Register Saturation in Superscalar and VLIW Codes

Transcription

1 Rgistr Sturtion in Suprslr n VLIW Cos Si Am Ali Touti INRIA, Domin Voluu, BP , L Csny x, Frn Si-Am-Ali.Touti@inri.r Astrt. T rgistrs onstrints n tn into ount uring t suling ps o n yli t pnn grp (DAG) : ny sul must minimiz t rgistr rquirmnt. In tis wor, w mtmtilly stuy n xtn t ppro wi onsists o omputing t xt uppr-oun o t rgistr n or ll t vli suls, inpnntly o t untionl unit onstrints. A prvious wor (URSA) ws prsnt in [5,4]. Its im ws to som sril rs to t originl DAG su tt t worst rgistr n os not x t numr o vill rgistrs. W writ n pproprit mtmtil ormlism or tis prolm n xtn t DAG mol to t into ount ly r rom n writ into rgistrs wit multipl rgistrs typs. Tis ormultion prmits us to provi in tis ppr ttr uristis n strtgis (nrly optiml), n w prov tt t URSA tniqu is not suiint to omput t mximl rgistr rquirmnt, vn i its solution is optiml. 1 Introution n Motivtion In Instrution Lvl Prlllism (ILP) ompilrs, o suling n rgistr llotion r two mor tss or o optimiztion. Co suling onsists o mximizing t xploittion o t ILP or y t o. On tor tt iniits su us is t rgistrs onstrints. A limit numr o rgistrs proiits n unoun numr o vlus simultnously liv. I t rgistr llotion is rri out or suling, ls pnnis r introu us o t rgistrs rus, prouing ngtiv impt on t vill ILP xpos to t sulr. I suling is rri out or, spill o migt introu us o n insuiint numr o rgistrs. A ttr ppro is to m o suling n rgistr llotion intrt wit otr in omplx omin pss, ming t rgistr llotion n t suling uristis vry orrlt. In tis rtil, w prsnt our ontriution to voiing n xssiv numr o vlus simultnously liv or ll t vli suls o DAG, prviously stui in [4,5]. Our pr-pss nlyzs DAG (wit rspt to ontrol low) to u t mximl rgistr n or ll suls. W ll tis limit t rgistr sturtion (RS) us t rgistr n n r tis limit ut nvr x it. W provi ttr uristis to omput n ru it i it xs t numr R. Willm (E.): CC 2001, LNCS 2027, pp , Springr-Vrlg Brlin Hilrg 2001

2 214 S.A.A. Touti o vill rgistrs y introuing nw rs. Exprimntl rsults sow tt in most ss our strtgis r nrly optiml. Tis rtil is orgniz s ollows. Stion 2 prsnts our DAG mol wi n us or ot suprslr n VLIW prossors (UAL n NUAL smntis [15]). T RS prolm is tortilly stui in St. 3. I it xs t numr o vill rgistrs, uristi or ruing it is givn in St. 4. W v implmnt sotwr tools, n xprimntl rsults r sri in St. 5. Som rlt wor in tis il is givn in St. 6. W onlu wit our rmrs n prsptivs in St DAG Mol A DAG G =(V,E,δ) in our stuy rprsnts t t pnns twn t oprtions n ny otr sril onstrints. E oprtion u s stritly positiv ltny lt(u). T DAG is in y its st o oprtions V, its st o rs E = {(u, v)/ u,v V }, n δ su tt δ() is t ltny o t r in trms o prossor lo yls. A sul σ o G is positiv untion wi givs n intgr xution (issu) tim or oprtion : σ is vli =(u, v) E σ(v) σ(u) δ() W not y Σ(G) t st o ll t vli suls o G. Sin writing to n ring rom rgistrs oul ly rom t ginning o t oprtion sul tim (VLIW s), w in t two ly untions δ r n δ w su tt δ w (u) is t writ yl o t oprtion u, n δ r (u) is t r yl o u. In otr wors, u rs rom t rgistr il t instnt σ(u)+δ r (u), n writs in it t instnt σ(u)+δ w (u). To simpliy t writing o som mtmtil ormuls, w ssum tt t DAG s on sour ( ) n on sin ( ). I not, w introu two ititious nos (, ) rprsnting nops (vit t t n o t RS nlysis). W virtul sril r 1 =(,s) to sour wit δ( 1 ) = 0, n n r 2 =(t, ) rom sin wit t ltny o t sin oprtion δ( 2 )=lt(t). T totl sul tim o sul is tn σ( ). T null ltny o n r 1 is not inonsistnt wit our ssumption tt ltnis must stritly positiv us t virtul sril rs no longr rprsnt t pnnis. Furtrmor, w n voi introuing ts virtul nos witout ny onsqun on our tortil stuy sin tir purpos is only to simpliy som mtmtil xprssions. Wn stuying t rgistr n in DAG, w m irn twn t nos, pning on wtr ty in vlu to stor in rgistr or not, n lso pning on wi rgistr typ w r ousing on (int, lot, t.). W lso m irn twn gs, pning on wtr ty r low pnnis troug t rgistrs o t typ onsir : V R V is t sust o oprtions wi in vlu o t typ unr onsirtion (int, lot, t.), w simply ll tm vlus. W ssum tt

3 Rgistr Sturtion in Suprslr n VLIW Cos 215 t most on vlu o t typ onsir n in y n oprtion. T oprtions wi in multipl vlus r tn into ount i ty in t most on vlu o t typ onsir. E R E is t sust o rs rprsnting tru pnnis troug vlu o t typ onsir. W ll tm low rs. E S = E E R r ll sril rs. Figur 1. givs t DAG tt w us in tis ppr onstrut rom t o o prt (). In tis xmpl, w ous on t loting point rgistrs : t vlus n low rs r sown y ol lins. W ssum or instn tt r ours xtly t t sul tim n writ t t inl xution stp (δ r (u) =0,δ w (u) =lt(u) 1). () lo [i1], R () lo [i2], R () lo [i3], R () mult R, R, R () imult R, R, R, ir (g) toint R, ir g (i) i ir g, 4, ir i () mult stz R, ir i, R, R,g () iv R, ir, R () g? stnz R, 1, R, g () g g? su R, 1, R g i () o or suling n rgistr llotion () t DDG G () PK(G) Fig. 1. DAG mol Nottion n Dinitions on DAGs In tis ppr, w us t ollowing nottions or givn DAG G =(V,E): Γ + G (u) ={v V/(u, v) E} sussors o u ; Γ G (u) ={v V/(v, u) E} prssors o u ; =(u, v) E sour() =u trgt() =v. u, v r ll npoints ; u, v V : u<v pt (u,...,v)ing; u, v V : u v (u <v) (v <u). u n v r si to prlll ; u V u = {v V/v = u v<u} u s snnts inluing u ; u V u = {v V/v = u u<v} u s snnts inluing u. two rs, r nt i ty sr n npoint ; A V is n ntiin in G u, v A u v ; AM is mximl ntiin A ntiin in G A AM ; t xtn DAG G\ E o G gnrt y t rs st E V 2 is t grp otin rom G tr ing t rs in E. As onsqun, ny vli sul o G is nssrily vli sul or G : G = G\ E = Σ(G ) Σ(G)

4 216 S.A.A. Touti lt I 1 =[ 1, 1 ] N n I 2 =[ 2, 2 ] N two intgr intrvls. W sy tt I 1 is or I 2, not y I 1 I 2, i 1 < 2. 3 Rgistr Sturtion 3.1 Rgistr N o Sul Givn DAG G =(V,E,δ), vlu u V R is liv ust tr t writing lo yl o u until its lst ring (onsumption). T vlus wi r not r in G or r still liv wn xiting t DAG must pt in rgistrs. W nl ts vlus y onsiring tt t ottom no onsums tm. W in t st o onsumrs or vlu u V R s { {v V/(u, v) ER } i (u, v) E Cons(u) = R otrwis Givn sul σ Σ(G), t lst onsumption o vlu is ll t illing t n not : u V R ill σ (u) = mx v Cons(u) ( σ(v)+δr (v) ) All t onsumrs o u wos ring tim is qul to t illing t o u r ll t illrs o u. W ssum tt vlu writtn t instnt t in rgistr is vill on stp ltr. Tt is to sy, i oprtion u rs rom rgistr t instnt t wil oprtion v is writing in it t t sm tim, u os not gt v s rsult ut gts t vlu prviously stor in tis rgistr. Tn, t li intrvl L σ u o vlu u oring to σ is ]σ(u)+δ w (u), ill σ (u)]. Givn t li intrvls o ll t vlus, t rgistr n o σ is t mximum numr o vlus simultnously liv : RN σ (G) = mx vs σ(i) 0 i σ( ) wr vs σ (i) ={u V R /i L σ u} is t st o vlus liv t tim i Builing rgistr llotion (ssign pysil rgistr to vlu) wit R vill rgistrs or sul wi ns R rgistrs n sily on wit polynomil-omplxity lgoritm witout introuing spill o nor inrsing t totl sul tim [16]. 3.2 Rgistr Sturtion Prolm T RS is t mximl rgistr n or ll t vli suls o t DAG : RS(G) = mx RN σ(g) σ Σ(G) W ll σ sturting sul i RN σ (G) =RS(G). In tis stion, w stuy ow to omput RS(G). W will s tt tis prolm oms own to

5 Rgistr Sturtion in Suprslr n VLIW Cos 217 nswring t qustion wi oprtion must ill tis vlu? Wn looing or sturting suls, w o not worry out t totl sul tim. Our im is only to prov tt t rgistr n n r t RS ut nnot x it. Minimizing t totl sul tim is onsir in St. 4 wn w ru t RS. Furtrmor, or t purpos o uiling sturting suls, w v provn in [16] tt to mximiz t rgistr n, looing or only on suitl illr o vlu is suiint rtr tn looing or group o illrs : or ny sul tt ssigns mor tn on illr or vlu, w n oviously uil notr sul wit t lst t sm rgistr n su tt tis vlu is ill y only on onsumr. So, t purpos o tis stion is to slt suitl illr or vlu to sturt t rgistr rquirmnt. Sin w o not ssum ny sul, t li intrvls r not in so w nnot now t wi t vlu is ill. Howvr, w n u wi onsumrs in Cons(u) r impossil illrs or t vlu u. Iv 1,v 2 Cons(u) n pt (v 1 v 2 ), v 1 is lwys sul or v 2 wit t lst lt(v 1 ) prossor yls. Tn v 1 n nvr t lst r o u (rmmr tt w ssum stritly positiv ltnis). W n onsquntly u wi onsumrs n potntilly ill vlu (possil illrs). W not pill G (u) t st o t oprtions wi n ill vlu u V R : pill G (u) = { v Cons(u)/ v Cons(u) ={v} } On n tt ll oprtions in pill G (u) r prlll in G. Any oprtion wi os not long to pill G (u) n nvr ill t vlu u. Lmm 1. Givn DAG G =(V,E,δ), tn u V R σ Σ(G) v pill G (u) : σ(v)+δ r (v) =ill σ (u) (1) v pill G (u) σ Σ(G) : ill σ (u) =σ(v)+δ r (v) (2) Proo. A omplt proo is givn in [17], pg 13. A potntil illing DAG o G, not PK(G) =(V,E PK ), is uilt to mol t potntil illing rltions twn oprtions, (s Fig. 1.), wr : E PK = {(u, v)/ u V R v pill G (u)} Tr my mor tn on oprtion nit or illing vlu. Lt us gin y ssuming illing untion wi nors n oprtion v pill G (u) to t illr o u V R. I w ssum tt (u) is t uniqu illr o u V R,w must lwys vriy t ollowing ssrtion : v pill G (u) {(u)} σ(v)+δ r (v) <σ ( (u) ) + δ r ( (u) ) (3) Tr is mily o suls wi nsurs tis ssrtion. To in tm, w xtn G y nw sril rs tt nor ll t potntil illing oprtions o vlu u to sul or (u). Tis ls us to in n xtn DAG ssoit to not G = G\ E wr : ( ) } E = {=(v,(u))/u V R v pill G (u) {(u)} witδ()=δ r (v) δ r (u) +1

6 218 S.A.A. Touti g 1 1 i 1 () PK(G) wit () G () DV (G) () B(G) Fig. 2. Vli illing untion n iprtit omposition Tn, ny sul σ Σ(G ) nsurs Proprty 3. T onition o t xistn o su sul ins t onition o vli illing untion : is vli illing untion G is yli Figur 2 givs n xmpl o vli illing untion. Tis untion is sown y ol rs in prt (), wr trgt ills its sours. Prt () is t DAG ssoit to. Provi vli illing untion, w n u t vlus wi n nvr simultnously liv or ny σ Σ(G ). Lt R (u) = u V R t st o t snnt vlus o u V. Lmm 2. Givn DAG G =(V,E,δ) n vli illing untion, tn : 1. t snnt vlus o (u) nnot simultnously liv wit u : u V R σ Σ(G ) v R (u) L u σ L v σ (4) 2. tr xists vli sul wi ms t otr vlus non snnt o (u) simultnously liv wit u, i.. u V R σ Σ(G ) : v R v R (u) L u σ L v σ φ (5) v pill G (u) Proo. A omplt proo is givn in [17], pg 23. W in DAG wi mols t vlus tt n nvr simultnously liv oring to. T isoint vlu DAG o G ssoit to, n not DV (G) =(V R,E DV ) is in y : E DV = { (u, v)/u, v V R v R (u) }

7 Rgistr Sturtion in Suprslr n VLIW Cos 219 Any r (u, v) indv (G) mns tt u s li intrvl is lwys or v s li intrvl oring to ny sul o G, s prt Fig Tis inition prmits us to stt in t ollowing torm tt t rgistr n o ny sul o G is lwys lss tn or qul to mximl ntiin in DV (G). Also, tr is lwys sul wi ms ll t vlus in tis mximl ntiin simultnously liv. Torm 1. Givn DAG G =(V,E,δ) n vli illing untion tn : σ Σ(G ):RN σ (G) AM σ Σ(G ):RN σ (G) = AM wr AM is mximl ntiin in DV (G) Proo. A omplt proo is givn in [17], pg 18. Torm 1 llows us to rwrit t RS ormul s RS(G) = mx AM vli illing untion wr AM is mximl ntiin in DV (G). W rr to t prolm o ining su illing untion s t mximizing mximl ntiin prolm (MMA). W ll solution or t MMA prolm sturting illing untion, n AM its sturting vlus. Unortuntly, w v provn in [17] (pg 24) tt ining sturting illing untion is NP-omplt. 3.3 A Huristi or Computing t RS Tis stion prsnts our uristis to pproximt n optiml y notr vli illing untion. W v to oos illing oprtion or vlu su tt w mximiz t prlll vlus in DV (G). Our uristis ous on t potntil illing DAG PK(G), strting rom sour nos to sins. Our im is to slt group o illing oprtions or group o prnts to p s mny snnt vlus liv s possil. T min stps o our uristis r : 1. ompos t potntil illing DAG PK(G) into onnt iprtit omponnts ; 2. or iprtit omponnt, sr or t st sturting illing st (in low) ; 3. oos illing oprtion witin t sturting illing st (in low). W ompos t potntil illing DAG into onnt iprtit omponnts (CBC) in orr to oos ommon sturting illing st or group o prnts. Our purpos is to v mximum numr o ilrn n tir snnts vlus simultnously liv wit tir prnts vlus. A CBC =(S,T,E ) is prtition o sust o oprtions into two isoint sts wr : 1 Tis DAG is simplii y trnsitiv rution.

8 220 S.A.A. Touti E E PK is sust o t potntil illing rltions ; S V R is t st o t prnt vlus, su tt prnt is ill y t lst on oprtion in T ; T V is t st o t ilrn, su tt ny oprtion in T n potntilly ill t lst vlu in S. A iprtit omposition o t potntil illing grp PK(G) is t st (s Fig. 2.) B(G) ={ =(S,T,E )/ E PK B(G) : E } Not tt B(G) s, s S t, t T s s t t in PK(G). A sturting illing st SKS() o iprtit omponnt =(S,T,E )is sust T T su tt i w oos illing oprtion rom tis sust, tn w gt mximl numr o snnt vlus o ilrn in T simultnously liv wit prnt vlus in S. Dinition 1 (Sturting Killing St). Givn DAG G =(V,E,δ), sturting illing st SKS() o onnt iprtit omponnt B(G) is sust T T, su tt : 1. illing onstrints : t T Γ (t) =S 2. minimizing t numr o snnt vlus o T min R t t T Unortuntly, omputing SKS is lso NP-omplt ([17], pg 81). A Huristi or Fining SKS Intuitivly, w soul oos sust o ilrn in iprtit omponnt tt woul ill t grtst numr o prnts wil minimizing t numr o snnt vlus. W in ost untion ρ tt nls us to oos t st nit il. Givn iprtit omponnt =(S,T,E ) n st Y o (umult) snnt vlus n st X o non (yt) ill prnts, t ost o il t T is : ρ X,Y (t) = Γ (t) X R t Y i R t Y φ Γ (t) X otrwis T irst s nls us to slt t il wi ovrs t most non ill prnts wit t minimum snnt vlus. I tr is no snnt vlu, tn w oos t il tt ovrs t most non ill prnts. Algoritm 1 givs moii gry uristi tt srs or n pproximtion SKS n omputs illing untion in polynomil tim. Our uristi s t ollowing proprtis.

9 Rgistr Sturtion in Suprslr n VLIW Cos 221 Algoritm 1 Gry-: uristis or t MMA prolm Rquir: DAG G =(V,E,δ) or ll vlus u V R o (u) = {ll vlus r initilly non ill} n or uil B(G) t iprtit omposition o PK(G). or ll iprtit omponnt =(S,T,E ) B(G) o X := S {ll prnts r initilly unovr} Y := φ {initilly, no umult snnt vlus} SKS () :=φ wil X φ o {uil t SKS or } slt t il t T wit t mximl ost ρ X,Y (t) SKS () :=SKS () {t} X := X Γ (t){rmov ovr prnts} Y := Y R t {upt t umult snnt vlus} n wil or ll t SKS () o {in rsing ost orr} or ll prnt s Γ (t) o i (s) = tn {ill non ill prnts o t} (s) :=t n i n or n or n or Torm 2. Givn DAG G =(V,E,δ), tn : 1. Gry- lwys prous vli illing untion ; 2. PK(G) is n invrt tr = Gry- is optiml. Proo. Complt proos or ot (1) n (2) r givn in [17], pgs 31 n 44 rsp. Sin t pproximt illing untion is vli, Torm 1 nsurs tt w n lwys in vli sul wi rquirs xtly AM rgistrs. As onsqun, our uristi os not omput n uppr oun o t optiml rgistr sturtion n tn t optiml RS n grtr tn t on omput y Gry-. A onsrvtiv uristi wi omputs solution xing t optiml RS nnot nsur t xistn o vli sul wi rs t omput limit, n n it woul imply n osolt RS rution pross n wst o rgistrs. T vliity o illing untion is y onition us it nsurs tt tr xists rgistr llotion wit xtly AM rgistrs. As summry, r r our stps to omput t RS : 1. pply Gry- on G. T rsult is vli illing untion ; 2. onstrut t isoint vlu DAG DV (G); 3. in mximl ntiin AM o DV (G) using Dilwort omposition [10] ; Sturting vlus r tn AM n RS (G) = AM RS(G).

10 222 S.A.A. Touti 2/2 2/3 3/1 1/1 4 () PK(G) wit () DV (G) Fig. 3. Exmpl o omputing t rgistr sturtion Figur 3. sows sturting illing untion omput y Gry- : ol rs not tt trgt ills its sours. E illr is ll y its ost ρ. Prt () givs t isoint vlu DAG ssoit to. T Sturting vlus r {,,,,,, }, so t RS is 7. 4 Ruing t Rgistr Sturtion In tis stion w uil n xtn DAG G = G\ E su tt t RS is limit y stritly positiv intgr (numr o vill rgistrs) wit t rspt o t ritil pt. Lt R tis limit. Tn : σ Σ(G) :RN σ (G) RS(G) R W v provn in [16] tt tis prolm is NP-r. In tis stion w prsnt uristi tt s sril rs to prvnt som sturting vlus in AM (oring to sturting illing untion ) rom ing simultnously liv or ny sul. Also, w must r to not inrs t ritil pt i possil. Srilizing two vlus u, v V R mns tt t ill o u must lwys rri out or t inition o v, or vi-vrs. A vlu sriliztion u v or two vlus u, v V R is in y : i v pill G (u) tn t sril rs { =(v,v)/ v pill G (u) {v} wit δ() =δ r (v ) δ w (v) } ls t sril rs { =(u,v)/ u pill G (u) (v <u ) wit δ() =δ r (u ) δ w (v) } To o not violt t DAG proprty (w must not introu yl), som sriliztions must iltr out. T onition or pplying u v is tt v pill G (u) : (v <v ). W os t st sriliztion witin t st o ll t possil ons y using ost untion ω(u v) =(ω 1,ω 2 ), su tt : ω 1 = µ 1 µ 2 is t prition o t rution otin witin t sturting vlus i w rry out tis vlu sriliztion, wr

11 Rgistr Sturtion in Suprslr n VLIW Cos 223 µ 1 is t numr o sturting vlus sriliz tr u i w rry out t sriliztion ; µ 2 is t prit numr o u s snnt vlus tt n om simultnously liv wit u ; ω 2 is t inrs in t ritil pt. Our uristi is sri in Algoritm 2. It itrts t vlu sriliztions witin t sturting vlus until w gt t limit R or until no mor sriliztions r possil (or non is xpt to ru t RS). On n tt i tr is no possil vlu sriliztion in t originl DAG, our lgoritm xits t t irst itrtion o t outr wil-loop. I it sus, tn ny sul o G ns t most R rgistrs. I not, it still rss t originl RS, n tus limits t rgistr n. Introuing n minimizing t spill o is notr NP-omplt prolm stui in [8,3,2,9,14] n not rss in tis wor. Now, w xplin ow to omput t prition prmtrs µ 1,µ 2,ω 2.W not G i t xtn DAG o stp i, i its sturting untion, n AM i its sturting vlus n Ri u t snnt vlus o u in G i : 1. (u v) nsurs tt i+1 (u) <vin G i+1. Aoring to Lmm 2, µ 1 = Ri v AM i is t numr o sturting vlus in G i wi nnot simultnously liv wit u in G i+1 ; 2. nw sturting vlus oul introu into G i+1 :iv pill Gi (u), w or i+1 (u) =v. Aoring to Lmm 2, µ 2 = Ri v Ri v v pill Gi (u) is t numr o vlus wi oul simultnously liv wit u in G i+1. µ 2 = 0 otrwis ; 3. i w rry out (u v) ing i, t introu sril rs oul nlrg t ritil pt. Lt lp i (v,v) t longst pt going rom v to v in G i. T nw longst pt in G i+1 going troug t sriliz nos is : mx introu =(v,v) δ()>lp i(v,v) lp i (,v )+lp i (v, )+δ() I tis pt is grtr tn t ritil pt in G i, tn ω 2 is t irn twn tm, 0 otrwis. At t n o t lgoritm, w pply gnrl vriition stp to nsur t potntil illing proprty provn in Lmm 1 or t originl DAG. W v provn in Lmm 1 tt t oprtions wi o not long to pill G (u) nnot ill t vlu u. Atr ing t sril rs to uil G, w migt violt tis ssrtion us w introu som rs wit ngtiv ltnis. To ovrom tis prolm, w must gurnt t ollowing ssrtion : u V R, v Cons(u) pill G (u) v pill G (u)/v <vin G = lp G (v,v) >δ r (v ) δ r (v) (6)

12 224 S.A.A. Touti Algoritm 2 Vlu Sriliztion Huristi Rquir: DAG G =(V,E,δ) n stritly positiv intgr R G := G omput AM, sturting vlus o G ; wil AM > R o onstrut t st U o ll missil sriliztions twn sturting vlus in AM wit tir osts (ω 1,ω 2); i /(u v) U/ω 1(u v) > 0 tn {no mor possil RS rution} xit ; n i X := {(u v) U/ω 2(u v) =0}{t st o vlu sriliztions tt o not inrs t ritil pt} i X φ tn oos vlu sriliztion (u v) inx wit t minimum ost R ω 1 ; ls oos vlu sriliztion (u v) inx wit t minimum ost ω 2 ; n i rry out t sriliztion (u v) ing ; omput t nw sturting vlus AM o G ; n wil nsur potntil illing oprtions proprty { longst pts twn pill oprtions} In t, tis prolm ours i w rt pt in G rom v to v wr v, v pill G (u). I ssrtion (6) is not vrii, w sril r =(v,v) wit δ() =δ r (v ) δ r (v) + 1 s illustrt in Fig initil DAG (1) (2) (3) nsur pill proprty or n Fig. 4. C Potntil Killrs Proprty Exmpl 1. Figur 5 givs n xmpl or ruing t RS o our DAG rom 7 to 4 rgistrs. W rmin tt t sturting vlus o G r AM = {,,,,,, }. Prt () sows ll t possil vlu sriliztions witin ts sturting vlus. Our uristi slts s nit, sin it is xpt to limint 3 sturting vlus witout inrsing t ritil pt. T mximl introu longst pt troug tis sriliztion is (,,,,, ) = 8, wi is lss tn t originl ritil pt (26). T xtn DAG G is prsnt in prt () wr t vlu sriliztion is introu : w t t sril rs (, ) n (, ) wit -4 ltny. Finlly, w t sril rs (, ) n

13 Rgistr Sturtion in Suprslr n VLIW Cos g i () ll t possil vlu sriliztions () G () PK(G) wit () DV (G) Fig. 5. Ruing rgistr sturtion (, ) wit unit ltny to nsur t pill G () proprty. T wol ritil pt os not inrs n RS is ru to 4. Prt () givs sturting illing untion or G, sown wit ol rs in PK(G). DV (G) is prsnt in prt () to sow tt t nw RS oms 4 loting point rgistrs. 5 Exprimnttion W v implmnt t RS nlysis using t LEDA rmwor. W rri out our xprimnts on vrious loting point numril loops tn rom vrious nmrs (livrmor, wtson, sp-p, t.). W ous in ts os on t loting point rgistrs. T irst xprimnttion is vot to ing Gry iiny. For tis purpos, w v in n implmnt in [16] n intgr linr progrmming mol to omput t optiml RS o DAG. W us CPLEX to rsolv ts linr progrmming mols. T totl numr o xprimnt DAGs is 180, wr t numr o nos gos up to 120 n t numr o vlus gos up to 114. Exprimntl rsults sow tt our uristis giv qusi-optiml solutions. T worst xprimntl rror is 1, wi mns tt t optiml RS is in worst s grtr y on rgistr tn t on omput y Gry-. T son xprimnttion is vot to ing t iiny o t vlu sriliztion uristis in orr to ru t RS. W v lso in n implmnt in [16] n intgr linr progrmming mol to omput t optiml rution o t RS wit minimum ritil pt inrs (NP-r prolm). T totl numr o xprimnt DAGs is 144, wr t numr o nos gos up to 80 n t numr o vlus gos up to 76. In lmost ll ss, our uristis mngs to gt t optiml solutions. Optiml ru RS ws in t worst ss lss y on rgistr tn our uristis rsults. Sin RS omputtion in t vlu sriliztion uristis is on y Gry-, w its worst xprimntl rror (1 rgistr) wi ls to totl mximl rror o two rgistrs. All optiml vs. pproximt rsults r ully til in [16]. Sin our strtgis rsult in goo iiny, w us tm to stuy t RS vior in DAGs. Exprimnttion on only loop ois sows tt t RS is low, rnging rom 1 to 8. W v unroll ts loops wit irnt

14 226 S.A.A. Touti unrolling tors going up to 20 tims. T im o su unrolling is to gt lrg DAGs, inrs t rgistrs prssur n xpos mor ILP to i mmory ltnis. W rri out wi rng o o xprimnts to stuy t RS n its rution wit wit vrious limits o vill rgistrs (going rom 1 up to 64). W xprimnt 720 DAGs wr t numr o o nos gos up to 400 n t numr o vlus gos up to 380. Full rsults r til in [17,16]. T irst rmr u rom our ull xprimnts is tt t RS is lowr tn t numr o vill rgistrs in lot o ss. T RS nlysis ms it possil to voi t rgistrs onstrints in o suling : most o ts os n sul witout ny intrtion wit t rgistr llotion, wi rss t ompil-tim omplxity. Son, in most ss our uristis sus in ruing it until ring t trgt limit. In w ss w los som ILP us o t intrinsi rgistr prssur o t DAGs : ut sin spill o rss t prormn rmtilly us o t mmory ss ltnis, tro twn spilling n inrsing t ovrll sul tim n on in w ritil ss. Finlly, in t ss wr t RS is lowr tn t numr o vill rgistrs, w n us t xtr non us rgistrs y ssigning to tm som glol vrils n rry lmnts wit t gurnt tt no spill o oul introu tr y t sulr n t rgistr llotor. 6 Rlt Wor n Disussion Comining o suling n rgistr llotion in DAGs ws stui in mny wors. All t tniqus sri in [11,6,13,7,12] us tir uristis to uil n optimiz sul witout xing rtin limit o vlus simultnously liv. T ul notion o t RS, ll t rgistr suiiny, ws stui in [1]. Givn DAG, t utors gv uristi wi oun t minimum rgistr n ; t omputtion ws O(log 2 V ) tor o t optiml. Not tt w n sily us t RS rution to omput t rgistr suiiny. Tis is on in prti y stting R = 1 s t trgt limit or t RS rution. Our wor is n xtnsion to URSA [4,5]. T minimum illing st tniqu tri to sturt t rgistr rquirmnt in DAG y ping t vlus liv s lt s possil : t utors pro y ping s mny ilrn liv s possil in iprtit omponnt y omputing t minimum st wi ill ll t prnt s vlus. First, sin t utors i not ormliz t RS prolm, w n sily giv xmpls to sow tt minimum illing st os not sturt t rgistr n, vn i t solution is optiml [17]. Figur. 6 sows n xmpl wr t RS omput y our uristis (Prt ) is 6 wr t optiml solution or URSA yils RS o 5 (prt ). Tis is us URSA i not t into ount t snnt vlus wil omputing t illing sts. Son, t vliity o t illing untions is n importnt onition to omput t RS n unortuntly ws not inlu in URSA. W v provn in [17] tt non vli illing untions n xist i no r is tn. Finlly, t URSA DAG

15 Rgistr Sturtion in Suprslr n VLIW Cos 227 mol i not irntit twn t typs o t vlus n i not t into ount lys in rs rom n writs into t rgistrs il. g i g i i g () originl DAG () DV (G) su {, } () DV (G) su {} is t sturting illing st is t optiml minimum illing st Fig. 6. URSA rw 7 Conlusion n Futur Wor In our wor, w mtmtilly stuy n in t RS notion to mng t rgistrs prssur n voi spill o or t suling n rgistr llotion psss. W xtn URSA y ting into ount t oprtions in ot Unit n Non Unit Assum Ltnis (UAL n NUAL [15]) smntis wit irnt typs (vlus n non vlus) n vlus (lot, intgr, t.). T orml mtmtil moling n tortil stuy prmit us to giv nrly optiml strtgis n prov tt t minimum illing st is insuiint to omput t RS. Exprimnttions sow tt t rgistrs onstrints n osolt in mny os, n my tror ignor in orr to simpliy t suling pross. T uristis w us mng to ru t RS in most ss wil som ILP is lost in w DAGs. W tin tt ruing t RS is ttr tn minimizing t rgistr n : tis is us minimizing t rgistr n inrss t rgistr rus, n t ILP loss must inrs s onsqun. Our DAG mol is suiintly gnrl to mt ll urrnt rittur proprtis (RISC or CISC), xpt or som ritturs wi support issuing pnnt instrutions t t sm lo yl, wi woul rquir rprsnttion using null ltny. Stritly positiv ltnis r ssum to prov t pill oprtion proprty (Lmm 1) wi is importnt to uil our uristis. W tin tt tis rstrition soul not mor rw nor n importnt tor in prormn grtion, sin null ltny oprtions o not gnrlly ontriut to t ritil xution pts. In t utur, w will xtn our wor to loops. W will stuy ow to omput n ru t RS in t s o yli suls li sotwr piplining (SWP) wr t li intrvls om irulr.

16 228 S.A.A. Touti Rrns 1. A. Agrwl, P. Klin, n R. Rvi. Orring Prolms Approximt : Rgistr Suiiny, Singl Prossor Suling n Intrvl Grp Compltion. intrnl rsr rport CS-91-18, Brown Univrsity, Provin, Ro Isln, Mr P. Brgnr, P. Dl, D. Engrtsn, n M. O K. Spill Co Minimiztion vi Intrrn Rgion Spilling. ACM SIG-PLAN Notis, 32(5): , My D. Brnstin, D. Q. Golin, M. C. Golumi, H. Krwzy, Y. Mnsour, I. Nson, n R. Y. Pintr. Spill Co Minimiztion Tniqus or Optimizing Compilrs. SIGPLAN Notis, 24(7): , July Proings o t ACM SIGPLAN 89 Conrn on Progrmming Lngug Dsign n Implmnttion. 4. D. Brson, R. Gupt, n M. So. URSA: A unii RSour llotor or rgistrs n untionl units in VLIW ritturs. In Conrn on Aritturs n Compiltion Tniqus or Fin n Mium Grin Prlllism, pgs , Orlno, Flori, Jn D. A. Brson. Uniition o Rgistr Allotion n Instrution Suling in Compilrs or Fin-Grin Prlll Arittur. PD tsis, Pittsurg Univrsity, D. G. Brl, S. J. Eggrs, n R. R. Hnry. Intgrting Rgistr Allotion n Instrution Suling or RISCs. ACM SIGPLAN Notis, 26(4): , Apr T. S. Brsir. FRIGG: A Nw Appro to Comining Rgistr Assignmnt n Instrution Suling. Mstr tsis, Miign Tnologil Univrsity, D. Clln n B. Kolnz. Rgistr Allotion vi Hirril Grp Coloring. SIGPLAN Notis, 26(6): , Jun Proings o t ACM SIGPLAN 91 Conrn on Progrmming Lngug Dsign n Implmnttion. 9. G. J. Citin. Rgistr llotion n spilling vi grp oloring. ACM SIG-PLAN Notis, 17(6):98 105, Jun P. Crwly n R. P. Dilwort. Algri Tory o Lttis. Prnti Hll, Englwoo Clis, J. R. Goomn n W.-C. Hsu. Co Suling n Rgistr Allotion in Lrg Bsi Blos. In Conrn Proings 1988 Intrntionl Conrn on Supromputing, pgs , St. Mlo, Frn, July C. Norris n L. L. Pollo. A Sulr-Snsitiv Glol Rgistr Allotor. In IEEE, itor, Supromputing 93 Proings: Portln, Orgon, pgs , 1109 Spring Strt, Suit 300, Silvr Spring, MD 20910, USA, Nov IEEE Computr Soity Prss. 13. S. S. Pintr. Rgistr Allotion wit Instrution Suling: A Nw Appro. SIGPLAN Notis, 28(6): , Jun M. Poltto n V. Srr. Linr sn rgistr llotion. ACM Trnstions on Progrmming Lngugs n Systms, 21(5): , Spt Slnsr, B. Ru, n S. Ml. Aiving Hig Lvls o instrution-lvl Prlllism wit Ru Hrwr Complxity. Tnil Rport HPL , Hwlt Pr, S.-A.-A. Touti. Optiml Rgistr Sturtion in Ayli Suprslr n VLIW Cos. Rsr Rport, INRIA, Nov tp.inri.r/inria/prots/3/touti/optirs.ps.gz. 17. S.-A.-A. Touti n F. Tomsst. Rgistr Sturtion in Dt Dpnn Grps. Rsr Rport RR-3978, INRIA, July tp.inri.r/inria/pulition/puli-ps-gz/rr/rr-3978.ps.gz.