35 th Design Automation Conference Copyright 1998 ACM
|
|
- Domenic Clarke
- 5 years ago
- Views:
Transcription
1 Eint Booln ivision n Sustitution Shih-Chih Chng vi Ihsin Chng pt. o CS n IE Ntionl Chung Chng Univrsit Ultim Intronnt Th. Chi-Yi, TAIWAN, ROC Sunnvl, CA Astrt Booln ivision, n hn Booln sustitution, prous ttr rsult thn lgri ivision n sustitution. Howvr, u to th lk o n int Booln ivision lgorithm, Booln sustitution hs rrl n us. W prsnt n int Booln ivision n sustitution lgorithm. Our thniqu is s on th philosoph o runn ition n rmovl. B ing multipl wirs/gts in spiliz w, w tilor th philosoph onto th Booln ivision n sustitution prolm. rom th viwpoint o tritionl ivision/sustitution, our lgorithm n prorm sustitution not onl in sum-o-prout orm or ut lso in prout-o-sum orm. Our lgorithm n lso nturll tk ll tps o on't rs into onsirtion. As r s sustitution is onrn, w lso isuss th s whr w r llow to ompos not onl th ivin ut lso th ivisor. Exprimnts r prsnt n th rsult is promising. 1 Introution In multi-lvl logi snthsis, n importnt stp in minimizing th r o iruit is sustitution [4] (or rsustitution [10]). Sustitution rrs to th stp whr untion is simpli in omplxit using n itionl input tht ws not prviousl in th untion's immit nins. Sustitution n ru th omplxit o untion us prt o th untion is rpl th itionl input tht rprsnts som xisting untion in th iruit. Th xprssion o th xisting untion is thror shr n rus. To prorm sustitution, th onpt o ivision pls mjor rol. Givn two Booln untions n, i w n xprss in th orm = Q + R, whr n + rsptivl rprsnt th Booln AN n Booln OR oprtors, thn w s tht n ivi n tht untions Q n R r, rsptivl, th quotint n th rminr. Sustitution n lgri or Booln, pning on i th unrlining ivision is lgri or Booln. In lgri ivision [4], logi xprssions r trt s lgri polnomils, with som rstritions pl on th mnipultions o th polnomils. In prtiulr, th prout o two untions G is lgri onl i no vril pprs in oth n G. As onsqun o th rstrition, rtin x-98/0006/$ th sign Automtion Conrn Copright 1998 ACM Booln intitis suh s xx = 0 n xx = x o not xist. As n xmpl, givn = + + n ivisor = +, through lgri ivision w otin = ( + )+. Through Booln ivision, whih n xploit ll th proprtis in Booln lgr [1], w otin = ( + )( + ). Assuming no with untion + xists in th iruit, with lgri sustitution w thn hv = +, whil with Booln sustitution w hv=(+ ). In this xmpl, untion hs 5 litrls 1 or sustitution. Algri sustitution rus th numr o litrls to 4, whil Booln sustitution rus it to 3. Booln ivision, n hn Booln sustitution, in thor prous ttr rsults. Howvr, thr os not xist gnrl n int Booln ivision lgorithm. In trms o th ov xmpl, this mns tht th st rsult o ruing to 3 litrls is vr iult to hiv. In this ppr, w rst prsnt nw thniqu to prorm Booln ivision. Our thniqu is s on th onpt o runn ition n rmovl (RAR) isuss in [2][3][5][8]. Th si philosoph o th RAR thniqu is to rst som runn n thn rmov othr runnis lswhr, with th gol tht th rmov ons ru th iruit siz mor thn th on. With x stup tht is spill ongur, w tilor th RAR philosoph onto th Booln ivision prolm. Unlik tritionl RAR thniqus, whih rquir runn hking on th potntil wir to, our lgorithm is tilor in w tht w know priori tht our intrst potntil wir is runnt. Also, lthough quit tiv on ing on runn n thn rmoving othr runnis, th tritionl RAR thniqus hv littl suss on tring to multipl wirs/gts. In our lgorithm, th tritionl RAR philosoph istilor to multipl wirs/gts in spi w prtiulrl or th Booln ivision prolm. As r s sustitution is onrn, knowing how to prorm ivision is onl th rst stp. Th son stp is to hoos potntil ivisors. Tritionll, sustitution on untion is on going through th xisting nos in th iruit n trting h o thm 1 In tor orm [4], s oppos to sum-o-prout orm. AC98-06/98 Sn rniso, CA USA
2 s potntil ivisor o. ivision is tri on h potntil ivisor n sustitution is rri out whn th tril is vorl. Sin it is up to th unrlining ivision lgorithm to onlu i ivisor is goo or not, th lgorithm m miss som \goo" ivisors. In th xmpl mntion rlir, lt us s th no with untion + os not xist n, inst, no with untion = + + x xists. Sin untion os not pn on vril x, tritionl ivision lgorithm woul quikl onlu tht th quotint o untion ivi is 0, n thror no sustitution woul our. Howvr, i w slightl hng th iruit strutur omposing + + x to two nos n 1 = + n n 2 = n 1 +x, untion n thn sustitut with no n 1. W will us th trm si ivision to rr to th snrio whr th givn ivisor is not llow to ompos, n th trm xtn ivision or th snrio whr th ivisor is llow to ompos, rtinl with som purpos in min. In th ov xmpl whr untion = ++ is ivi = ++x, wwoul s tht unr si ivision th quotintis0. or th sm n w woul lso s tht unr xtn ivision th suxprssion + n xtrt out s nw ivisor, n with th nw ivisor + th quotint is+. rom this viwpoint, ll th tritionl ivision lgorithms prorm onl si ivision, whil our lgorithm prsnt in this ppr prorms xtn ivision. Tritionl sustitution pprohs oprt on h no's intrnl sum-o-prout t strutur, n hn n onl prorm sustitution/ivision in th sum-o-prout orm. In ontrst, our lgorithm oprts on iruit strutur irtl. Givn n initil iruit, th rst stp o our lgorithm is to ompos h no's intrnl sum-o-prout orm into two-lvl AN n OR gts. Th iruit thn, in gnrl, hs lvl o AN gts, ollow lvl o OR gts, n so on. As rsult, in ition to th tritionl sum-o-prout tp o sustitution, our lgorithm n lso prorm sustitution in th vor o prout-o-sum orm. In othr wors, in two-lvl orm, whthr th ivin/ivisor r unh o ANs ollow n OR, or unh o ORs ollow n AN r ompltl smmtri to us. or xmpl, lt = ( + )( + )( + ) n = ()() xisting nos. With our lgorithm w n quikl sustitut into n otin = +, i.., = +. Prorming sustitution in suh mnnr is ompltl not possil in th tritionl pprohs us o th strong tthmnt to th unrlining sum-o-prout xprssion, whil in our thniqu prorming sustitution through sumo-prout orm or prout-o-sum orm r sill th sm. 2 Runn ition n rmovl g1 g2 g3 g4 g5 g6 g7 () g8 g9 o1 o2 Th most rlt work to our Booln sustitution lgorithm is th thniqu o runn ition n rmovl (RAR). Hr w provi til rviw. In [2][3][5][8], th thniqu o RAR is propos n ppli to gnrl multi-lvl logi optimiztion. Th g1 g2 g3 g5 () igur 1: Th RAR thniqu si philosoph in RAR is to som runn rst n thn tr to rmov othr runnis lswhr, with th gol tht th rmov ons ru th iruit siz mor thn th on. W rviw th thniqu with n xmpl iruit. ig. 1(), without th ott wir, shows n irrunnt iruit. Th ott wir g5!g9 is runnt wir, i.., ing th wir os not hng th iruit's untionlit. Howvr, on this wir is, th two thik wirs, g1!g4 n g6!g7, om runnt. In this s, w n rmov ths two runnt wirs without hnging th iruit's untionlit. Atr rmoving ths two wirs, w thn hv th iruit shown in ig. 1(), whih is smllr in siz. In gnrl, th RAR thniqu rst is, s on som ost untion, som xisting irrunnt wir tht is th trgt to rmov. Thn th thniqu srhs or som non-xisting wir, somtims ll nit onntion, tht on n rmov th trgt wir. inll th thniqu hks i th nit onntion is runnt, i.., i ing th non-xisting wir prsrvs th iruit's untionlit. Onl whn th nit onntion is vri s runnt, w n thn th onntion n urthr rmov th trgt wir. Not tht most o th RAR thniqus onl tr to inrmntll on wir t tim. u to high srh sp, orts tht tr to multipl wirs/gts n rmov vn mor wirs/gts hv onl littl suss (.g.: [2]). 3 Bsi ivision Givn untion n ivisor, w us th trm si ivision to rr to th snrio whr th ivisor is not llow to ompos, n us th trm xtn ivision to rr to th snrio whr th ivisor n rl ompos, with som optimiztion gol in min. In this stion w ous on si ivision. 3.1 SOS n POS o untion W rst n som nitions. A prout trm, or u, is st o litrls AN' togthr. A sum trm is st o litrls OR' togthr. A untion 1 ontins untion 2 i th on-st o 1 ontins th on-st o 2. As n xmpl, untion (u) ontins untion (u) ; untion (sum trm) + ontins untion (sum trm). urthrmor, w n SOS n POS o untion s ollows: SOS: Givn untion in two-lvl sum-oprout orm, w s untion G, lso in sumo-prout orm, is sum-o-suprout, or SOS, g8 g9 o1 o2
3 o i vr u in is ontin t lst on u in G. POS: Givn untion in two-lvl prout-osum orm, w s untion G, lso in prouto-sum orm, is prout-o-susum, or POS, o i vr sum trm in ontins t lst on sum trm in G. or xmpl, = + is SOS o = ++ us vr u in is ontin ithr u or u in. or nothr xmpl, 0 = + + x is lso SOS o th ov, sin ing mor us to os not hng th originl ontinmnt rltionship in. On th othr hn, untion E = + is not SOS o, sin u is not ontin in n u in untion E. On th POS si, or xmpl, =()() ispos o =(+)( + )( + ) us vr sum trm in ontins ithr sum trm or sum trm in. or nothr xmpl, untion 0 =()()(x + ) is lso POS o th ov, sin ing mor sum trms to os not hng th originl ontinmnt rltionship in. On th othr hn, untion E = ()(+) is not POS o, sin sum trm + os not ontin n sum trm in untion E. Th onpts o SOS n POS pl ntrl rol in our lgorithm, n w now look t som o thir simpl proprtis. Lmm 1 Lt untion G SOS o untion. Thn = G. Lmm 2 Lt untion G POS o untion. Thn = + G. Ths two lmms stlish th groun whr w n tilor th thniqu o runn ition n rmovl (RAR) onto our sustitution prolm. To illustrt th onpt, w tk th xmpl o = + + n = + rom Stion 1. Sin is SOS o, Lmm 1, th nw untion nw = ( + )( + + ) must quivlnt to th originl untion. rom th RAR viwpoint, w hv sussull \" runn into th iruit. ousing on th originl prt insi nw,w thn tr to rmov s mn runnis s possil, n n quikl rriv =( +)( + ). Smmtri to th SOS s, w n prorm similr oprtions on POS. Lt =(+)( + )( + ) n =()(). Sin is POS o, Lmm 2, th nw untion nw =()()+(+)( + )( + ) must quivlnt to th originl untion. ousing on rmoving runnis rom th originl prt insi nw,w thn quikl hv = Prorming si ivision Givn untion n ivisor, in this stion w prsnt n lgorithm tht prorms si Booln ivision, i.., = Q + R. Th st w to xplin our lgorithm is to isuss it with n xmpl. ig. 2() shows two nos, whih orrspon to = n = +. Sin our ntrl i is s on th SOS onpt, th rst stp to () () Q () R \R R () () igur 2: Bsi ivision 0 0 g 1 g 2 g 3 g 4 g 5 g 6 Q init 1 Q init prorm ivi is to tk out rom ll th us tht r not ontin n u in, n suh us will our nl rminr trm R. Among th our us in, is th onl suh u sin 6 n 6. ig. 2() shows th iruit strutur tr w orm th rminr, whr w us ott irl R to init th rminr rgion n \R to not th rsulting untion with u tkn out rom. Sin vr u in \R is now ontin t lst on u in, is SOS o \R. B Lmm 1, \R woul st unhng i AN' with. This t is shown in ig. 2() with n xtr ol AN gt n th shit o \R rom or this AN gt to tr this AN gt. rom th viwpoint o th RAR thniqu, w hv sussull runn n th iruit still hs th sm untionlit. Now th rgion mrk th irl Q init is highl runnt. Th nl stp is to prorm runn rmovl on th Q init rgion n w rh th nl rsult shown in ig. 2(), whih is o th orm = Q + R. To show howrunn rmovl is on, w uplit th iruit snpshot shown in ig. 2() to ig. 2() n rmrk som nos. Lt us illustrt how wir!g 2, th thik wir in ig. 2(), is tt s runnt wir. or wir!g 2 stuk-t-1 ult to tstl, must 0 to tivt th ult. or th ult t to propgt through gt g 2, must 0. or th ult t to propgt through gt g 6, g 1 must 1. Sin = 0 n = 0 implis + = 0, gt g 1 must 0, whih is onit. A onit uring th implition pross mns th ult!g 2 stuk-t-1 is untstl, n thror wir!g 2 n rpl onstnt 1. Our si ivision lgorithm works s illustrt th ov xmpl. In summr, our lgorithm onsists o thr stps. Th rst stp o our lgorithm is to ompos th ivin so tht th us tht mk th ivisor not SOS o orm th rminr R. Th son stp is to AN with \R, whih os not hng th untionlit o\r Lmm 1. Th thir stp is to rmov ll th r- \R R
4 unnis insi th \R rgion. Not tht it is th runn ition n rmovl stps tht mk our thniqu Booln. Compring to th tritionl RAR thniqus, howvr, mjor irn lis on th t tht w know priori tht th wirs/gts r runnt us o th SOS proprt in Lmm 1. In othr wors, unlik th tritionl RAR thniqus, w o not n to hk i th wir/gt r runnt or not. urthrmor, s mntion in Stion 2, thr hs n littl suss in works tring to gnrliz th RAR thniqu to ing multipl wirs/gts. Wht our lgorithm os is ssntill tilor vrsion o th RAR philosoph onto th sustitution prolm, with x ongurtion o multipl wirs/gts ition. Also not tht sin th wirs/gts r known to runnt priori, th most tim-onsuming stp in our lgorithm is onl on th runn rmovl stp. With irnt implition mthos (.g.:[6][7][9]) w n tull just th tro twn th run tim n th qulit o rsult. or xmpl, w n limit our implition pross onl insi smll rgion, th \R rgion plus th rgion. As r s sustitution is onrn, most o th ronvrgns n implition onits woul our in this smll rgion. Limiting th implition pross insi this smll rgion woul grtl ru th tim rquir s oppos to tritionl runn rmovl pross. On th othr hn, w n rtinl spn mor tim to prorm implitions to gts outsi this smll rgion, n thr nturll inorporting n xtrnl or intrnl on't rs into onsirtion. In th xtrm s, w n vn opt som quit xhustiv implition thniqu suh s rursiv lrning [7] to inorport lrg mount o intrnl on't rs. W o not isuss th tils hr ut simpl point out th xistn o suh xiilit on vrious implition lgorithms. inll, s n sn rom th ov xmpl, our lgorithm oprts on iruit strutur irtl, rthr thn mnipulting xprssions lik tritionl pprohs. As mntion rlir, w thror r not limit to oing sustitution onl in trms o th tritionl sum-o-prout viwpoint. With th POS onpt, w n lso prorm sustitution on two untions whn th r oth in th prout-o-sum orm. Inst o using th SOS onpt n Lmm 1, w n us th POS onpt n Lmm 2, n th sm philosoph s illustrt ov woul ppl irtl. As simpl xmpl, imgin iruit tht is intil to th on shown in ig. 2 with ll th AN gts hng to OR gts n vi vrs. With our lgorithm it is s s s ws illustrt in this stion, whil in tritionl sustitution thniqu ll th sum-o-prout xprssions orm omplt nw prolm whos rsult is iult to prit. Sin onptull SOS n POS r smmtri, throughout th rmining o this ppr w o not go into th tils o th s or POS. 4 Extn ivision Th prvious stion prsnt our lgorithm tht prorms si ivision, whr ivisor is not llow to ompos. Givn untion n ivisor, unr si ivision w sk to rxprss s = Q + R. This mns w r onl llow to ompos ut not on. In this stion w prsnt n lgorithm tht prorms wht w ll xtn ivision. Givn untion n ivisor, unr xtn ivision w r llow to ompos not onl ut lso, with th purpos o minimizing th numr o litrls in sustitution. In ssn, w rst wnt to sprt th us in into two groups, th or ivisor C n th rmining ivisor R. On this sprtion is trmin, w ompos th originl ivisor into two nos suh tht = C + R. omposing into nw no or th or ivisor C mns tht C, suxprssion tht ws originll m in th givn ivisor, isnow xpos n n us or sustitution. W thn ppl our si ivision lgorithm in th prvious stion on untion n or ivisor C to otin th rsult. or xmpl, givn untion = n ivisor = + + x, w ompos th ivisor into th or ivisor C = + n th rmining ivisor R = x. Appling our si ivision lgorithm on n C, w thn otin th sm rsult s illustrt in th prvious stion. It shoul lr tht th most importnt thing hr is to intlligntl trmin th or ivisor C, sin on C is trmin n xtn ivision rus to si ivision. Rll tht uring our si ivision lgorithm, it is th stp o runn rmovl tht rll prorms th minimiztion pross. Looking k in ig. 2(), whnvr w rmov wir rom th us in th Q init rgion, w tivl ru litrl in th nl quotint. Wht w woul lik to hv is or ivisor tht is l to rmov th most wirs. To trmin th or ivisor C with givn untion n givn ivisor, our si i is to hv h wir in th us o \vot" or nit or ivisor. or h wir w in th us o,w prorm implitions to s whih us in ivisor r l to rmov wir w. or xmpl, lt untion = n ivisor = , whos iruit strutur is shown in ig. 3(). In ig. 3(), w nm ivisor 's v us 1 ; 2 ; 3 ; 4 n 5 ; w lso nm untion 's our us x 1 ;x 2 ;x 3 n x 4, whih r rsptivl rivn gts g 1 ;g 2 ;g 3 n g 4. Consir wir!g 1 stuk-t-1 ult. W hv th ollowing implitions. = 0 (tivt ult) =) 1 = 0 =1&= 1 (llow ult thru g1) =) 4 =1 x2= 0 (llow ult thru g5) & =0 =) =0 =0 =) 2=0 Assum or now tht w somhow hv trmin or ivisor C. This or ivisor, in our spiliz ongurtion or si ivision, s into gt similr to th ol AN gt g 6 in ig. 2() o th prvious stion. This mns tht i w wnt n ult t in th Q init rgion to propgt through th ol AN gt, this or ivisor C must hv vlu 1 uring th ult's implition pross. In th s o xtn ivision, i th or ivisor tht w vntull tr-
5 () x 1 g 1 g 2 x 2 x 3 g 3 x 4 g 4 g 5 2=0 1 =0 () Q init igur 3: Extn ivision min hs implition vlu 0 or prtiulr ult, th ult must untstl us onit will our with th rquir ssignmnt o 1 mntion ov. W illustrt this point ontinuing th xmpl or wir!g 1 stuk-t-1 ult. W ous on th rsults tht ppr on th i 's si whos implition vlus r 0. In this s, w hv 1 = 0 n 2 =0. Assuming w vntull hoos s our nl or ivisor, i.., C = = +, thn our si ivision lgorithm in th prvious stion woul hng th iruit strutur to th on shown in ig. 3(), whr C = + is onnt to th ol AN gt. ollowing th si ivision lgorithm in th prvious stion, w woul tr to rmov smn wirs s possil in th Q init rgion. Whn w gin prorm implitions or th ult!g 1 stuk-t-1 ult, shown with ross in ig. 3(), w know tht 1 n 2, n hn C, ll hv implition vlu 0. This rts onit us, s stt rlir, or th ult t o!g 1 stuk-t-1 to propgt through th ol AN gt, C must ssign 1. In othr wors, i w ohoos s our or ivisor, w xpt wir!g 1 to rmov in th susqunt si ivision. Now, in trmining th or ivisor, irnt wirs hv irnt implition vlus on th i 's si in ig. 3(). In som sns, this mns tht h wir \vots" or nit or ivisor. In th ov xmpl, wir!g 1 vots or nit or ivisor This shoul om lr i w look t th omplt sitution tr h wir prorms implitions on th xmpl iruit shown in ig. 3(). Tl 1() lists ll th i 's tht hv implition vlu 0 or h wir. W xplin th intrprttion o Tl 1() xmpls. Th mning o th son row is tht w xpt wir! g 1 to rmov i w hoos s th or ivisor. or simpliit, w lso s tht wir!g 1 vots or nit or ivisor Similrl, th mning o th ourth row is tht w o not xpt wir!g 2 to rmov, rgrlss o whtvr or ivisor w () 1 2 wir i =0!g1!g1!g1!g2!g2!g3!g3!g4!g4 ;4;5 4;5 4;5 1 4;5 wir i =0!g1!g1!g1!g2!g2!g3!g3!g4!g4 ;4;5 4;5 4;5 () initil () nl Tl 1: Vot tl hoos, n hn hs no nit or ivisor. Th rmining ntris o Tl 1() n intrprt in similr w. Th ov voting shm monstrts our ritri or hoosing goo or ivisor. rom th RAR thniqu's viwpoint, howvr, on mor thing w n to mk sur is tht nit or ivisor is in runnt wir whih w n vntull \" to th iruit. This is on hking i th nit or ivisor vot wir w is SOS o th u tht is onnt to wir w. or xmpl, rom th rst ntr in Tl 1(), th nit or ivisor o wir!g 1 is = +. Th u tht is onnt to wir!g 1 is x 1 =. Sin th nit or ivisor + is SOS o u, w know vntull i w or ivisor + into th iruit, th wir will runnt wir n thror th iruit untionlit woul not hng. In Tl 1(), th onl nit or ivisors tht o not hol or this onition r wirs!g 3 n!g 4. Th nit or ivisor or wir!g 3 is = +, whih is not SOS o th orrsponing u x 3 =. On th s o wir!g 4, nit or ivisor 1 = is not SOS o th orrsponing u x 4 =. W thror n to lt ths two ntris in Tl 1() n w hv our nl vot tl, shown in Tl 1(). To nliz th hoi o th or ivisor, vrious huristis n us. W ru th ov hoi prolm to mximl liqu prolm in grph thor. u to sp limit, w omit th tils hr. Appling our xtn ivision lgorithm to th sustitution prolm, w wnt to point out tht w n tull o mor thn wht th ov isussion shows. In th ov ormultion, w ous onl on on xisting no. In th s o sustitution, w tull hv rom to slt our or ivisor rom mong mn iruit nos. As n xmpl o how this gnrliztion works, imgin th givn ivisor in th ov xmpl, = , os not xist in our iruit n inst, two nos 1 = + + n 2 = + xist, s shown in ig. 3(). Whn untion = is givn n w wnt to srh or goo ivisor twn 1 n 2 with xtn ivision, w n tmporril prtn tht ll th v us r rom th sm no, n thror th ow is intil to th xmpl shown in this stion. Eh wir in th us o 1 n 2 vots or nit or ivisor n w hvnintil vot
6 nh init sis si xt. xt.+gc lit. lit. pu lit. pu lit. pu lit. pu C C C C C C C C lu px lu s rg i i i rot t trm x x s s s s s s totl % 1.2% 9.5% 9.8% 10.5% Tl 2: Exprimntl rsults tl s shown in Tl 1(). Th onl slight moition w n is in th nl mximl liqu ormultion, whr w n to mol th t tht som us in th son olumn o Tl 1() originll om rom irnt no. Not tht, s is lso th s or si ivision, w n prorm xtn ivision in trms o sum-o-prout orm s wll s prout-o-sum orm. Inst o ousing on th us tht hv implition vlu 0, w woul thn ous on th sum trms tht hv implition vlu 1. Th rst o th lgorithm pplis similrl. 5 Exprimntl rsults W hv implmnt our lgorithm n ppli it to th sustitution prolm. Our implmnttion hs thr ongurtions: si ivision, xtn ivision without onsiring glol intrnl on't rs, n xtn ivision with glol intrnl on't rs tkn into onsirtion. W prorm xprimnts on MCNC n ISCAS nhmrks within SIS [10] nvironmnt. W rst run th sript limint 0; simpli; gx; on h nhmrk to otin th initil iruit. W thn ompr our lgorithm with th lgri rsustitution \rsu -" in SIS. Tl 2 shows th omprison twn SIS n our rsults. Th rst olumn shows th nm o th iruit. Th son olumn shows th initil litrl ount. Th olumns ll \sis" is th rsult o running \rsu -" ommn, with suolumns \lit." n \pu" rporting th numr o litrls n CPU tim, rsptivl. All litrl ounts r in tor orm. Tk th iruit C2670 s n xmpl, tr running th ov sript, initill th iruit h 939 litrls, shown in th son olumn. Atr running \rsu -" th litrl ountws ru to 849 litrls. With our si ivision it is ru to 840 (Column \si"). Th xtn ivision ru it to 831 (Column \xt."). Th xtn ivision with glol on't rs tkn into ount rought itown to 828 (Column \xt.+gc"). Th lst two rows show th summtion o h olumn n th prntg o improvmnt ompr to th initil litrl ount. As th tl inits, our ivision lgorithms outprorm th tritionl ivision n sustitution. In trms o run tim, \rsu -", th si ivision, n th xtn ivision without glol on't rs spnt similr CPU tims. Th xtn ivision with glol on't rs spnt th most tim. 6 Conlusion In this ppr w rst prsnt n int lgorithm, s on th philosoph o runn ition n rmovl (RAR), or prorming Booln ivision. With th onpt o SOS n POS, w tilor th RAR philosoph to th Booln ivision prolm. Th tiloring nls us to multipl wirs/gts in spiliz ongurtion n rmov mor wirs/gts. Appling our Booln ivision lgorithm, our lgorithm n prorm sustitution not onl in th tritionl sum-o-prout orm, ut lso in prout-o-sum orm. W thn gnrliz our si ivision to wht w ll xtn ivision. Extn ivision llows us to ompos not onl on th ivin ut lso on th ivisor. urthrmor, our thniqu is l to nturll inorport ll tps o on't rs into onsirtion. W lso prsnt som xprimntl rsults to vri th tivnss o our lgorithm. Rrns [1] R.K. Brton n C. MMulln, \Th omposition n toriztion o Booln Exprssions," Pro. ISCAS, pp , [2] S.C.Chng n M.Mrk-Sowsk, \Prtur n simpli: multi-lvl Booln ntwork optimizr," Pro. ICCA, pp. 2-6, Nov [3] S.C. Chng, L. VnGinnkn, n M. Mrk-Sowsk, \st Booln Optimiztion Rwiring," Pro. ICCA, pp , [4] G. Mihli, \Snthsis n Optimiztion o igitl Ciruits," M Grw Hill Txt, [5] L.A.Entrn n K.T.Chng, \Comintion n squntil logi optimiztion runn ition n rmovl," IEEE TCA, Vol. 14, no. 7, pp , Jul [6] T. Kirkn n M.R. Mrr, \A Topologil Srh Algorithm or ATPG," Pro. AC, pp , [7] W. Kunz n.k. Prhn, \Rursiv Lrning: An Attrtiv Altrntiv to th ision Tr or Tst Gnrtion in igitl Ciruits," Pro. ITC, pp , [8] W. Kunz n.k. Prhn, \Multi-Lvl Logi Optimiztion Implition Anlsis," Pro. ICCA, pp. 6-13, [9] M. Shulz n E. Auth, \Avn Automti Tst Pttrn Gnrtion n Runn Intition Thniqus," Pro. TCS, pp , [10] E.Sntovih t. l. \SIS: A Sstm or Squntil Ciruit Snthsis" Mmornum UCB M92/41, UC, Brkl.
Present 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 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(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 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. 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 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 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 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 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 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 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 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 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 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 informationa b v a v b v c v = a d + bd +c d +ae r = p + a 0 s = r + b 0 4 ac + ad + bc + bd + e 5 = a + b = q 0 c + qc 0 + qc (a) s v (b)
Outlin MULTIPLE-LEVEL LOGIC OPTIMIZATION Gionni D Mihli Stnfor Unirsit Rprsnttions. Tonom of optimition mthos: { Gols: r/l. { Algorithms: lgri/booln. { Rul-s mthos. Empls of trnsformtions. Booln n lgri
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 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 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 informationFundamental Algorithms for System Modeling, Analysis, and Optimization
Fundmntl Algorithms for Sstm Modling, Anlsis, nd Optimiztion Edwrd A. L, Jijt Rohowdhur, Snjit A. Sshi UC Brkl EECS 144/244 Fll 2011 Copright 2010-11, E. A. L, J. Rohowdhur, S. A. Sshi, All rights rsrvd
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationDesign Optimization Based on Diagnosis Techniques
Dsign Optimiztion Bs on Dignosis Thniqus Anrs Vnris Mgy S. Air Irhim N. Hjj Univrsity of Toronto Motorol Univrsity of Illinois ECE Dprtmnt 77 W. Prmr ECE Dprtmnt n CSL Toronto, ON M5S 34 Austin, T 78729
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationA Graph-Based Synthesis Algorithm for AND/XOR Networks 1. Yibin Ye Kaushik Roy. a minimum ESOP of a function except for those with
A Grph-Bs Synthsis Algorithm or AND/XOR Ntworks Yibin Y Kushik Roy Shool o Eltril n Computr Enginring Puru Univrsity, Wst Lytt, IN 4797-285, USA Abstrt In this ppr, w introu Shr Multipl Root XORbs Domposition
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 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 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 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 informationJournal of Solid Mechanics and Materials Engineering
n Mtrils Enginring Strss ntnsit tor of n ntrf Crk in Bon Plt unr Uni-Axil Tnsion No-Aki NODA, Yu ZHANG, Xin LAN, Ysushi TAKASE n Kzuhiro ODA Dprtmnt of Mhnil n Control Enginring, Kushu nstitut of Thnolog,
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 informationThe Plan. Honey, I Shrunk the Data. Why Compress. Data Compression Concepts. Braille Example. Braille. x y xˆ
h ln ony, hrunk th t ihr nr omputr in n nginring nivrsity of shington t omprssion onpts ossy t omprssion osslss t omprssion rfix os uffmn os th y 24 2 t omprssion onpts originl omprss o x y xˆ nor or omprss
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 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 informationReducing the Depth of Quantum Circuits Using Additional Circuit Lines
Ruing th Dpth of Quntum Ciruits Using Aitionl Ciruit Lins Nil Assi 1, Rort Will 1,2, Mthis Sokn 1,2, n Rolf Drhslr 1,2 1 Institut of Computr Sin, Univrsity of Brmn Group of Computr Arhittur, D-28359 Brmn,
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 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 informationSelf-Adjusting Top Trees
Th Polm Sl-jsting Top Ts ynmi ts: ol: mintin n n-tx ost tht hngs o tim. link(,w): ts n g twn tis n w. t(,w): lts g (,w). pplition-spii t ssoit with gs n/o tis. ont xmpls: in minimm-wight g in th pth twn
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 information8Algebraic UNCORRECTED SAMPLE PAGES. techniques. What you will learn. Australian curriculum. Chapter 8A 8B 8C 8D 8E 8F
8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K Chptr Wht you will lrn 8Algri thniqus Epning inomil prouts Prt squrs n irn o prt squrs Ftorising lgri prssions Ftorising th irn o two squrs Ftoristion y grouping Ftorising
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 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 informationMultipoint Alternate Marking method for passive and hybrid performance monitoring
Multipoint Altrnt Mrkin mtho or pssiv n hyri prormn monitorin rt-iool-ippm-multipoint-lt-mrk-00 Pru, Jul 2017, IETF 99 Giuspp Fiool (Tlom Itli) Muro Coilio (Tlom Itli) Amo Spio (Politnio i Torino) Riro
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 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 informationFSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *
CmSc 365 Thory of Computtion Finit Stt Automt nd Rgulr Exprssions (Chptr 2, Sction 2.3) ALPHABET oprtions: U, conctntion, * otin otin Strings Form Rgulr xprssions dscri Closd undr U, conctntion nd * (if
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 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 informationSection 3: Antiderivatives of Formulas
Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin
More informationMore Foundations. Undirected Graphs. Degree. A Theorem. Graphs, Products, & Relations
Mr Funtins Grphs, Pruts, & Rltins Unirt Grphs An unirt grph is pir f 1. A st f ns 2. A st f gs (whr n g is st f tw ns*) Friy, Sptmr 2, 2011 Ring: Sipsr 0.2 ginning f 0.4; Stughtn 1.1.5 ({,,,,}, {{,}, {,},
More information5/7/13. Part 10. Graphs. Theorem Theorem Graphs Describing Precedence. Outline. Theorem 10-1: The Handshaking Theorem
Thorm 10-1: Th Hnshkin Thorm Lt G=(V,E) n unirt rph. Thn Prt 10. Grphs CS 200 Alorithms n Dt Struturs v V (v) = 2 E How mny s r thr in rph with 10 vrtis h of r six? 10 * 6 /2= 30 1 Thorm 10-2 An unirt
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 informationa b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...
Progrssiv Printing T.M. CPITLS g 4½+ Th sy, fun (n FR!) wy to tch cpitl lttrs. ook : C o - For Kinrgrtn or First Gr (not for pr-school). - Tchs tht cpitl lttrs mk th sm souns s th littl lttrs. - Tchs th
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 informationChem 104A, Fall 2016, Midterm 1 Key
hm 104A, ll 2016, Mitrm 1 Ky 1) onstruct microstt tl for p 4 configurtion. Pls numrt th ms n ml for ch lctron in ch microstt in th tl. (Us th formt ml m s. Tht is spin -½ lctron in n s oritl woul writtn
More informationChapter 18. Minimum Spanning Trees Minimum Spanning Trees. a d. a d. a d. f c
Chptr 8 Minimum Spnning Trs In this hptr w ovr importnt grph prolm, Minimum Spnning Trs (MST). Th MST o n unirt, wight grph is tr tht spns th grph whil minimizing th totl wight o th gs in th tr. W irst
More informationBinomials and Pascal s Triangle
Binomils n Psl s Tringl Binomils n Psl s Tringl Curriulum R AC: 0, 0, 08 ACS: 00 www.mthltis.om Binomils n Psl s Tringl Bsis 0. Intif th prts of th polnomil: 8. (i) Th gr. Th gr is. (Sin is th highst
More informationCS553 Lecture Register Allocation 1
Low-Lvl Issus Lst ltur Livnss nlysis Rgistr llotion Toy Mor rgistr llotion Wnsy Common suxprssion limintion or PA2 Logistis PA1 is u PA2 hs n post Mony th 15 th, no lss u to LCPC in Orgon CS553 Ltur Rgistr
More informationDecimals DECIMALS.
Dimls DECIMALS www.mthltis.o.uk ow os it work? Solutions Dimls P qustions Pl vlu o imls 0 000 00 000 0 000 00 0 000 00 0 000 00 0 000 tnths or 0 thousnths or 000 hunrths or 00 hunrths or 00 0 tn thousnths
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 informationAquauno Video 6 Plus Page 1
Connt th timr to th tp. Aquuno Vio 6 Plus Pg 1 Usr mnul 3 lik! For Aquuno Vio 6 (p/n): 8456 For Aquuno Vio 6 Plus (p/n): 8413 Opn th timr unit y prssing th two uttons on th sis, n fit 9V lklin ttry. Whn
More information