliition n Brigs Computtion...1) Inpnnt Componnt Computtion: W propos inpnnt omponnt omputtion on th omposition grph to ru th ILP prolm siz without losing optimlity. In lyout grph o rl sign, w osrv mny isolt lustrs. Thror, w n rk own th whol sign into svrl inpnnt omponnts, n pply si ILP ormultion or h on. Th ovrll solution n tkn s th union o ll th omponnts without ting th glol optimlity. Th runtim o ILP ormultion rss rmtilly with th rution o vrils n onstrints, n th oloring ssignmnt n tivly lrt. Inpnnt omponnt omputtion is wll-known thniqu whih hs n ppli in mny prvious stuis [4][5][7][9]...) Lyout Grph Simpliition: W n simpliy th lyout grph y rmoving ll nos with gr lss thn or qul to two. At th ginning, ll nos with gr lss thn or qul to two r tt n rmov tmporrily rom th lyout grph. This rmoving pross will ontinu until ll th nos r t lst gr-thr. Th lyout grph simpliition lgorithm is shown in Algorithm 1. I ll th nos in th lyout grph n push onto th stk, Algorithm 1 n solv TPL lyout omposition optimlly in linr tim. As n xmpl shown in Fig.5, vry no n push onto stk n inlly olor whn Algorithm 1 Lyout Grph Simpliition n Color Assignmnt Rquir: Lyout Grph G to simplii, stk S 1: whil n G s.t. gr(n) o : S.push(n); : G.lt(n); 4: n whil 5: Domposition grph onstrution. 6: TPL lyout omposition or nos not simplii. 7: whil!s.mpty() o 8: n = S.pop(); 9: G.(n); 10: Assign n lgl olor. 11: n whil () () Fig. 6: Brigs Computtion. () Atr rigs omputtion, ll g s rig. () In two su-grphs rry out ILP ormultion. () Rott olors in on su-grph to rig. is popp o. Not tht vn whn som nos nnot simplii, lyout grph simpliition n ru prolm siz rmtilly. Aitionlly, w osrv tht this lgorithm n lso prtition th lyout grph into svrl su-grphs...) Brigs Computtion: A rig o grph is n g whos rmovl isonnts th grph into two omponnts. I th two omponnts r inpnnt, rmoving th rig n ivi th whol ILP into two inpnnt ILP ormultions. Thorm : Prtitioning omposition grph y rmoving rigs os not introu nw stiths. An xmpl o th rigs omputtion is shown in Fig. 6. First o ll, onlit g is oun to rig. Rmoving th rig ivis th omposition grph into two sis. Atr ILP s olor ssignmnt, i no n no r ssign th sm olor, w n rott olors o ll nos in on su-grph. Similr mtho n opt whn rig is stith g. W opt n O( V + E ) lgorithm [0] to in rigs in omposition grph. Using ov thr lrtion thniqus, th ILP ormultion n still hiv optiml solutions. In othr wors, our lrtion lgorithms n kp optimlity. Du to pg limit, w skip th til isussion hr. 4. SDP BASED ALGORITHM Although th lrt lgorithms n simpliy th prolm siz in mny wys, ILP my still too slow or () 4
5 () () () () () () (g) (h) Fig. 5: This lyout n irtly ompos y lyout grph simpliition. () Input lyout. () Corrsponing lyout grph. ()()() Itrtivly rmov n push in nos with gs no mor thn. ()(g)(h)(i) Itrtivly pop up n rovr no, n ssign ny lgl olor. (j) Finl omposition rsult. (i) (j) 1 (-, ) 1 (-,- ) (1, 0) Fig. 7: Thr vtors (1, 0), ( 1, ), ( 1, ) rprsnt thr irnt olors. lrg prolms whih nnot simplii tivly. In this stion w provi n pproximtion lgorithm to otin mor rpi solutions. First, novl vtor progrmg or TPL lyout omposition is ormult. Thn w rlx th vtor progrmg into Smiinit Progrmg (SDP). Givn this solution rom Smiinit Progrmg, w n otin th TPL omposition rsults in polynomil tim Vtor Progrmg or TPL Lyout Domposition In TPL omposition, thr r thr possil olors. W st unit vtor v i or vry no i. I ij is onlit g, w wnt nos v i n v j to r prt. I ij is stith g, w hop or nos v i n v j to th sm. As shown in Fig. 7, w ssoit ll th nos with thr irnt unit vtors: (1, 0), ( 1, ) n ( 1, ). Not tht th ngl twn ny two vtors o th sm olor is 0, whil th ngl twn vtors with irnt olors is π/. Aitionlly, w in th innr prout o two m- imnsion vtors v i n v j s ollows: m v i v j = v ik v jk k=1 whr h vtor v i n rprsnt s (v i1, v i,... v im ). Thn or th vtors v i, v j {(1, 0), ( 1, ), ( 1, )}, w hv th ollowing proprty: { 1, vi = v v i v j = j v i v j 1 Bs on th ov proprty, w n ormult th TPL lyout omposition s th ollowing vtor progrm: ( v i v j + 1 ) + α (1 v i v j ) () ij CE ij SE s.t. v i {(1, 0), ( 1, ), ( 1, )} () Formul () is quivlnt to mthmtil ormul (1). Sin th TPL omposition is NP-hr, this vtor progrmg is lso NP-hr. In th nxt prt, w will rlx () to smiinit progrmg, whih n solv in polynomil tim. 4.. Smiinit Progrmg Approximtion Constrint () rquirs solutions o () isrt. Atr rmoving this onstrint, w gnrt ormul (4) s ollows: ij CE ( y i y j + 1 ) + α ij SE (1 y i y j ) (4) s.t. y i y i = 1, i V (4) y i y j 1, ij CE (4) 5
6 This ormul is rlxtion o () sin w n tk ny sil solution v i = (v i1, v i ) to prou sil solution o (4) y stting y i = (v i1, v i, 0, 0,, 0), i.. y i y j = 1 n y i y j = v i v j in this solution. Thus i Z R is th vlu o n optiml solution o ormul (4) n OP T is n optiml vlu o ormul (), it must stisy: Z R OP T. In othr wors, solution o (4) is n pproximtion to tht in (). Sin w only r out th vlu o y i, progrm (4) n urthr simplii y liting th onstnts in th ojtiv untion: ( y i y j ) α ( y i y j ) (5) ij CE s.t. (4) (4) ij SE Without isrt onstrint (), progrms (4) n (5) r not NP-hr now. To solv (5) in polynomil tim, w will show tht it is quivlnt to smiinit progrmg. Smiinit progrmg (SDP) is similr to linr progrmg whih hs linr ojtiv untion n linr onstrints. Howvr, squr symmtri mtrix o vrils n onstrin to positiv smiinit. Although smiinit progrms r mor gnrl thn linr progrms, oth o thm n solv in polynomil tim. Bsis, th rlxtion s on th smiinit progrmg hs ttr thortil rsults thn thos s on LP [16]. Consir th ollowing stnr smiinit progrm: SDP: A X (6) X ii = 1, i V (6) X ij 1, ij CE (6) X 0 (6) whr A X is th innr prout twn two mtris A n X, i.. i j A ijx ij. Hr A ij is th ntry tht lis in th i-th row n th j-th olumn o mtrix A. Constrint (6) mns mtrix X shoul positiv smiinit. 1, ij CE A ij = α, ij SE (7) 0, othrwis Similrly, X ij is th i-th row n th j-th olumn ntry o X. Not tht th solution o SDP is rprsnt s positiv smiinit mtrix X, whil solutions o vtor progrmg r stor in list o vtors. Howvr, w n show tht thy r quivlnt. Lmm : A symmtri mtrix X is positiv smiinit i n only i X = V V T or som mtrix V. Givn positiv smiinit mtrix X, using th Cholsky omposition w n in orrsponing mtrix V in O(n ) tim. Thorm : Th smiinit progrm (6) n th vtor progrm (5) r quivlnt. Proo: Givn solutions o (5) { v 1, v, v m }, th orrsponing mtrix X is in s X ij = v i v j. In th othr irtion, s on Lmm, givn mtrix X rom (6), w n in mtrix V stisying X = V V T y using th Cholsky omposition. Th rows o V r vtors {v i } tht orm th solutions o (5). 4.. Mpping Algorithm Solutions o progrm (6) r ontinuous, whil optiml solutions in () r isrt. In this sustion w mp th ontinuous solutions into isrt ons. In th mtrix X gnrt y SDP, i X ij is los to 1, thn nos i n j shoul in th sm msk, whil i X ij is los to 0.5, no i n no j tn to in irnt msks. Our mpping lgorithm is givn in Algorithm, whih ins th rltiv rltionships mong th nos n mps thm into thr irnt msks. First som triplts r onstrut n sort to stor ll X ij inormtion. Thn, w rry out our mpping lgorithm in two stps. In th irst stp, i X ij is los to 1, th no i n th no j will in th sm msk, whil i X ij is los to 0.5, thy will ll to in irnt msks. Hr th vtors UnionLvl[k] n SpLvl[k] r som usr in prmtrs: UnionLvl[] r los to 1 n SpLvl[] r los to 0.5. In th son stp, w ontinu to union th no i n th no j with mximum X ij until ll nos r ssign into thr msks. W us th isjoint-st t strutur to group nos into thr msks. Implmnt with union y rnk n pth omprssion, th running tim pr oprtion o isjoint-st is lmost onstnt [1]. Lt n th numr o nos, n th numr o triplts is n. Sorting ll th triplts rquirs O(n logn). Sin ll triplts r sort, h o thm n visit t most on. Bus th runtim o h oprtion n inish lmost in onstnt tim, th omplxity o Algorithm is O(n logn). Algorithm Mpping Algorithm 1: Solv th progrm (6), gt mtrix X. : Ll h non-zro ntry X i,j s triplt (X ij, i, j). : sort ll (X ij, i, j) y X ij. 4: or k = 1 to R o 5: or h tripl (X ij, i, j) o 6: i X ij > UnionLvl[k] && Comptil(i, j) thn 7: Union(i, j); 8: n i 9: n or 10: or h tripl (x ij, i, j) o 11: i X ij < SpLvl[k] thn 1: Sprt(i, j); 1: n i 14: n or 15: n or 16: whil Msks numr > o 17: Pik tripl with mximum X ij n Comptil(i, j); 18: Union (i, j); 19: n whil 6
7 1 () 4 5 Fig. 8: Exmpl to olor omposition grph. ()Input omposition grph. ()Using smiinit progrmg n Algorithm, w ssign nos into irnt olors (msks) An Exmpl o th SDP Bs Algorithm Fig. 8 shows n xmpl o th omposition grph. This grph inlus 7 onlit gs n 1 stith g, n n olor with olors. Morovr, th grph is not - olorl sin it ontins o yls. Hr w show how th smiinit progrmg n us to solv TPL lyout omposition prolm. I w st α = 0.1, thn mtrix A is s ollow: A = 1 () Atr solving th smiinit progrmg (6), w n gt mtrix X s ollowing: X = hr w only show th uppr prt o th mtrix X. From th mtrix X w n in tht no 1 n no 4 shoul in th sm olor (us X 14 = 1.0), n no n no 5 shoul lso in th sm olor (us X 5 = 1.0). Howvr, sin X 1, X 1 n X 15 r los to 0.5, nos, n 5 nnot ssign in th sm olor s no 1. Using Algorithm, w n mp ll th nos into thr olors: {1, 4}, {} n {, 5}. Th inl mpping rsult is shown in Fig. 8(). 5. EXPERIMENTAL RESULTS W implmnt our lgorithm in C++ n tst it on n Intl Cor.0GHz Linux mhin with G RAM. OpnAss. [] is us or intring with GDSII irtly. W hoos CBC [] s our solvr or th intgr linr progrmg, n CSDP [4] s th solvr or th smiinit progrmg. ISCAS-85 & 89 nhmrks r sl own n moii to rlt th 16nm thnology no. Th mtl on lyr is us or xprimntl purposs, us it is on o th most omplx lyrs in trms o lyout omposition. Th imum with n sping om 5nm n 0nm. Th 5 imum olorl istn is st s 85nm, n th imum ovrlpping mrgin or stith insrtion is 10nm. Th prmtr α is st s Comprison First, w show th tivnss o th lyout grph simpliition n th rigs omputtion. Tl II omprs th Norml ILP n th Alrt ILP, whr th Norml ILP only uss inpnnt omponnt omputtion thniqu, whil th Alrt ILP uss ll thr lrtion thniqus. Columns SE# n CE# not th stith g numr n onlit g numr rsptivly. From Tl II w n s tht lyout grph simpliition n rigs omputtion r quit tiv: th stith g numr n ru y 90%, whil th onlit numr n ru y 9%. Th olumns st# n n# show th stith numr n th onlit numr in th inl sign. CPU(s) is omputtionl tim in sons. Compr with th Norml ILP, th Alrt ILP n hiv th sm rsults in roun 18% o th runtim. Not tht i no lrtiv thniqu is us, th runtim or ILP is unptl vn or smll iruits lik C4. From Tl II w n s tht oth ILP ormultions n hiv th optiml solutions, us o th sm onlit numr n stith numr. Son, w vriy th qulity n iiny o our pproximtion lgorithm s on smiinit progrmg. Tl II lso omprs th Alrt ILP n th SDP s lgorithm, whr SDP Bs nots th smiinit progrmg s lgorithm. Not tht ths two mthos shr th sm omposition grph, i.. oth th stith g numr n th onlit g numr in thir omposition grph r qul. As w n s, using SDP s mtho th runtim n urthr ru y 4% n th stith numr n ru y 7%. Th tro or this lrtion is th 9% mor onlits. 5.. Eiiny In orr to urthr vlut th slility o our SDP s mtho, w rt our itionl nhmrks (C1-C4) to tst two omposition lgorithms on vry ns lyouts. Tl III lists th omprison o th Sp-up ILP n th SDP s mtho on ths vry ns lyouts. As w n s, ompr with th Sp-up ILP, SDP s mtho n ru stith numr y 10% whil introus 5% mor onlits. Furthrmor, SDP s mtho n hiv 140 sp-up. Th rson or th rmtilly lrtion is tht: or low nsity lyout, th omposition prolm n ivi into mny su-prolms, n typilly h suprolm ontins no mor thn 0 nos. Whil or high nsity lyout, thr r mor nos in h su-prolm, whr SDP n muh str thn ILP. 6. CONCLUSION In this ppr, w propos gnrl intgr linr progrmg (ILP) ormultion or th TPL lyout omposition to simultnously imiz th onlits n stiths. To 7
8 TABLE II: Runtim n Prormn Comprisons Ciruit Comp# Norml ILP Alrt ILP SDP Bs SE# CE# st# n# CPU(s) SE# CE# st# n# CPU(s) st# n# CPU(s) C C C C C C C C C ?? > C ?? > S S ?? > S ?? > S ?? > S ?? > vg rtio TABLE III: Comprison on Vry Dns Lyouts Ciruit SE# CE# Alrt ILP SDP Bs st# n# CPU(s) st# n# CPU(s) C C C C vg rtio improv slility, w vlop thr lrtion thniqus without losing solution qulity: lyout grph simpliition, inpnnt omponnt omputtion n rigs omputtion. Furthrmor, w propos novl smiinit progrmg lgorithm to improv slility or vry ns lyouts. Exprimntl rsults show tht our mthos r vry tiv. Sin this is th irst systmti ttmpt on TPL lyout omposition or gnrl lyouts, w xpt to s lot o rsrhs s TPL my opt y inustry in th nr utur. REFERENCES [1] J.-S. Yng n D. Z. Pn, Ovrly wr intronnt n tig vrition moling or oul pttrning thnology, in IEEE/ACM Intrntionl Conrn on Computr-Ai Dsign (ICCAD), 008, pp [] Y. Boroovsky, Lithogrphy 009 ovrviw o opportunitis, in Smion Wst, 009. [] V. O. Anton, N. Ptr, H. Juy, G. Ronl, n N. Rort, Pttrn split ruls! siility stuy o rul s pith omposition or oul pttrning, in Pro. o SPIE, 007. [4] A. B. Khng, C.-H. Prk, X. Xu, n H. Yo, Lyout omposition or oul pttrning lithogrphy, in ACM/IEEE Intrntionl Conrn on Computr Ai Dsign (ICCAD), 008, pp [5] K. Yun, J.-S. Yng, n D. Pn, Doul pttrning lyout omposition or simultnous onlit n stith imiztion, in ACM Intrntionl Symposium on Physil Dsign (ISPD), 009, pp [6] Y. Xu n C. Chu, GREMA: grph rution s iint msk ssignmnt or oul pttrning thnology, in ACM/IEEE Intrntionl Conrn on Computr Ai Dsign (ICCAD), 009, pp [7] J.-S. Yng, K. Lu, M. Cho, K. Yun, n D. Pn, A nw grph-thorti, multi-ojtiv lyout omposition rmwork or oul pttrning lithogrphy, in IEEE/ACM Asi n South Pii Dsign Automtion Conrn (ASPDAC), 010. [8] Y. Xu n C. Chu, A mthing s omposr or oul pttrning lithogrphy, in ACM Intrntionl Symposium on Physil Dsign (ISPD), 010, pp [9] K. Yun n D. Pn, WISDOM: Wir spring nhn omposition o msks in oul pttrning lithogrphy, in IEEE/ACM Intrntionl Conrn on Computr-Ai Dsign (ICCAD), 010, pp. 8. [10] S.-Y. Chn n Y.-W. Chng, Ntiv-onlit-wr wir prturtion or oul pttrning thnology, in IEEE/ACM Intrntionl Conrn on Computr-Ai Dsign (ICCAD), 010, pp [11] C.-H. Hsu, Y.-W. Chng, n S. R. Nssi, Simultnous lyout migrtion n omposition or oul pttrning thnology, in IEEE/ACM Intrntionl Conrn on Computr-Ai Dsign (ICCAD), 009, pp [1] C. Cork, J.-C. Mr, n L. Brns, Comprison o tripl-pttrning omposition lgorithms using prioi tiling pttrns, in Pro. o SPIE, 008. [1] Y. Chn, P. Xu, L. Mio, Y. Chn, X. Xu, D. Mo, P. Blno, C. Bnhr, R. Hung, n C. S. Ngi, Sl-lign tripl pttrning or ontinuous i sling to hl-pith 15nm, in Pro. o SPIE, 011. [14] B. Mrki, H. D. Chn, Y. Chn, A. Wng, J. Ling, K. Spr, T. Mnrkr, X. Chn, P. Xu, P. Blnko, C. Ngi, C. Bnhr, n M. Nik, Innovtiv sl-lign tripl pttrning or 1x hl pith using singl spr position-spr th stp, in Pro. o SPIE, 011. [15] Q. Li, NP-ompltnss rsult or positiv lin-y-ill sp pross, in Pro. o SPIE, 010. [16] L. Vnnrgh n S. Boy, Smiinit progrmg, SIAM Rv., vol. 8, pp , Mrh [17] D. Krgr, R. Motwni, n M. Sun, Approximt grph oloring y smiinit progrmg, J. ACM, vol. 45, pp , Mrh [18] M. R. Gry, D. S. Johnson, n L. Stokmy, Som simplii NPomplt grph prolms, Thortil Computr Sin, vol. 1, pp. 7 67, [19] S. Khnn, N. Linil, n S. Sr, On th hrnss o pproximting th hromti numr, in Thory n Computing Systms, 199., Proings o th n Isrl Symposium on th, Jun. 199, pp [0] R. E. Trjn, A not on ining th rigs o grph, Inormtion Prossing Lttrs, vol., pp , [1] T. T. Cormn, C. E. Lisrson, n R. L. Rivst, Introution to lgorithms. Cmrig, MA, USA: MIT Prss, [] [Onlin]. Avill: [] [Onlin]. Avill: [4] B. Borhrs, CSDP, C lirry or smiinit progrmg, Optimiztion Mthos n Sotwr, vol. 11, pp. 61 6,
arxiv: v1 [cs.ar] 11 Feb 2014
Lyout Domposition or Tripl Pttrning Lithogrphy Bi Yu, Kun Yun, Boyng Zhng, Duo Ding, Dvi Z. Pn ECE Dpt. Univrsity o Txs t Austin, Austin, TX USA 7871 Cn Dsign Systms, In., Sn Jos, CA USA 9514 Emil: {i,
More informationCOMPLEXITY OF COUNTING PLANAR TILINGS BY TWO BARS
OMPLXITY O OUNTING PLNR TILINGS Y TWO RS KYL MYR strt. W show tht th prolm o trmining th numr o wys o tiling plnr igur with horizontl n vrtil r is #P-omplt. W uil o o th rsults o uquir, Nivt, Rmil, n Roson
More informationConstructive Geometric Constraint Solving
Construtiv Gomtri Constrint Solving Antoni Soto i Rir Dprtmnt Llngutgs i Sistms Inormàtis Univrsitt Politèni Ctluny Brlon, Sptmr 2002 CGCS p.1/37 Prliminris CGCS p.2/37 Gomtri onstrint prolm C 2 D L BC
More information1 Introduction to Modulo 7 Arithmetic
1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w
More information, each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management
nrl tr T is init st o on or mor nos suh tht thr is on sint no r, ll th root o T, n th rminin nos r prtition into n isjoint susts T, T,, T n, h o whih is tr, n whos roots r, r,, r n, rsptivly, r hilrn o
More informationCSC Design and Analysis of Algorithms. Example: Change-Making Problem
CSC 801- Dsign n Anlysis of Algorithms Ltur 11 Gry Thniqu Exmpl: Chng-Mking Prolm Givn unlimit mounts of oins of nomintions 1 > > m, giv hng for mount n with th lst numr of oins Exmpl: 1 = 25, 2 =10, =
More informationPaths. Connectivity. Euler and Hamilton Paths. Planar graphs.
Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,
More informationOutline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)
More informationPlanar Upward Drawings
C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th
More informationAn undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V
Unirt Grphs An unirt grph G = (V, E) V st o vrtis E st o unorr gs (v,w) whr v, w in V USE: to mol symmtri rltionships twn ntitis vrtis v n w r jnt i thr is n g (v,w) [or (w,v)] th g (v,w) is inint upon
More informationLayout Decomposition for Quadruple Patterning Lithography and Beyond
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
More informationThe University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008
Th Univrsity o Syny MATH2969/2069 Grph Thory Tutoril 5 (Wk 12) Solutions 2008 1. (i) Lt G th isonnt plnr grph shown. Drw its ul G, n th ul o th ul (G ). (ii) Show tht i G is isonnt plnr grph, thn G is
More informationCycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!
Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik
More informationPresent state Next state Q + M N
Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I
More information(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely
. DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,
More informationGraph Isomorphism. Graphs - II. Cayley s Formula. Planar Graphs. Outline. Is K 5 planar? The number of labeled trees on n nodes is n n-2
Grt Thortil Is In Computr Sin Vitor Amhik CS 15-251 Ltur 9 Grphs - II Crngi Mllon Univrsity Grph Isomorphism finition. Two simpl grphs G n H r isomorphi G H if thr is vrtx ijtion V H ->V G tht prsrvs jny
More information12/3/12. Outline. Part 10. Graphs. Circuits. Euler paths/circuits. Euler s bridge problem (Bridges of Konigsberg Problem)
12/3/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 Ciruits Cyl 2 Eulr
More information5/9/13. Part 10. Graphs. Outline. Circuits. Introduction Terminology Implementing Graphs
Prt 10. Grphs CS 200 Algorithms n Dt Struturs 1 Introution Trminology Implmnting Grphs Outlin Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 2 Ciruits Cyl A spil yl
More informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous
More informationWhy the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.
Why th Juntion Tr lgorithm? Th Juntion Tr lgorithm hris Willims 1 Shool of Informtis, Univrsity of Einurgh Otor 2009 Th JT is gnrl-purpos lgorithm for omputing (onitionl) mrginls on grphs. It os this y
More informationOutline. Computer Science 331. Computation of Min-Cost Spanning Trees. Costs of Spanning Trees in Weighted Graphs
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim s Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #34 Introution Min-Cost Spnnin Trs 3 Gnrl Constrution 4 5 Trmintion n Eiiny 6 Aitionl
More informationCSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018
CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs
More informationExam 1 Solution. CS 542 Advanced Data Structures and Algorithms 2/14/2013
CS Avn Dt Struturs n Algorithms Exm Solution Jon Turnr //. ( points) Suppos you r givn grph G=(V,E) with g wights w() n minimum spnning tr T o G. Now, suppos nw g {u,v} is to G. Dsri (in wors) mtho or
More informationb. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?
MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth
More informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
s s of s Computr Sin & Enginring 423/823 Dsign n Anlysis of Ltur 03 (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) s of s s r strt t typs tht r pplil to numrous prolms Cn ptur ntitis, rltionships twn
More informationOutline. Binary Tree
Outlin Similrity Srh Th Binry Brnh Distn Nikolus Austn nikolus.ustn@s..t Dpt. o Computr Sins Univrsity o Slzur http://rsrh.uni-slzur.t 1 Binry Brnh Distn Binry Rprsnttion o Tr Binry Brnhs Lowr Boun or
More informationSimilarity Search. The Binary Branch Distance. Nikolaus Augsten.
Similrity Srh Th Binry Brnh Distn Nikolus Augstn nikolus.ugstn@sg..t Dpt. of Computr Sins Univrsity of Slzurg http://rsrh.uni-slzurg.t Vrsion Jnury 11, 2017 Wintrsmstr 2016/2017 Augstn (Univ. Slzurg) Similrity
More information0.1. Exercise 1: the distances between four points in a graph
Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 pg 1 Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 u: W, 3 My 2017, in lss or y mil (grinr@umn.u) or lss S th wsit or rlvnt mtril. Rsults provn in th nots, or in
More informationAlgorithmic and NP-Completeness Aspects of a Total Lict Domination Number of a Graph
Intrntionl J.Mth. Comin. Vol.1(2014), 80-86 Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph Girish.V.R. (PES Institut of Thnology(South Cmpus), Bnglor, Krntk Stt, Ini) P.Ush (Dprtmnt
More informationCSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp
CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140
More informationA High-Performance Triple Patterning Layout Decomposer with Balanced Density
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
More informationModule graph.py. 1 Introduction. 2 Graph basics. 3 Module graph.py. 3.1 Objects. CS 231 Naomi Nishimura
Moul grph.py CS 231 Nomi Nishimur 1 Introution Just lik th Python list n th Python itionry provi wys of storing, ssing, n moifying t, grph n viw s wy of storing, ssing, n moifying t. Bus Python os not
More informationECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS
C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h
More informationGarnir Polynomial and their Properties
Univrsity of Cliforni, Dvis Dprtmnt of Mthmtis Grnir Polynomil n thir Proprtis Author: Yu Wng Suprvisor: Prof. Gorsky Eugny My 8, 07 Grnir Polynomil n thir Proprtis Yu Wng mil: uywng@uvis.u. In this ppr,
More informationCS 241 Analysis of Algorithms
CS 241 Anlysis o Algorithms Prossor Eri Aron Ltur T Th 9:00m Ltur Mting Lotion: OLB 205 Businss HW6 u lry HW7 out tr Thnksgiving Ring: Ch. 22.1-22.3 1 Grphs (S S. B.4) Grphs ommonly rprsnt onntions mong
More information12. Traffic engineering
lt2.ppt S-38. Introution to Tltrffi Thory Spring 200 2 Topology Pths A tlommunition ntwork onsists of nos n links Lt N not th st of nos in with n Lt J not th st of nos in with j N = {,,,,} J = {,2,3,,2}
More informationGraphs. Graphs. Graphs: Basic Terminology. Directed Graphs. Dr Papalaskari 1
CSC 00 Disrt Struturs : Introuon to Grph Thory Grphs Grphs CSC 00 Disrt Struturs Villnov Univrsity Grphs r isrt struturs onsisng o vrs n gs tht onnt ths vrs. Grphs n us to mol: omputr systms/ntworks mthml
More informationUsing the Printable Sticker Function. Using the Edit Screen. Computer. Tablet. ScanNCutCanvas
SnNCutCnvs Using th Printl Stikr Funtion On-o--kin stikrs n sily rt y using your inkjt printr n th Dirt Cut untion o th SnNCut mhin. For inormtion on si oprtions o th SnNCutCnvs, rr to th Hlp. To viw th
More informationDesigning A Concrete Arch Bridge
This is th mous Shwnh ri in Switzrln, sin y Rort Millrt in 1933. It spns 37.4 mtrs (122 t) n ws sin usin th sm rphil mths tht will monstrt in this lsson. To pro with this lsson, lik on th Nxt utton hr
More informationEE1000 Project 4 Digital Volt Meter
Ovrviw EE1000 Projt 4 Diitl Volt Mtr In this projt, w mk vi tht n msur volts in th rn o 0 to 4 Volts with on iit o ury. Th input is n nlo volt n th output is sinl 7-smnt iit tht tlls us wht tht input s
More informationA 4-state solution to the Firing Squad Synchronization Problem based on hybrid rule 60 and 102 cellular automata
A 4-stt solution to th Firing Squ Synhroniztion Prolm s on hyri rul 60 n 102 llulr utomt LI Ning 1, LIANG Shi-li 1*, CUI Shung 1, XU Mi-ling 1, ZHANG Ling 2 (1. Dprtmnt o Physis, Northst Norml Univrsity,
More informationMath 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.
Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right
More informationSeven-Segment Display Driver
7-Smnt Disply Drivr, Ron s in 7-Smnt Disply Drivr, Ron s in Prolm 62. 00 0 0 00 0000 000 00 000 0 000 00 0 00 00 0 0 0 000 00 0 00 BCD Diits in inry Dsin Drivr Loi 4 inputs, 7 outputs 7 mps, h with 6 on
More informationCS 461, Lecture 17. Today s Outline. Example Run
Prim s Algorithm CS 461, Ltur 17 Jr Si Univrsity o Nw Mxio In Prim s lgorithm, th st A mintin y th lgorithm orms singl tr. Th tr strts rom n ritrry root vrtx n grows until it spns ll th vrtis in V At h
More informationProblem solving by search
Prolm solving y srh Tomáš voo Dprtmnt o Cyrntis, Vision or Roots n Autonomous ystms Mrh 5, 208 / 3 Outlin rh prolm. tt sp grphs. rh trs. trtgis, whih tr rnhs to hoos? trtgy/algorithm proprtis? Progrmming
More informationOutline. Circuits. Euler paths/circuits 4/25/12. Part 10. Graphs. Euler s bridge problem (Bridges of Konigsberg Problem)
4/25/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 2 Eulr s rig prolm
More informationS i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA.
S i m p l i y i n g A l g r SIMPLIFYING ALGEBRA www.mthltis.o.nz Simpliying SIMPLIFYING Algr ALGEBRA Algr is mthmtis with mor thn just numrs. Numrs hv ix vlu, ut lgr introus vrils whos vlus n hng. Ths
More informationCS September 2018
Loil los Distriut Systms 06. Loil los Assin squn numrs to msss All ooprtin prosss n r on orr o vnts vs. physil los: rport tim o y Assum no ntrl tim sour Eh systm mintins its own lol lo No totl orrin o
More informationECE 407 Computer Aided Design for Electronic Systems. Circuit Modeling and Basic Graph Concepts/Algorithms. Instructor: Maria K. Michael.
0 Computr i Dsign or Eltroni Systms Ciruit Moling n si Grph Conptslgorithms Instrutor: Mri K. Mihl MKM - Ovrviw hviorl vs. Struturl mols Extrnl vs. Intrnl rprsnttions Funtionl moling t Logi lvl Struturl
More informationGraphs. CSC 1300 Discrete Structures Villanova University. Villanova CSC Dr Papalaskari
Grphs CSC 1300 Disrt Struturs Villnov Univrsity Grphs Grphs r isrt struturs onsis?ng of vr?s n gs tht onnt ths vr?s. Grphs n us to mol: omputr systms/ntworks mthm?l rl?ons logi iruit lyout jos/prosss f
More information# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.
How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0
More informationAnnouncements. Not graphs. These are Graphs. Applications of Graphs. Graph Definitions. Graphs & Graph Algorithms. A6 released today: Risk
Grphs & Grph Algorithms Ltur CS Spring 6 Announmnts A6 rls toy: Risk Strt signing with your prtnr sp Prlim usy Not grphs hs r Grphs K 5 K, =...not th kin w mn, nywy Applitions o Grphs Communition ntworks
More informationSolutions to Homework 5
Solutions to Homwork 5 Pro. Silvia Frnánz Disrt Mathmatis Math 53A, Fall 2008. [3.4 #] (a) Thr ar x olor hois or vrtx an x or ah o th othr thr vrtis. So th hromati polynomial is P (G, x) =x (x ) 3. ()
More informationSection 10.4 Connectivity (up to paths and isomorphism, not including)
Toy w will isuss two stions: Stion 10.3 Rprsnting Grphs n Grph Isomorphism Stion 10.4 Conntivity (up to pths n isomorphism, not inluing) 1 10.3 Rprsnting Grphs n Grph Isomorphism Whn w r working on n lgorithm
More informationCS200: Graphs. Graphs. Directed Graphs. Graphs/Networks Around Us. What can this represent? Sometimes we want to represent directionality:
CS2: Grphs Prihr Ch. 4 Rosn Ch. Grphs A olltion of nos n gs Wht n this rprsnt? n A omputr ntwork n Astrtion of mp n Soil ntwork CS2 - Hsh Tls 2 Dirt Grphs Grphs/Ntworks Aroun Us A olltion of nos n irt
More informationQUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt o Computr n Inormtion Sins CSCI 2710 (Trno) Disrt Struturs TEST or Sprin Smstr, 2005 R this or strtin! This tst is los ook
More informationNefertiti. Echoes of. Regal components evoke visions of the past MULTIPLE STITCHES. designed by Helena Tang-Lim
MULTIPLE STITCHES Nrtiti Ehos o Rgl omponnts vok visions o th pst sign y Hln Tng-Lim Us vrity o stiths to rt this rgl yt wrl sign. Prt sping llows squr s to mk roun omponnts tht rp utiully. FCT-SC-030617-07
More informationOpenMx Matrices and Operators
OpnMx Mtris n Oprtors Sr Mln Mtris: t uilin loks Mny typs? Dnots r lmnt mxmtrix( typ= Zro", nrow=, nol=, nm="" ) mxmtrix( typ= Unit", nrow=, nol=, nm="" ) mxmtrix( typ= Int", nrow=, nol=, nm="" ) mxmtrix(
More informationlearning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms
rp loritms lrnin ojtivs loritms your sotwr systm sotwr rwr lrn wt rps r in mtmtil trms lrn ow to rprsnt rps in omputrs lrn out typil rp loritms wy rps? intuitivly, rp is orm y vrtis n s twn vrtis rps r
More informationCSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review
rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht
More informationNP-Completeness. CS3230 (Algorithm) Traveling Salesperson Problem. What s the Big Deal? Given a Problem. What s the Big Deal? What s the Big Deal?
NP-Compltnss 1. Polynomil tim lgorithm 2. Polynomil tim rution 3.P vs NP 4.NP-ompltnss (som slis y P.T. Um Univrsity o Txs t Dlls r us) Trvling Slsprson Prolm Fin minimum lngth tour tht visits h ity on
More informationSolutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1
Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):
More informationRegister Allocation. How to assign variables to finitely many registers? What to do when it can t be done? How to do so efficiently?
Rgistr Allotion Rgistr Allotion How to ssign vrils to initly mny rgistrs? Wht to o whn it n t on? How to o so iintly? Mony, Jun 3, 13 Mmory Wll Disprity twn CPU sp n mmory ss sp improvmnt Mony, Jun 3,
More informationCS61B Lecture #33. Administrivia: Autograder will run this evening. Today s Readings: Graph Structures: DSIJ, Chapter 12
Aministrivi: CS61B Ltur #33 Autogrr will run this vning. Toy s Rings: Grph Struturs: DSIJ, Chptr 12 Lst moifi: W Nov 8 00:39:28 2017 CS61B: Ltur #33 1 Why Grphs? For xprssing non-hirrhilly rlt itms Exmpls:
More information(a) v 1. v a. v i. v s. (b)
Outlin RETIMING Struturl optimiztion mthods. Gionni D Mihli Stnford Unirsity Rtiming. { Modling. { Rtiming for minimum dly. { Rtiming for minimum r. Synhronous Logi Ntwork Synhronous Logi Ntwork Synhronous
More informationNew challenges on Independent Gate FinFET Transistor Network Generation
Nw hllngs on Inpnnt Gt FinFET Trnsistor Ntwork Gnrtion Viniius N. Possni, Anré I. Ris, Rnto P. Ris, Flip S. Mrqus, Lomr S. Ros Junior Thnology Dvlopmnt Cntr, Frl Univrsity o Plots, Plots, Brzil Institut
More informationQUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt of Computr n Informtion Sins CSCI 710 (Trnoff) Disrt Struturs TEST for Fll Smstr, 00 R this for strtin! This tst is los ook
More informationRegister Allocation. Register Allocation. Principle Phases. Principle Phases. Example: Build. Spills 11/14/2012
Rgistr Allotion W now r l to o rgistr llotion on our intrfrn grph. W wnt to l with two typs of onstrints: 1. Two vlus r liv t ovrlpping points (intrfrn grph) 2. A vlu must or must not in prtiulr rhitturl
More informationMAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017
MAT3707/201/1/2017 Tutoril lttr 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Smstr 1 Dprtmnt o Mtmtil Sins SOLUTIONS TO ASSIGNMENT 01 BARCODE Din tomorrow. univrsity o sout ri SOLUTIONS TO ASSIGNMENT
More informationTrees as operads. Lecture A formalism of trees
Ltur 2 rs s oprs In this ltur, w introu onvnint tgoris o trs tht will us or th inition o nroil sts. hs tgoris r gnrliztions o th simpliil tgory us to in simpliil sts. First w onsir th s o plnr trs n thn
More informationNumbering Boundary Nodes
Numring Bounry Nos Lh MBri Empori Stt Univrsity August 10, 2001 1 Introution Th purpos of this ppr is to xplor how numring ltril rsistor ntworks ffts thir rspons mtrix, Λ. Morovr, wht n lrn from Λ out
More informationGraph Contraction and Connectivity
Chptr 17 Grph Contrtion n Conntivity So r w hv mostly ovr thniqus or solving prolms on grphs tht wr vlop in th ontxt o squntil lgorithms. Som o thm r sy to prllliz whil othrs r not. For xmpl, w sw tht
More informationCOMP108 Algorithmic Foundations
Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht
More informationA Simple Code Generator. Code generation Algorithm. Register and Address Descriptors. Example 3/31/2008. Code Generation
A Simpl Co Gnrtor Co Gnrtion Chptr 8 II Gnrt o for singl si lok How to us rgistrs? In most mhin rhitturs, som or ll of th oprnsmust in rgistrs Rgistrs mk goo tmporris Hol vlus tht r omput in on si lok
More informationWeighted graphs -- reminder. Data Structures LECTURE 15. Shortest paths algorithms. Example: weighted graph. Two basic properties of shortest paths
Dt Strutur LECTURE Shortt pth lgorithm Proprti of hortt pth Bllmn-For lgorithm Dijktr lgorithm Chptr in th txtook (pp ). Wight grph -- rminr A wight grph i grph in whih g hv wight (ot) w(v i, v j ) >.
More informationPage 1. Question 19.1b Electric Charge II Question 19.2a Conductors I. ConcepTest Clicker Questions Chapter 19. Physics, 4 th Edition James S.
ConTst Clikr ustions Chtr 19 Physis, 4 th Eition Jms S. Wlkr ustion 19.1 Two hrg blls r rlling h othr s thy hng from th iling. Wht n you sy bout thir hrgs? Eltri Chrg I on is ositiv, th othr is ngtiv both
More information16.unified Introduction to Computers and Programming. SOLUTIONS to Examination 4/30/04 9:05am - 10:00am
16.unii Introution to Computrs n Prormmin SOLUTIONS to Exmintion /30/0 9:05m - 10:00m Pro. I. Kristin Lunqvist Sprin 00 Grin Stion: Qustion 1 (5) Qustion (15) Qustion 3 (10) Qustion (35) Qustion 5 (10)
More informationProperties of Hexagonal Tile local and XYZ-local Series
1 Proprtis o Hxgonl Til lol n XYZ-lol Sris Jy Arhm 1, Anith P. 2, Drsnmik K. S. 3 1 Dprtmnt o Bsi Sin n Humnitis, Rjgiri Shool o Enginring n, Thnology, Kkkn, Ernkulm, Krl, Ini. jyjos1977@gmil.om 2 Dprtmnt
More informationA PROPOSAL OF FE MODELING OF UNIDIRECTIONAL COMPOSITE CONSIDERING UNCERTAIN MICRO STRUCTURE
18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS A PROPOSAL OF FE MODELING OF UNIDIRECTIONAL COMPOSITE CONSIDERING UNCERTAIN MICRO STRUCTURE Y.Fujit 1*, T. Kurshii 1, H.Ymtsu 1, M. Zo 2 1 Dpt. o Mngmnt
More informationIntegration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals
Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion
More informationAnnouncements. These are Graphs. This is not a Graph. Graph Definitions. Applications of Graphs. Graphs & Graph Algorithms
Grphs & Grph Algorithms Ltur CS Fll 5 Announmnts Upoming tlk h Mny Crrs o Computr Sintist Or how Computr Sin gr mpowrs you to o muh mor thn o Dn Huttnlohr, Prossor in th Dprtmnt o Computr Sin n Johnson
More informationGREEDY TECHNIQUE. Greedy method vs. Dynamic programming method:
Dinition: GREEDY TECHNIQUE Gry thniqu is gnrl lgorithm sign strtgy, uilt on ollowing lmnts: onigurtions: irnt hois, vlus to in ojtiv untion: som onigurtions to ithr mximiz or minimiz Th mtho: Applil to
More informationN=4 L=4. Our first non-linear data structure! A graph G consists of two sets G = {V, E} A set of V vertices, or nodes f
lulu jwtt pnlton sin towr ounrs hpl lpp lu Our irst non-linr t strutur! rph G onsists o two sts G = {V, E} st o V vrtis, or nos st o E s, rltionships twn nos surph G onsists o sust o th vrtis n s o G jnt
More informationGreedy Algorithms, Activity Selection, Minimum Spanning Trees Scribes: Logan Short (2015), Virginia Date: May 18, 2016
Ltur 4 Gry Algorithms, Ativity Sltion, Minimum Spnning Trs Sris: Logn Short (5), Virgini Dt: My, Gry Algorithms Suppos w wnt to solv prolm, n w r l to om up with som rursiv ormultion o th prolm tht woul
More informationarxiv: v1 [cs.ds] 20 Feb 2008
Symposium on Thortil Aspts of Computr Sin 2008 (Borux), pp. 361-372 www.sts-onf.org rxiv:0802.2867v1 [s.ds] 20 F 2008 FIXED PARAMETER POLYNOMIAL TIME ALGORITHMS FOR MAXIMUM AGREEMENT AND COMPATIBLE SUPERTREES
More informationDETAIL B DETAIL A 7 8 APPLY PRODUCT ID LABEL SB838XXXX ADJ FOUR POST RACK SQUARE HOLE RAIL B REVISION
RVISION RV SRIPTION Y T HNG NO NOT OR PROUT LL JJH // LR TIL PPLY PROUT I LL TIL INSI UPPR ROSS MMR ON PR RK IS J OUR POST RK SQUR HOL RIL IS MN MS G NUT, PNL RNG 99 PPLY PROUT I LL INSI UPPR ROSS MMR
More informationEXAMPLE 87.5" APPROVAL SHEET APPROVED BY /150HP DUAL VFD CONTROL ASSEMBLY CUSTOMER NAME: CAL POLY SLO FINISH: F 20
XMPL XMPL RVISIONS ZON RV. SRIPTION T PPROV 0.00 THIS IS N PPROVL RWING OR YOUR ORR. OR MNUTURING N GIN, THIS RWING MUST SIGN N RTURN TO MOTION INUSTRIS. NY HNGS M TO THIS RWING, TR MNUTURING HS GUN WILL
More informationA Low Noise and Reliable CMOS I/O Buffer for Mixed Low Voltage Applications
Proings of th 6th WSEAS Intrntionl Confrn on Miroltronis, Nnoltronis, Optoltronis, Istnul, Turky, My 27-29, 27 32 A Low Nois n Rlil CMOS I/O Buffr for Mix Low Voltg Applitions HWANG-CHERNG CHOW n YOU-GANG
More informationQuartets and unrooted level-k networks
Phylogntis Workshop, Is Nwton Institut or Mthmtil Sins Cmrig 21/06/2011 Qurtts n unroot lvl-k ntworks Philipp Gmtt Outlin Astrt n xpliit phylognti ntworks Lvl-k ntworks Unroot lvl-1 ntworks n irulr split
More information1. Determine whether or not the following binary relations are equivalence relations. Be sure to justify your answers.
Mth 0 Exm - Prti Prolm Solutions. Dtrmin whthr or not th ollowing inry rltions r quivln rltions. B sur to justiy your nswrs. () {(0,0),(0,),(0,),(,),(,),(,),(,),(,0),(,),(,),(,0),(,),(.)} on th st A =
More informationCSE303 - Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata
CSE303 - Introduction to th Thory of Computing Smpl Solutions for Exrciss on Finit Automt Exrcis 2.1.1 A dtrministic finit utomton M ccpts th mpty string (i.., L(M)) if nd only if its initil stt is finl
More informationMULTIPLE-LEVEL LOGIC OPTIMIZATION II
MUTIPE-EVE OGIC OPTIMIZATION II Booln mthos Eploit Booln proprtis Giovnni D Mihli Don t r onitions Stnfor Univrsit Minimition of th lol funtions Slowr lgorithms, ttr qulit rsults Etrnl on t r onitions
More informationComputational Biology, Phylogenetic Trees. Consensus methods
Computtionl Biology, Phylognti Trs Consnsus mthos Asgr Bruun & Bo Simonsn Th 16th of Jnury 2008 Dprtmnt of Computr Sin Th univrsity of Copnhgn 0 Motivtion Givn olltion of Trs Τ = { T 0,..., T n } W wnt
More informationJonathan Turner Exam 2-10/28/03
CS Algorihm n Progrm Prolm Exm Soluion S Soluion Jonhn Turnr Exm //. ( poin) In h Fioni hp ruur, u wn vrx u n i prn v u ing u v i v h lry lo hil in i l m hil o om ohr vrx. Suppo w hng hi, o h ing u i prorm
More informationTOPIC 5: INTEGRATION
TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function
More informationSteinberg s Conjecture is false
Stinrg s Conjtur is als arxiv:1604.05108v2 [math.co] 19 Apr 2016 Vinnt Cohn-Aa Mihal Hig Danil Král Zhntao Li Estan Salgao Astrat Stinrg onjtur in 1976 that vry planar graph with no yls o lngth our or
More informationModule 2 Motion Instructions
Moul 2 Motion Instrutions CAUTION: Bor you strt this xprimnt, unrstn tht you r xpt to ollow irtions EXPLICITLY! Tk your tim n r th irtions or h stp n or h prt o th xprimnt. You will rquir to ntr t in prtiulr
More informationAnalysis for Balloon Modeling Structure based on Graph Theory
Anlysis for lloon Moling Strutur bs on Grph Thory Abstrt Mshiro Ur* Msshi Ym** Mmoru no** Shiny Miyzki** Tkmi Ysu* *Grut Shool of Informtion Sin, Ngoy Univrsity **Shool of Informtion Sin n Thnology, hukyo
More informationDUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski
Dut with Dimons Brlt DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Lsli Roglski Photo y Anrw Wirth Supruo DUETS TM from BSmith rt olor shifting fft tht mks your work tk on lif of its own s you mov! This
More informationClustering for Processing Rate Optimization
Clustring for Prossing Rt Optimiztion Chun Lin, Ji Wng, n Hi Zhou Eltril n Computr Enginring Northwstrn Univrsity Evnston, IL 60208 Astrt Clustring (or prtitioning) is ruil stp twn logi synthsis n physil
More information