for Performance Potsdam, NY Urbana, IL can be connected to the inputs and outputs of the LUT to

Transcription

1 Thnoloy Mppin o Squntil Ciruits or LUT-s FPGAs or Prormn Pihn Pn C. L. Liu Dpt. o Eltril & Computr En. Dpt. o Computr Sin Clrkson Univrsity Univrsity o Illinois t Urn-Chmpin Potsm, NY Urn, IL Astrt In this ppr, w stuy th thnoloy mppin prolm or squntil iruits or LUT-s FPGAs. Th onvntionl pproh or this prolm is s on thnoloy mppin lorithm or omintionl iruits whil ssumin th positions o th ip-ops r x. W propos nw pproh in whih FFs n ritrrily rposition y rtimin. W prsnt n int thnoloy mppin lorithm tht prous mppin solution with th minimum lok prio or iruit without loops unr th unit ly mol. Th lorithm is lso xtn to th nrl ly mol, in whih s it prous mppin solution with lok prio t most n intronnt or LUT ly wy rom th minimum on. Not tht th lorithm n lso us or iruits with loops y rmovin som o th FFs to rk th loops or th pplition o th lorithm. Th supriority o our pproh is urthr monstrt xprimntlly. Kywors: FPGAs, thnoloy mppin, rtimin, loi rplition, look-up tl, squntil iruits, lok prio 1 Introution Fil prormml t rrys (FPGAs) hv volv rpily to om n importnt ASIC thnoloy. Most onspiuous turs o FPGAs r low mnuturin ost or low volum sins, short sin yl, n rprormmility. Ths turs mk FPGAs prtiulrly ttrtiv or suh pplitions s sin prototypin n hrwr multion. In this ppr w onn ourslvs to look-up tl s FPGA rhitturs [19]. A LUT-s FPGA onsists o n rry o prormml loi loks (PLBs) tothr with prormml intronntions. Th or o PLB is k-input LUT (k-lut) whih n implmnt ny omintionl loi with up to k inputs n sinl output, Th work ws prtilly support y th Ntionl Sin Fountion unr rnt MIP whr k is positiv intr rnin usully rom 3 to 9. Thr r lso w ip-ops (FFs) in h PLB whih n onnt to th inputs n outputs o th LUT to rliz squntil hvior. Th thnoloy mppin prolm or LUT-s FPGAs is to prou, or ivn iruit, n quivlnt iruit omprisin o k-luts. This prolm hs n stui xtnsivly. Howvr, lmost ll mppin lorithms wr sin or omintionl iruits. Mppin lorithms or omintionl iruits hv n propos or vrious optimiztion ritri: prormn [3, 2, 7, 11, 15, 20], r [4, 5, 6, 8, 13, 14, 18], routility [1, 16]. W mntion hr prtiulrly FlowMp [3]. For k-oun omintionl iruit, FlowMp is l to prou mppin solution with th minimum ly unr th unit ly mol. Th onvntionl pproh to thnoloy mppin o squntil iruits is to us mppin lorithm or omintionl iruits to mp th omintionl loi twn FFs. Spilly, th FFs in iruit r rmov to otin omintionl ntwork. Thn th omintionl ntwork is mpp. Finlly, th FFs r pl k. This pproh hs two ovious shortomins: (i) it ils to onsir sinl pnnis ross FF ounris, n (ii) it os not onsir th possiility o xposin th omintionl loi twn th FFs in irnt wys. Not tht th FFs in iruit n rposition y thniqu ll rtimin [10]. Rntly, oupl o irt thnoloy mppin mthos or squntil iruits wr propos [12, 17]. Howvr, thy lso ssum tht th initil positions o th FFs r x, thouh th huristi lorithm in [17] uss rtimin s post-prossin stp. 1.1 Motivtions W noti tht vn with n optiml mppin lorithm or omintionl iruits, th onvntionl pproh my not prou n optiml mppin solution or squntil iruit. As n xmpl, onsir th iruit in Fiur 1(). Suppos k = 4. I th FFs r not rposition, it n vri tht ithr or must n input to ny 4-LUT or t. As rsult, ny mppin solution hs lok prio t lst two n uss t lst our LUTs. On possil mppin solution is init y th sh polyons in Fiur 1(). On th othr hn, i w mov oth 1 n 2 kwr (y rtimin) s shown in Fiur 1(), whv th mppin solution with on LUT shown in Fiur 1(),

2 i1 1 2 () () () Fiur 1: Avnt o rtimin. 1 y x () () () Fiur 2: Avnt o loi rplition. whr th LUT is orm y ll ts in Fiur 1(). Noti tht this mppin solution hs lok prio on. W urthr noti tht to ully xploit th potntil o rtimin, loi rplition is lso nssry. Rplition n hlp prou mppin solutions whih r othrwis impossil. Consir th iruit in Fiur 2(). Assum k = 4. First, w noti tht on th two input s to h o th ts, n, i FF is rmov (y rtimin) rom on, ntiv FF will introu on th othr. For instn, i w movff 1out o x, ntiv FF will introu on y. Hn, th rsultnt iruit is symmtril. As rsult, w n ssum tht th FF positions r ll x. In this s, it n vri tht t lst two o th outputs o,, n must inputs to ny 4-LUT or t. Consquntly, ny mppin solution must us t lst six LUTs n hv lok prio t lst two. Howvr, i w uplit (to om n 0 ), (to om n 0 ), n (to om n 0 ), thn rtim th FFs ross ts 0, 0, n 0 s shown in Fiur 2(), w n mp ll th ts (inluin th uplit ons) to sinl 4-LUT s shown in Fiur 2(). Not tht this mppin solution hs lok prio on. 1.2 Contriutions In this ppr w xmin th thnoloy mppin prolm or squntil iruits or LUT-s FPGAs in th most nrl sttin. Conptully, our prolm ormultion n sri y th irm in Fiur 3. Tht is, th sp o mppin solutions onsists o ll th iruits tht n otin y rtimin n rplitin th ivn initil iruit, thn mppin th omintionl loi twn th FFs, ollow y nothr rtimin n rplition. Not tht in th onvntionl pproh, Stp 1 n Stp 3 in Fiur 3 r missin. Th ojtiv o th thnoloy mppin prolm rss in this ppr is to otin mppin solution with smll lok prio, whih is th mximum ly twn ny two sussiv FFs in iruit. W will prsnt n int lorithm tht prous mppin solution with th minimum lok prio or looplss squntil iruit, unr th unit ly mol. Furthrmor, th lorithm is xtn to th nrl ly mol. In tht s, it prous mppin solution with lok prio provly los to th minimum on. ivn squntil iruit rplition & rtimin mppin th omintionl loi rplition & rtimin solutions in solution sp Fiur 3: Solution sp tr intrtin rtimin n rplition. W shoul mphsiz tht wht Fiur 3 shows is th nition o th solution sp, not th stps tht our lorithm tks to solv th prolm. In t, to try to otin mppin solution y rryin out th stps in th ur squntilly is wht n lorithm s on th onvntionl pproh os, n it invitly rrivs t su-optiml solution sin thr is no wy to know wht rtimin n rplition to us or mppin th omintionl loi twn FFs. Anothr wy to unrstn our pproh is

3 tht it intrts rtimin, rplition, n mppin loi lolly to otin th st mppin solutions. It is ovious tht th solution sp in our ormultion is normous sin thr r too mny wys to rtim n rplit iruit. On ontriution o this work is intiyin muh smllr sust o th solution sp tht is suint or otinin minimum lok prio mppin solution. An importnt issu in thnoloy mppin or LUT-s FPGAs is th ormtion o LUTs. Thouh this issu is rltivly simpl or omintionl iruits, it is omplit or squntil iruits us o th inlusion o rtimin n rplition. Anothr ontriution o this work is to riv mtho or ormin LUTs in squntil iruits y introuin th onpt o xpn iruits. Our lorithm is lso s on th nrl i o no llin [9]. W will introu llin shm tht tks into onsirtion oth lys n FFs. This work is nothr sinint stp towr th unrstnin o thnoloy mppin or LUT-s FPGAs. 2 Prliminris n prolm nition A (squntil) iruit n mol s n wiht irt (multi-)rph. Th nos r th primry inputs (PIs), th primry outputs (POs), n th omintionl prossin lmnts (PEs). (A PE is ithr t or k-lut pnin on whthr th iruit is th initil on or mppin solution.) Th s rprsnt th intronntions. Thr is n rom u to v with wiht t i th output o u, tr pssin throuh t FFs, is n input to v. W will us w() to not th wiht o. Fornovin squntil iruit N, w us N v to not th suiruit inu y th nos tht n rh v. Th output o N v is tht o v. Th lok prio o N is th mximum totl ly (o ts n intronntions) on th omintionl pths (pths without FFs) in N. Rtimin is thniqu o rpositionin th FFs without hnin th untionlity or th strutur o th iruit [10]. Rtimin noyvlu i mns rmovin i FFs rom h nout n in i FFs to h nin o th no. Fiur 4 shows th s in whih i = 1 n 1. In nrl, ll nos in iruit n rtim simultnously ( rtimin o th iruit). It n shown tht th rtim iruit n th oriinl iruit hv th sm untionlity i no rtimin is prorm t th PIs n POs (i.., th rtimin vlus or th PIs n POs r ll zro). v +1-1 Fiur 4: Rtimin no. Th rtim lok prio o N is n to th minimum lok prio o ll th iruits tht n otin y rtimin N. Eint lorithms or omputin th rtim lok prio s wll s rtimin tht hivs th rtim lok prio r prsnt in [10]. v Rr in to Fiur 3. Suppos N is th iruit to mpp. W ssum tht N is k-oun, i.., h t hs t most k inputs. Lt N 0 iruit otin rom N usin rplition n rtimin. Lt N 00 mppin solution o th omintionl loi in N 0, n S th iruit otin y plin th rmov FFs k ton 00 n ollow y nothr rtimin 1. S is mppin solution o N. Th thnoloy mppin prolm rss in this ppr is s ollows: Prolm 1 Fin mppin solution with th minimum lok prio. Finlly, w list svrl rph-thorti onpts. Lt G DAG with on sink ut possily svrl sours. A ut (X;X) is prtition o th nos in G suh tht th sink is in X n ll th sours r in X. Th -st E(X;X) o th ut is th st o s rom X to X, th no-st V (X;X) o th ut is th st o nos in X whih r onnt to on or mor nos in X. Th on o th ut is th surph inu y X. I jv(x; X)j k, (X;X) is urthr ll k-sil ut, or k-ut or short. 3 Formtion o LUTs Rll tht in omintionl iruit, k-lut or no v is orm y th on o k-ut in N v [3]. In this stion, w r oin to riv wy to xmin k-luts or nos in squntil iruits. In our prolm ormultion, squntil iruit n onptully rtim n rplit ritrrily. As rsult, thr is no x iruit or LUT ormtion. To ovrom this iulty, w will stuy n quivlnt prolm in whih only sust o spil mppin solutions is onsir. A mppin solution in whih th output sinls o th LUTs r rom th oriinl iruit is rrr to s simpl mppin solution. Not tht u to rtimin, th output sinl o LUT in mppin solution is in nrl sinl in th oriinl iruit ly or vn y w lok yls. Th st o simpl mppin solutions my not ontin mppin solution with th minimum lok prio. Howvr, w hv th ollowin rsult: Thorm 1 Thr is simpl mppin solution whos rtim lok prio is qul to th minimum lok prio o ll mppin solutions. 2 I w hv simpl mppin solution with th minimum rtim lok prio mon ll simpl mppin solutions, w n thn rtim th simpl mppin solution usin rtimin tht hivs th lok prio to otin mppin solution with th minimum lok prio. As rsult, inst o stuyin Prolm 1 w will stuy th ollowin quivlnt prolm: Prolm 2 Fin simpl mppin solution whos rtim lok prio is minimum. 1 Thouh w n lso pply rplition hr, it turns out to unnssry.

4 Th importn o simpl mppin solutions is tht y rstritin to thm, w only n to stuy how to orm LUTs or th nos in N. To orm LUT or no v, strihtorwr pproh is to us th on o ut in N v. Thn FFs within th on r mov to th ounris o th on usin rtimin n loi rplition. Bsi th prolm tht th numr o inputs to th nl LUT is not irtly rlt to th siz o th no-st o th th ut (this mks it iult to slt uts in th rst pl), this pproh my not nrt ll possil LUTs or v. As n xmpl, or th iruit in Fiur 5() i w rplit (to om 1 n 2) n (to om 1 n 2), thn rtimin 2 y 1,w rriv t th iruit in Fiur 5(). Th loi lint y th sh polyon orms 3-LUT s shown in Fiur 5(). Howvr, this LUT nnot riv without rplition rst. () () Fiur 5: A LUT orm only throuh rplition. Wnowintrou th onpt o xpn iruits. W will us xpn iruits to riv LUTs. Th xpn iruit E v or v is onstrut y rplitin th nos in N v. Th intuition hin th onstrution is s ollows: For pth rom u to v in N v: () 1 2 (u =)u 1! u 2! t! u t+1(= v); w pply rplition to rt uniqu orrsponin pth in th xpn iruits s ollows: u ! u ! 0 t! u 0 t+1; P whr i = w(j), ijt u i i is rplit opy ou i, 0 i opy o i(with th sm wiht), or 1 i t. Lt r(u; v) =w(p)jp, pth rom u to v in N. Tht is, r(u; v) is th st o irnt pth wihts o ll th pths rom u to v in N. E v is otin y sussivly rplitin th nos in N v in topoloil orr strtin rom v. At th innin, N v v is th sm s N v xpt w rnm v s v 0. Suppos w hv onstrut Nv T, n x is in T or h (u; x) inn v, thn N T [u v is otin rom N T v y rplitin u into jr(u; v)j opis s ollows: (1) For h 2 r(u; v), introu no u. (2) For h (u; x 1 ) with wiht t in N T v, n (u ;x 1 ) with wiht t whr = 1 + t. (3) For h (y; u) with wiht t in N T v, n (y; u ) with wiht t or vry 2 r(u; v). (4) Rmov no un th s inint to it. Whn T onsists o ll th nos in N v, th rsultin iruit is E v. As n xmpl, or t in th iruit in Fiur 2(), Fiur 6() shows its xpn iruit. Th importn o th xpn iruits is tht k-lut or no n riv rom h k-ut in th xpn iruit or th no. As n xmpl, or th 6-ut lint y th sh irl in Fiur 6(), th orrsponin 6-LUT is shown in Fiur 6(). In nrl, i u is in th no-st o k-ut in E v, u tr pssin throuh FFs is n input to th orrsponin k-lut. Th loi o th k-lut is simply th on o th k-ut lss th FFs. W n urthr show tht ths r th only k-luts tht n to xmin. In summry, whv Thorm 2 To xmin ll k-luts or no v, it su- s to xmin ll th k-luts tht n riv rom th k-uts in E v. Finlly, w stimt th numrs o nos n s in E v. Hrtr, w will us n to not th numr o nos n to not th numr o FFs in N. Sin N is k-oun, th numr o s in N is O(kn). Oviously, th mximum vlu in r(u; v) or ny pir o nos u n v is t most. Consquntly, th numr o istint vlus in r(u; v) is t most. Thror, th numronos in E v is O(n) n th numr o s is O(kn) (not tht E v is lso k-oun). In prti, w xpt ths numrs to muh smllr. 4 An optiml lok prio mppin lorithm In this stion w prsnt n lorithm or solvin Prolm 2. W minly ous on th unit ly mol, nmly, LUTs hv on unit ly n intronntions hv zro ly. Ltr, w will riy sri how to xtn th lorithm to th nrl ly mol. As ws mntion rlir, w ssum tht th initil iruit N is looplss n k-oun. Th lorithm is s on solvin th ision vrsion o Prolm 2: Prolm 3 Givn trt lok prio, n simpl mppin solution whos rtim lok prio is or lss, i suh mppin solution xists. I whv n lorithm or solvin Prolm 3, w n o inry srh on to solv Prolm 2. In th rminr o this stion, w r oin to prsnt n lorithm or solvin Prolm 3. Our lorithm mploys llin prour n is s on ntwork ow thniqus. Bor prsntin th lorithm, w xmin this qustion: Givn iruit M n n intr, whthr M n rtim to lok prio or lss. Thouh th lorithms in [10] n us to nswr this qustion, w wnt mtho tht is mor suitl or solvin our prolm. To this n, w onstrut rph whos topoloy is th sm

5 i2 i 1 i2 6-ut 6-LUT () () Fiur 6: Otinin LUTs in xpn iruits. s tht o M, ut with th wihts in rn: For n u! v, th nw wiht is w() + 1. Th l-vlu o v is n to th mximum wiht oth pths rom th PIs to v orin to th nw wihts. W hv, Thorm 3 M n rtim to lok prio or lss i th l-vlu o h PO is lss thn or qul to +1. Our lorithm or solvin Prolm 3 hs two phss: th llin phs n th mppin phs. In th llin phs, w omput ll or h noinn. Th ll o no is th minimum l-vlu o th k-luts or th no mon ll simpl mppin solutions o th iruit. With th lls, w n sily trmin whthr N hs simpl mppin solution with rtim lok prio or lss: I th ll o h PO is lss thn or qul to + 1, thr is suh mppin solution. Othrwis, thr is not. I th mppin solution xists, w nrt on suh solution in th mppin phs. In th nxt two sustions, w sri urthr tils o th two phss, sprtly. 4.1 Th llin phs Fornovin N, lt l opt (v) not its ll, nmly, th minimum l-vlu o th k-luts or v mon ll simpl mppin solutions o N. W trmin l opt (v) or h no v in N in this phs o th lorithm. Givn k-ut (X; X) ine v, th minimum l-vlu o th orrsponin k-lut is, mxl opt (u) +1ju!x2E(X; 0 X): (1) By Thorm 2, whv tht l opt (v) is qul to th minimum o th quntity in (1) mon ll k-uts in E v. Oviously, w n omput th lls o th nos in N in topoloil orr strtin rom th PIs. For h PI v, l opt (v) = 0. Suppos w nowwnt to trmin l opt (v) o non-pi no v, ivn tht th lls o ll its prssors hv lry n trmin. To omput l opt (v), w in onsir th ision prolm, tht is, Prolm 4 Dtrmin whthr l opt (v) L or ivn intr L. W us ntwork ow thniqus to solv Prolm 4. A ow ntwork G is onstrut rom E v y pplyin to E v stnr ntwork trnsormtion, ll no-splittin. Eh no xpt v 0 in E v is split into two nos with riin twn thm. A suprsour is n onnt to ll th sours in th ntwork. Th riin or no u hs unit pity i l(u) +1 L. All othr s in G hs innit pity. Th ollowin rsult n rily shown: Lmm 1 l nw L i G hs ut with pity lss thn or qul to k. 2 Bs on th lssil Mx-ow Min-ut Thorm, w n us n umntin pth lorithm or solvin th mx- ow prolm to trmin whthr G hs ut with pity lss thn or qul to k in O(k je(g)j)=o(k 2 n) tim. Thouh l opt (v) n now trmin usin inry srh, w tully only n to solv Prolm 4 on. This is us o th ollowin rsult: Lmm 2 l opt (v) = L v 1 or L v, whr L v = mxl opt (u) w()+1ju!v is in N v. 2 To trmin l opt (v), w hk whthr l opt (v) L v 1. I so, l opt (v) =L v 1; othrwis, l opt (v) =L v. Bs on ths isussions, w hv th ollowin rsult: Thorm 4 Th minimum l-vlus o ll nos in N n trmin ino(k 2 n 2 )tim Th mppin phs Th purpos o this phs is to nrt mppin solution with lok prio or lss i it is trmin tht thr is on. This phs is rltivly simpl. Th rst stp is to nrt simpl mppin solution whos rtim lok prio is or lss. To o so, w kp two lists D n U. D is th st o nos in N or whih whv inlu thir k-luts in th simpl mppin solution n U is th st o nos whos k-luts r inputs to som k-luts in D n hv not yt n inlu. At th innin, D onsists o th PIs n U onsists o th POs. At h itrtion, no

6 v in U is rmov n to D. Lt th k-lut tht rlizs l opt (v) L vs trmin in th llin phs. Thn, i u tr pssin throuh FFs is n input to L v, w rt n rom L u to L v with wiht, n u to U i it is not in D or U. This pross stops whn U oms mpty. Lt S not th rsultin mppin solution. Not tht thr is t most on LUT in S or h nov in N. Finlly, to otin mppin solution with lok prio or lss, w simply pply th ollowin rtimin to S: r(l v)= 0 vispiorpo lopt (v) 1 othrwis. Th rtim solution is urnt to hv lok prio or lss, provi tht th ll o h PO is no mor thn Extnsion to th nrl ly mol In th nrl ly mol, h hs n ssoit ly vlu (not ()). Not tht ll k-luts hv th sm ly (not 2). For n in mppin solution, its ly is ssum to th mximum o th lys o ll s in th initil iruit tht r mpp to this. Th lorithm or th nrl ly mol is lmost th sm s th on or th unit ly mol xpt th ollowin ormul will us to omput l opt (v): min mxl(u) + ()+2ju!x2E(X; 0 X) ; (X; X) whr th minimiztion is ovr ll k-uts in E v. W n show tht th xtn lorithm n prou mppin solution with lok prio tht n wy rom th minimum on y t most th mximum o 2 n th intronntion lys. 5 Exprimntl rsults W implmnt our thnoloy mppin lorithm (rrr to s SqMp) n rri out som xprimnts. In this stion, w sri our xprimnts n summriz th rsults. Sin thr r no looplss squntil nhmrk iruits, w riv our tst xmpls rom th multi-lvl omintionl n squntil nhmrk iruits in th LGSynth91 suit rom MCNC. For omintionl nhmrk iruit, w to it on st o piplin. Th rsultin squntil iruit ws thn rtim to its rtim lok prio to otin tst xmpl. For squntil nhmrk iruit, w rmov st o ts to ut th loops, thn in rtim it to its rtim lok prio to otin tst xmpl. Som stnr SIS ommns suh s sp up, swp, n th omp wr us in th onstrution o th tst xmpls. For omprison, w implmnt thnoloy mppin lorithm s on th onvntionl pproh ll ComMp. ComMp mps squntil iruit y rst rmovin ll th FFs. Thn, it mps th rsultin omintionl loi usin FlowMp ly optiml thnoloy mppin lorithm or omintionl iruits. Finlly, th rmov FFs r onnt k to orm mppin solution o th oriinl iruit. ComMp lso uss rtimin s post-prossin stp y rtimin th initil mppin solution to its rtim lok prio. Th rsultin iruit is thn th nl output o ComMp. In our urrnt implmnttion, nithr ComMp nor SqMp ontins ny post-prossin oprtions or LUT rution. W tst oth ComMp n SqMp on st o xmpl iruits usin 5-LUTs. Th rsults r summriz in Tl 1. In Tl 1, unr olumn initil w list th numr o ts, th numr o FFs, n th lok prio () o h tst xmpl; unr olumn ComMp, w list th numr o LUTs, th numr o FFs, n th lok prio o th mppin solution prou y ComMp. Th sm quntitis r lso list or SqMp in olumn SqMp. Not tht th lok prios o th mppin solutions prou y SqMp r optiml. From th tl, it is lr tht quit otn ComMp prou su-optiml solutions. It n lso sn tht th mppin solutions prou y SqMp usully hv wr LUTs thn tht prou y ComMp. This ws lso xpt sin SqMp my xtn ross FF ounris to orm LUTs. Th runnin tims o SqMp wr out 5 tims tht o ComMp or th tst xmpls. 6 Summry In this ppr, whv stui th FPGA thnoloy mppin prolm or squntil iruits. Th prolm is stui in th most nrl sttin. In our ormultion, th mppin solution sp is muh lrr thn wht th onvntionl pproh (s on mppin lorithm or omintionl iruits) is l to xplor. W hv prsnt n optiml lok prio mppin lorithm or looplss squntil iruits. This lorithm n lso us or iruits with loops y rmovin som o th FFs to rk th loops or th pplition o our lorithm. Anothr wy to unrstn our lorithm is tht it omins rtimin, rplition, n mppin omintionl loi lolly, whil in th onvntionl pproh, thy r rri out sprtly. Our lorithm hs n implmnt. Exprimntl rsults lso onrm th supriority o our pproh ovr th onvntionl on. Currntly, w r in th pross o xtnin our rsults to squntil iruits with loops. W r lso onsirin usin th is vlop hr or LUT minimiztion. Rrns [1] N. Bht n D. Hill. Routl thnoloy mppin or FP- GAs. In ACM/SIGDA Workshop on FPGAs, ps 143{ 148, [2] J. Con n Y. Din. Byon th omintionl limit in pth minimiztion or LUT-s FPGA sins. In Dist Intl. Con. on Computr-Ai Dsin, ps 110{114, [3] J. Con n Y. Din. FlowMp: An optiml thnoloy mppin lorithm or ly optimiztion in lookup-tl s FPGA sins. IEEE Trns. on Computr-Ai Dsin, 13:1{11, 1994.

7 tst Initil ComMp SqMp xmpl #ts #FFs #LUTs #FFs #LUTs #FFs C C m omp ount u m i pm trm x mult s s s s totl Tl 1: Exprimntl rsults. [4] A. H. Frrhi n M. Srrzh. Complxity o th lookup-tl minimiztion prolm or FPGA thnoloy mppin. IEEE Trns. on Computr-Ai Dsin, 13:1319{1332, [5] R. J. Frnis, J. Ros, n K. Chun. Chortl: A thnoloy mppin or lookup tl-s l prormml t rrys. In Pro. ACM/IEEE Dsin Automtion Con., ps 613{619, [6] R. J. Frnis, J. Ros, n Z. Vrnsi. Chortl-r: Fst thnoloy mppin or lookup tl-s FPGAs. In Pro. ACM/IEEE Dsin Automtion Con., ps 227{ 233, [7] R. J. Frnis, J. Ros, n Z. Vrnsi. Thnoloy mppin or lookup tl-s FPGAs or prormn. In Dist Intl. Con. on Computr-Ai Dsin, ps 568{ 571, [8] K. Krplus. Xmp: A thnoloy mppr or tl-lookup FPGAs. In Pro. ACM/IEEE Dsin Automtion Con., ps 240{243, [9] E.L. Lwlr, K.N. Lvitt, n J. Turnr. Moul lustrin to minimiz ly in iitl ntworks. IEEE Trns. on Computrs, 18:47{57, [10] C. E. Lisrson, F. M. Ros, n J. B. Sx. Optimizin synhronous iruitry y rtimin. In Pro. 3r Clth Con. on VLSI, ps 87{116, [11] A. Mthur n C. L. Liu. Prormn rivn thnoloy mppin or lookup-tl s FPGAs usin th nrl ly mol. In ACM/SIGDA Workshop on Fil Prormml Gt Arrys, [12] R. Muri, R.K. Bryton, n A. Sniovnni-Vinntlli. Squntil synthsis or tl look up prormml t rrys. In Pro. ACM/IEEE Dsin Automtion Con., ps 224{229, [13] R. Muri, Y. Nishizki, N. Shnoy, R.K. Bryton, n A. Sniovnni-Vinntlli. Loi synthsis lorithms or tl look up prormml t rrys. In Pro. ACM/IEEE Dsin Automtion Con., ps 620{625, [14] R. Muri, N. Shnoy, R.K. Bryton, n A. Sniovnni- Vinntlli. Improv loi synthsis lorithms or tl look up rhitturs. In Dist Intl. Con. on Computr- Ai Dsin, ps 564{567, [15] P. Swkr n D. Thoms. Prormn irt thnoloy mppin or look-up tl s FPGAs. In Pro. ACM/IEEE Dsin Automtion Con., ps 208{212, [16] M. Shl, J. Kon, n P.K. Chn. Routility-rivn thnoloy mppin or lookup tl-s FPGA's. IEEE Trns. on Computr-Ai Dsin, 13:13{26, [17] U. Winmnn n W. Rosnstil. Thnoloy mppin or squntil iruits s on rtimin thniqus. In Pro. Europn Dsin Automtion Con., ps 318{323, [18] N.-S. Woo. A huristi mtho or FPGA thnoloy mppin s on th visiility. InPro. ACM/IEEE Dsin Automtion Con., ps 248{251, [19] Xilinx. Th Prormml Gt Arrys Dt Book. Xilinx, Sn Jos, CA, [20] H. Yn n D. F. Won. E-Mp: Optiml prormn rivn thnoloy mppin or itrtiv LUT s FPGA sins. In Dist Intl. Con. on Computr-Ai Dsin, ps 150{155, 1994.