An Effective Heuristic for Simple Offset Assignment with Variable Coalescing

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1 An Etiv Huristi or Simpl Ost Assinmnt with Vribl Colsin Hssn Slmy n J. Rmnujm Dprtmnt o Eltril n Computr Eninrin n Cntr or Computtion n Thnoloy Louisin Stt Univrsity, Bton Rou, LA 7080, USA {hslm,jxr}@.lsu.u Abstrt. In mny Diitl Sinl Prossors (DSPs) with limit mmory, prorms r lo in th ROM n thus it is vry importnt to optimiz th siz o th o to ru th mmory rquirmnt. Mny DSP prossors inlu rss nrtion units (AGUs) tht n prorm rss rithmti (uto-inrmnt n uto-rmnt) in prlll to instrution xution, n without th n or xtr instrutions. Muh rsrh hs bn onut to optimiz th lyout o th vribls in mmory to t th most bnit rom uto-inrmnt n utormnt. Th simpl ost ssinmnt (SOA) problm onrns th lyout o vribls or mhins with on rss ristr n th nrl ost ssinmnt (GOA) ls with multipl rss ristrs. Both ths problms ssum tht h vribl ns to b llot or th ntir urtion o prorm. Both SOA n GOA r NP-omplt. In this ppr, w prsnt huristi or SOA tht onsirs olsin two or mor non-intrrin vribls into th sm mmory lotion. SOA with vribl olsin is intn to rs th ost o rss rithmti instrutions s wll s to rs th mmory rquirmnt or vribls by mximizin th numbr o vribls mpp to th sm mmory slot. Rsults on svrl bnhmrks show th siniint improvmnt o our solution ompr to othr huristis. In ition, w hv pt simult nnlin to urthr improv th solution rom our huristi. Introution Emb prossors r oun in mny ltroni vis suh s tlphons, mrs, n lultors. Du to th tiht onstrints on th sin o mb systms, mmory is usully limit. In ontrst, th mmory rquirmnt or th xution o iitl sinl prossin n vio prossin os on n mb systm is siniint. Morovr, sin th prorm o rsis in th on-hip ROM, th siz o th o irtly trnslts into silion. So o minimiztion boms substntil ol in orr to optimiz th mount o mmory n. Mny Diitl Sinl Prossors (DSPs) suh s th TI Cx/C5x, Motorol 56xxx, Anlo Dvis 0x n ST D950 hv rss nrtion units (AGUs) [5]. Th AGU is rsponsibl or lultin th tiv rss. A typil AGU

2 Hssn Slmy n J. Rmnujm onsists o n rss ristr il n moiy ristr il s shown in Fiur. Th rhitturs o suh DSPs support only inirt mmory rssin. Sin th bs-plus-ost rssin mo is not support, n xtr instrution is n, in nrl, to (subtrt) n ost to (rom) th urrnt rss in th rss ristr to omput th nw rss. Howvr, suh rhitturs support uto-inrmnt n uto-rmnt o th rss ristr. Whn thr is n to n ost o or subtrt n ost o rom th urrnt rss, this n b on in prlll with th sm LOAD/STORE instrution usin uto-inrmnt or uto-rmnt; n this os not rquir n xtr rss rithmti instrution in th o. Exploitin this hrtristi will l to o omption n thus lss mmory us sin th lnth o th o in DSP irtly trnslts into rquir silion r. On mtho or minimizin th instrutions n or rss omputtion is to prorm o st ssinmnt o th vribls. Ost ssinmnt rrs to th problm o plin th vribls in th mmory to mximlly utiliz uto-inrmnt/rmnt n thus ru o siz. AR pointr Immit Constnt MR pointr q r = Arss Ristr Fil +/- Moiy Ristr Fil Etiv Arss Fi.. A typil Arss Gnrtion Unit (AGU) ontins moiy ristr il, rss ristr il n ALU Simpl ost ssinmnt (SOA) rrs to th s whr thr is only on rss ristr (AR), whrs nrl ost ssinmnt (GOA) rrs to th s whr thr r multipl rss ristrs []. In both SOA n GOA onsir in this ppr, th vlu o uto-inrmnt/rmnt is ; SOA n GOA r NP-omplt []. Svrl rsrhrs hv stui th ost ssinmnt problm n hv propos irnt huristis. In this ppr, w prsnt n tiv huristi or th simpl ost ssinmnt problm with vribl olsin. Colsin llows two or mor vribls to shr th sm mmory lotion provi tht thir liv rns o not ovr-

3 An Etiv Huristi or Simpl Ost Assinmnt lp. Bs on th liv rns o ll th vribls, n intrrn rph (IG) is onstrut in whih n (, b) inits tht vribls n b intrr n thus thy n not b mpp into th sm mmory lotion. Vribl olsin improvs th rsults by rsin th numbr o rss rithmti instrutions n s wll s th mmory rquirmnt or storin th vribls. Th rminr o th ppr is orniz s ollows. Stion prsnts rlt work in this r. Stion prsnts our lorithm or simpl ost ssinmnt with vribl olsin. Stion 4 ivs n xmpl tht shows how our lorithm works. Stion 5 prsnts th simult nnlin lorithm to urthr improv th rsults. Stion 6 summrizs th rsults. Finlly Stion 7 prsnts our onlusions. Rlt Work Th problm o simpl ost ssinmnt ws irst isuss by Brtly []. Thn Lio t l. [] show tht th SOA problm is NP-omplt n tht it is quivlnt to th Mximum Wiht Pth Covr (MWPC) problm. Thy propos huristis or both SOA n GOA. Givn n ss squn o th vribls, th ss rph hs no or h vribl with n o wiht w btwn nos n b mnin tht vribls n b ppr onsutivly w tims in th ss squn.in this ry huristi, s r slt in rsin orr o thir wihts provi tht hoosin n os not introu yl n it os not rsult in no o r mor thn two. Finlly, th ss rph onsirin only th slt s will trmin th plmnt o th vribls in th mmory. On possibl rsult o pplyin Lio s huristi to th ss squn in Fiur () is shown in Fiur (), whr th bol s r th slt s n th inl ost ssinmnt is [b]. Th ost o solution is th sum o th wihts o ll unslt s (i.., non-bol s). For th xmpl in Fiur (), th ost is whih rprsnts th non-bol tht rrs to th on rss rithmti oprtion n to o rom to in th ss squn sin vribls n r mpp to non-onsutiv mmory lotions. Luprs n Mrwl [9] xtn Lio s work by proposin ti-brk huristi or th SOA problm. Lio t l. i not stt wht hppns i two s hv qul wiht. Luprs n Mrwl us th ollowin ti-brk untion: i two s hv th sm wiht, thy pik th with th smllr vlu o th ti-brk untion T (, b) in or n (, b) s in qution 5. Atri t l. [] solv th SOA problm usin n inrmntl pproh. Thy tri to ovrom som o th problms with Lio s lorithm, minly in th s o qul wiht s s wll s th ry pproh o lwys sltin th mximum wiht s. Strtin with n initil ost ssinmnt (whih oul b th rsult o ny SOA huristi), thir inrmntl-soa tris to xplor mor points in th solution sp by onsirin th t o sltin urrntly unslt s.

4 4 Hssn Slmy n J. Rmnujm Luprs [7] ompr svrl lorithms or simpl ost ssinmnt. Ottoni t l. [] stui th simpl ost ssinmnt problm with vribl olsin (CSOA). Thir lorithm uss livnss inormtion to onstrut th intrrn rph. In th intrrn rph, th nos rprsnt vribls n n btwn two vribls mns tht thy intrr n thus thy n not b ols. Th uthors us th SOA huristi propos by Lio t l. [] nhn with th ti brk in [9], with th irn tht t h stp th lorithm hooss btwn (i) olsin two vribls; n (ii) sltin th with th mximum wiht s in Lio s lorithm. Thir lorithm ins th pir o nos tht n b ols with mximum sv whr sv rprsnts th tul svin rom olsin this pir o nos. At th sm tim, it ins th with th mximum wiht w tht n b slt usin Lio s lorithm. I thr r nits or both olsin n sltion, thn it will us olsin i sv is lrr thn w, othrwis us sltion. ()Th ss squn: b b b b (b) b () Fi.. () Ass squn. (b) Ass rph orrsponin to th ss squn. () Ost ssinmnt whr bol s rprsnt th slt s n th ost o suh ssinmnt is. In [], th uthors stui th ss o SOA with vribl olsin t th sm tim s []. Thir olsin lorithm irst sprts vlus into tomi units ll wbs by pplyin vribl rnmin. Thir propos huristi strts by pplyin pr-itrtion olsin ruls. Thn th lorithm piks th two vribls (i.., nos) with mximum svin or olsin provi tht thy rspt th vliity onitions. I th svin is positiv, thn th two nos r ols. Lio s SOA will thn b ppli to th nw ss rph. This pross will ontinu s lon s thr r two vribls tht n b ols. Svrl

5 An Etiv Huristi or Simpl Ost Assinmnt 5 othrs [0], [5], [6], [7], [9], [0] hv rss problms rlt to ost ssinmnt. CSOA: Ost ssinmnt with vribl olsin In simpl ost ssinmnt (SOA), h mmory lotion or slot is ssin only on vribl. Simpl ost ssinmnt with vribl olsin (CSOA) rrs to th s whr mor thn on vribl n b mpp into th sm mmory lotion. Vribl olsin is intn to rs th mmory rquirmnt by urthr rsin th numbr o rss rithmti instrutions s wll s by rsin th mmory rquirmnts or storin th vribls. Two vribls n b ols i thir liv rns o not ovrlp t ny tim whih mns tht t ny tim, thos two vribls r not n to b simultnously liv. In CSOA, n intrrn rph (IG) is onstrut by xminin th liv rns o ll th vribls. Eh no in th rph rprsnts vribl, n n btwn two nos mns thy intrr n thus thy nnot b ols. Two vribls n b ols i thy mt ll th ollowin onitions: th two vribls o not intrr; tr olsin, no no in th ss rph hs mor thn two slt s inint t it; (n) th rsultin ss rph is still yli onsirin only th slt s. So inst o lwys sltin n s in SOA, CSOA n ithr slt n or ols two vribls tht mt th thr onitions list bov. Our lorithm prsnt in Fiur intrts both sltion n olsin options in wy to minimiz th totl ost, whih is rprsnt by th numbr o rss rithmti instrutions, s wll s to rs th mmory rquirmnt or storin th vribls in mmory. Th lorithm tks s n input, th intrrn rph (IG) n th ss squn, n outputs th mppin o th vribls to mmory lotions possibly with olsin. From th ss squn, it onstruts th ss rph (AG) whih pturs th rquny o onsutiv ourrn o ny two vribls in th ss squn. Thn it sorts th s whos n-point vrtis intrr in rsin orr o thir wihts s ui or sltion. Sin on o th purposs o th huristi is to rs th mmory rquirmnt or storin th vribls, n (, b) suh tht (, b) / IG will not b onsir or sltion. Suh n will b nit or olsin whih mns tht wr s will b onsir or sltion n thus mor vribls will probbly b ols. Not tht th sltion o n my prvnt vribl olsin opportunitis in th utur. So only thos s whos npoints intrr will b onsir s nits or sltion in h itrtion o th lorithm. Any two vribls tht o not intrr r onsir s nits or olsin. In h itrtion, ll pirs o vribls tht mt th thr onitions or vribl olsin (mntion rlir) r nits or olsin. W in

6 6 Hssn Slmy n J. Rmnujm th ollowin vlus: Atul Gin(, b) Gin(, b) = P ossibl Loss(, b) Atul Gin(, b) = W (, b) + W (, x) + P ossibl Loss(, b) = + + x Aj() Aj(b) (b,x) Slt E (,x) / Slt Es y Aj() Aj(b) (b,y) / Slt Es (,y) Slt Es (,x) / IG,(b,x) IG (b,x) / Slt Es (b,y) / IG,(,y) IG (,y) / Slt Es () W (b, y) () (, x) (b, y) () A Gin vlu or h o ths nit pirs is lult tht pturs th bnit o olsin s wll s th possibl loss o utur opportunitis or olsin. Th vlu Gin(, b) is in s th tul svin tht rsults rom olsin vribls n b ivi by th possibl loss o utur olsin opportunitis u to olsin n b. Whn vribls n b r ols, ll s inint t n b o th orm (, x) n (b, x) will b mr, n i (, b) xists, it will b lt. Whn s (, x) n (b, x) r mr into (b, x), i t lst on o th s ws lry slt, thn (b, x) is lso onsir to b slt. Th vlu Gin(, b) is in s shown in Eqution n th vlu Atul Gin(, b) is in in Eqution. Th vlu Atul Gin(, b) is bsilly th sum o th wihts o th s inint t or b tht wr not slt bor n bm slt tr bin mr with slt plus th wiht o th (, b). Th vlu P ossibl Loss(, b) is in in Eqution s th sum o th s (, x) suh tht (, x) / IG, (b, x) is not slt, n (b, x) IG plus th sum o th s (b, y) suh tht (b, y)/ IG, (, y) is not slt, n (, y) IG. As pit in qution, P ossibl Loss(, b) onsirs only vrtis tht r nihbors to or b. Althouh othr initions o th loss n b us, w oun tht our inition pturs th possibl t o olsin on solutions tht n b onstrut. Evn thouh olsin involvs vrtis n not s, usin th numbr o s s th ssn or th loss in Eqution ls to bttr rsults. Th rtionl bhin this is tht n whos orrsponin vrtis intrr will probbly n up s slt n thus it my prvnt som olsin opportunitis n s rsult it my r th qulity o th inl solution. It is worth notin tht lthouh our huristi intrts both sltion n olsin, it ivs priority to olsin, whih n b lrly u rom th

7 An Etiv Huristi or Simpl Ost Assinmnt 7 inition o loss. W bliv this is on o th min rsons or our improvmnts in trms o th ost s wll s th mmory rquirmnt or storin th vribls. W ivi th vlu Atul Gin(, b) with th vlu P ossibl Loss(, b) to ount or th numbr o s whos orrsponin vribls wr intrrnr n now intrr s rsult rom olsin n b. Th rson bhin this is tht olsin two vribls with lr P ossibl Loss vlu my prvnt som utur olsin opportunitis n thus my prvnt hivin smllr ost ompr to olsin two vribls with smllr P ossibl Loss vlu. Amon ll th pirs tht r nits or olsin, our lorithm piks th pir with th mximum Gin. I th lorithm is bl to in pir or olsin s wll s n or sltion, thn it will ols i th Atul Gin rom olsin is rtr thn or qul to th wiht o th onsir or sltion; othrwis, it will slt th. On wy our huristi ttmpts to mximiz th numbr o vribls mpp to h mmory lotion is to llow th olsin o pirs o vribls with zro Gin vlu (i possibl) tr no mor vribls with positiv Gin n b ols. Colsin vribls without oo ui my prvnt possibl improvmnts ovr th stnr SOA solution. Consir th xmpl in Fiur 4. Fiur 4(b) shows Lio s ry solution. Th ost o this ost ssinmnt is 4. Fiur 4() shows th solution usin th lorithm in [] whos ost is lso 4. Althouh thr is potntil or improvmnt throuh vribl olsin, th lorithm in [] ils to ptur th improvmnt ovr Lio s solution. This is bus th lorithm in [] irst hooss to ols vrtis b n sin thy hv th mximum sv. Howvr, this hoi will prvnt ny utur olsin opportunitis. Our lorithm llvits this shortomin by lultin th P ossibl Loss(b, ) = 5 n thus Gin(b, ) = /5. So our lorithm irst piks n b or olsin sin Gin(, b) = ; (b, ) will not b onsir or sltion sin b n o not intrr. Th ost o th inl solution o our lorithm is zro, s shown in Fiur 4(). For sltion, w us two ti brk untions T n T in blow, T (, b) = r() + r(b) (4) T (, b) = W (, x) + W (b, y) (5) x Aj() y Aj(b) whr T (, b) is th sum o th r o n r o b in th ss rph. T (, b) is th Luprs ti brk untion in s th sum o th wihts o th s tht r inint t plus th sum o th wihts o th s tht r inint t b. I two s tht r nits or sltion hv th sm wiht thn w try to ti brk usin th untion T ; i T nnot brk th ti, w us T. An with smllr T or T will win th ti. I two pirs o vribls (, b) n (, ) tht r nits or olsin r suh tht Gin(, b)= Gin(, ), thn w irst try to brk th ti usin T 0 whih is th Atul Gin suh tht w hoos th pir with th bir Atul Gin. I both nit pirs hv th sm tul in, thn w ti brk usin T ollow by T, i n.

8 8 Hssn Slmy n J. Rmnujm - Colsn SOA Alorithm Input: th Ass squn. th Intrrn rph IG. Output: Ost ssinmnt. Buil th ss rph (AG) rom th ss squn. L = list o s (x,y) suh tht (x,y) IG in rsin orr o thir wihts usin T thn T or ti brk. Cols = ls. Slt = ls. Do Fin pir o nos (,b) or olsin tht stisy:. (, b) / IG.. AG will still b yli tr n b r ols onsirin slt s.. No no will n up with r > onsirin slt s. 4. (,b) hs mx Gin whr Gin is lult s in qution (). whr T 0, T, n T r th thr ti brk untions us in tht orr. I suh pir o nos is oun, thn Cols = tru. Amon th s tht blon to L pik th irst (,) suh tht:. Sltin (,) will not rsult in yli AG onsirin slt s.. Sltin (,) will not rsult in no with r > onsirin slt s. I suh n is oun, thn Slt = tru; rmov (,) rom L. I (Cols && Slt) I (Atul Gin(, b) = Wiht(, )) Upt ss rph AG with (, b) ols. Upt intrrn rph IG with (, b) ols. Upt list L Els Slt (,) Els i (Cols) Upt ss rph AG with (,b) ols Upt intrrn rph IG with (,b) ols Upt list L Els i (Slt) Slt (,) Whil (Cols Slt) Rturn ost ssinmnt Fi.. Our lorithm or Simpl Ost Assinmnt with vribl olsin.

9 An Etiv Huristi or Simpl Ost Assinmnt 9 b b () (b),b,,b,, () () Fi. 4. () Intrrn Grph. (b) Lio s SOA ry solution whr th ost = 4. ()Th solution rom th lorithm in [] o ost 4 whr it ils to ptur th potntil improvmnts rom olsin. () Th optiml solution rom our lorithm with ost = 0. 4 An Exmpl For th sk o lrity, onsir th xmpl in Fiur 5 whr Fiur 5() shows th intrrn rph (IG) n Fiur 5(b) shows th oriinl ss rph (AG). Fiurs 5()-(h) show how th ss rph is upt whn our huristi is ppli to this xmpl. Althouh not shown, whnvr two nos r ols, th intrrn rph (IG) will b upt to rlt th olsin o th nos s wll s to upt th intrrn s orinly. Tbl shows th stp-by-stp xution o our lorithm n th ritri us or hoosin th nits or sltion n or olsin. Not tht in Tbl w o not show olsin nits with zro Gin. Fiur 5(i) shows th inl solution with zro ost. I w run th lorithm in [] on th sm xmpl prsnt in Fiur 5, th ost o possibl inl solution (whih is shown in Fiur 6) is 4. 5 Simult Annlin Sin th ost ssinmnt problm is NP omplt, th huristi prsnt in Stion will vry likly prou suboptiml solution. So in orr to urthr improv th rsults, w us simult nnlin pproh. Simult Annlin (SA) [] is lobl stohsti mtho tht is us to nrt pproximt solutions to vry lr ombintoril problms. Th thniqu oriints rom th thory o sttistil mhnis, n is bs on th nloy btwn th

10 0 Hssn Slmy n J. Rmnujm b b,b () (b) (),b,b,b,, (),b () (),,b,, () (h) (i) Fi. 5. () Th Intrrn Grph. (b) Oriinl Ass Grph. ()-(h) Th ss rphs tr h itrtion o our lorithm. (i) Th inl ost ssinmnt, whih inurs zro ost.,b,,b, Fi. 6. On possibl inl solution or th xmpl shown in Fiur 5 usin th lorithm in []

11 An Etiv Huristi or Simpl Ost Assinmnt Tbl. A stp by stp run o our lorithm on th xmpl in Fiur 5 Itrtion Cols Cnit Sltion Dision vrtis AtulGin PossiblLoss Gin Wiht,b Cols(,b) b, 4 /4 (b,) Ti-brk T 0, / (,),, /, (b,), (b,) Slt (b,), (,),, (b,) Cols (,), (,) Ti-brk T 0, / (b,) Slt (b,) 4, (,) Ti-brk T (,) 5, (,) Slt (,) (,) 6, (,) Cols (,) nnlin pross o solis n th solution prour or lr ombintoril optimiztion problms. Th nnlin lorithm bins with n initil sibl oniurtion, n thn nihbor oniurtion is rt by prturbin th urrnt solution. I th ost o th nihborin solution is lss thn tht o th urrnt solution, th nihborin solution is pt; othrwis, it is pt or rjt with som probbility. Th probbility o ptin inrior solutions is untion o prmtr, ll th tmprtur T, n th hn in ost btwn th nihborin solution n th urrnt solution. Th tmprtur is rs urin th optimiztion pross, n th probbility o ptin n inrior solution rss with th rution o th tmprtur vlu. Th st o prmtrs ontrollin th initil tmprtur, stoppin ritrion, tmprtur rmnt btwn sussiv sts, n numbr o itrtions or h tmprtur is ll th oolin shul []. Typilly, t th binnin o th lorithm, th tmprtur T is lr n n inrior solution hs hih probbility o bin pt. Durin this prio, th lorithm. ts s rnom srh to in promisin rion in th solution sp. As th optimiztion prorsss, th tmprtur rss n thr is lowr probbility o ptin n inrior solution. Th lorithm thn bhvs lik own hill lorithm or inin th lol optimum o th urrnt rion. Sin simult nnlin rquirs siniint mount o tim in orr to onvr to oo solution, w i to us th inl solution rom our huristi s th initil solution or SA n thn rn SA or short prio o tim with

12 Hssn Slmy n J. Rmnujm low probbility o ptin b solution. Th nihbor untion n prorm on o th ollowin oprtions: Exhn th ontnt o two mmory lotions. Mov th ontnt o on mmory lotion. Unols ols no into two or mor nos. Cols two mmory lotions. 6 Rsults W implmnt our thniqus in th OstSton toolst [4] n w tst our lorithms on th MiBnh bnhmrks [4]. In Tbl, w ompr our rsults with our irnt thniqus us to solv th SOA problm, minly Luprs ti-brk [9], inrmntl with Luprs ti-brk INC-TB[9][7], Gnti lorithm GA[8], n Ottoni s CSOA []. W msur th prnt o th numbr o rss rithmti instrutions ompr to Lio s lorithm []. Our huristi rstilly rus th ost o simpl ost ssinmnt whn ompr to huristis tht o not llow vribl olsin sin vribl olsin inrss th proximity btwn vribls in mmory, thus it rus th numbr o upt instrutions. Column 6 shows tht our huristi ws bl to outprorm th CSOA huristi [] (rsults o whih r shown in Column 5) in ll th ss xpt or on bnhmrk. This improvmnt is u to th ui us in our hoi btwn nits or olsin whr w not only onsir th tul svin but lso n stimt o th possibl loss in utur olsin opportunitis. Also th i o just onsirin s whos npoints intrr or sltion inrss th opportunity or olsin nos with mximum Gin s in in Eqution. Th bility to ols pns on th slt s n vi-vrs. So n lorithm tht n hoos th riht nits or sltion n olsin, t th riht itrtion n i btwn thm, shoul onsir th inlun o suh ision on utur solutions. This is ount or in our lorithm by inin th possibl loss s ui or th possibl t o olsin on utur solutions. Th thr ti brk untions T 0, T, n T ply rol in hivin th lr improvmnts to th inl solution. W o not show th omprison to th thniqu in [] sin th uthors rport n vr ost rution o.% whn ompr to [9] whih is wors thn th rsults hiv in []. Our simult nnlin (SA) lorithm urthr improv th rsults by srhin th sibl rion or bttr solutions strtin rom th inl solution o our huristi. Rsults in Tbl olumn 6 shows tht th SA urthr improv th rsults in ll th ss in short CPU tim. In Tbl, w show th rution in mmory slots n to stor th vribls usin our lorithm ompr to tht o usin th lorithm prsnt in []. Rsults show tht our lorithm rstilly rus th mmory rquirmnt by mximizin th numbr o vribls tht r ssin to th sm mmory lotion n it outprorms th CSOA lorithm [] in ll th ss.

13 An Etiv Huristi or Simpl Ost Assinmnt Th rson bhin this rution is tht w in th Gin rom olsin in trms o possibl loss in olsin opportunitis s wll s u to th t tht w i not onsir th s (, b) suh tht (, b) / IG s nits or sltion n this will rsult in mor opportunitis or olsin. Howvr, th min rson or this improvmnt is tht our huristi llows zro Gin olsin btwn nos in th inl AG. Tht is, w ols pirs o vrtis (, b) (i possibl) suh tht Gin(, b) = 0. This zro Gin olsin will not ru th ost in trms o th numbr o rss rithmti instrutions but it will ontribut to mximizin th numbr o vribls mpp to mmory lotion. This xplins th hu irn btwn th improvmnts in Tbl n Tbl. Althouh huristi sin just to rs th mmory rquirmnt or storin th vribls n t bttr rsults thn thos in Tbl, it will b trimntl to th qulity o th inl solution in trms o th numbr o rss rithmti instrutions. So our huristi not only rss th ost (whih is in s th rution in th numbr o rss rithmti instrutions), but lso rss th numbr o mmory lotions n to stor th vribls. Tbl. Comprison btwn irnt thniqus or solvin th SOA problm whr olumn shows irnt bnhmrks, olumn shows th rsults by pplyin Lio s + Ti-brk [9], olumn shows th rsults o th GA in [8], olumn 4 shows th rsults i th Ti-brk [9] is ombin with th inrmntl SOA in [], n olumn 5 show th rsults in th s o SOA with vribl olsin [], olumn 6 shows our rsults whn pplyin our lorithm, olumn 7 shows th rsults usin simult nnlin. Bnhmrks TB (%) GA(%) INC-TB(%) CSOA(%) Our lorithm SA [9] [8] [9][7] [] (%) (%) pm pi sm jp mp pwit pp rst Conlusions Th problm o ost ssinmnt hs riv lot o ttntion rom rsrhrs u to its rt impt on o siz rution or DSPs. Ruin th o siz is bniil in th s o DSPs sin th o is irtly trnsorm into silion r. Sttistis show tht os or DSPs n hv up to 50% rss rithmti instrutions [8]. So th min i o th onoin rsrh in this il

14 4 Hssn Slmy n J. Rmnujm Tbl. Th numbr o mmory slots n usin our lorithm to th lorithm prsnt in []. Bnhmrks #Vribls #Mmory slots #Mmory slots [] our lorithm pm pi sm jp mp pwit pp rst is to rs th numbr o rss rithmti instrutions n thus th o siz. Th problm is stui s simpl ost ssinmnt (SOA) n s nrl ost ssinmnt (GOA), whr irnt thniqus n lorithms r us to tkl ths problms with irnt moiitions suh s th inlusion o th moiy-ristrs [9] s wll s th s whr th ost rn is rtr thn. In this ppr w prsnt huristi to solv th simpl ost ssinmnt with vribl olsin tht hooss btwn sltion n olsin in h itrtion by lultin th Atul Gin n P ossibl Loss or h pir o olsin nits. Rsults show tht our lorithm not only rss th numbr o rss rithmti instrutions, but lso rstilly rss th mmory rquirmnt or storin th vribls by mximizin th numbr o vribls tht r mpp to th sm mmory slot. Simult nnlin urthr improv th inl solution rom our huristi. Aknowlmnts. W r inbt to Sm Miki or rul rin o this ppr whih hs rsult in siniint improvmnt in th prsnttion o th ppr. In ition, w thnk th rrs or thir ommnts.w r inbt to Sm Miki or rul rin o this ppr whih hs rsult in siniint improvmnt in th prsnttion o th ppr. In ition, w thnk th rrs or thir ommnts. Rrns. S. Atri, J. Rmnujm, n M. Knmir. Improvin Ost Assinmnt or Emb Prossors. Lnus n Compilrs or Hih-Prormn Computin, S. Miki t l. (s.), Ltur Nots in Computr Sin, Sprinr, 00.. D.H. Brtly. Optimizin Stk Frm Asss or Prossors with Rstrit Arssin Mos. Sotwr-Prti n Exprin, vol., no., pp. 0-, 99.. S. Kirkptrik, C. D. Gltt Jr., n M. P. Vhi. Optimiztion by Simult Annlin. Sin, 0, 4598, , 98.

15 An Etiv Huristi or Simpl Ost Assinmnt 5 4. C. L, M. Potkonjk, n W. Mnion-Smith. Mibnh: A Tool or Evlutin n Synthsizin Multimi n Communitions Systms. In Pro. IEEE Intrntionl Symposium on Mirorhittur, pp. 0-5, R. Luprs. Co Gnrtion or Emb Prossors. In Pro. Intrntionl Systm Synthsis Symposium, R. Luprs. Co Optimiztion Thniqus or Emb Prossors. Kluwr Ami Publishrs, R. Luprs. Ost Assinmnt Showown: Evlution o DSP Arss Co Optimiztion Alorithms. th Intrntionl Conrn on Compilr Constrution (CC), Wrsw (Poln), Apr 00, Sprinr Ltur Nots on Computr Sin, LNCS R. Luprs, F. Dvi. A Uniorm Optimiztion Thniqu or Ost Assinmnt Problms. th Int. Systm Synthsis Symposium (ISSS), R. Luprs, P. Mrwl. Alorithms or Arss Assinmnt in DSP Co Gnrtion. Int. Conrn on Computr-Ai Dsin (ICCAD), B. Li, R. Gupt. Simpl Ost Assinmnt in Prsn o Subwor Dt. CASES, ACM Prss, 00.. S. Lio. Co Gnrtion n Optimiztion or Emb Diitl Sinl Prossors. Ph.D. Thsis, Dpt. o Eltril Eninrin n Computr Sin, Msshustts Institut o Thnoloy, S. Lio, S. Dvs, K. Kutzr, S. Tjin, A. Wn. Stor Assinmnt to Drs Co Siz. ACM SIGPLAN Conrn on Prormmin Lnu Dsin n Implmnttion (PLDI), D. Ottoni, G. Ottoni, G. Arujo, R. Luprs. Improvin Ost Assinmnt throuh simultnous Vribl Colsin. In Proins o th 7th Intrntionl Workshop on Sotwr n Compilrs or Emb Systms (SCOPES 0), in Sprinr LNCS 86, pp , Vinn, Austri, Sptmbr OstSton G. Ottoni, S. Rio, G. Arujo, S. Rjopln, n S. Mlik. Optiml Liv Rn Mr or Arss Ristr Allotion in Emb Prorms. In Pro. 0th Intrntionl Conrn on Compilr Constrution, CC 00, LNCS 07, pp Sprinr, April A. Ro n S. Pn. Stor Assinmnt Optimiztions to Gnrt Compt n Eiint Co on Emb DSPs. In SIGPLAN Conrn on Prormmin Lnu Dsin n Implmnttion, ps 8-8, A. Sursnm, S. Lio, n S. Dvs. Anlysis n Evlution o Arss Arithmti Cpbilitis in Custom DSP Arhitturs. In Dsin Automtion Conrn, pp. 87-9, S. Uynrynn, C. Chkrbrti: Arss Co Gnrtion or Diitl Sinl Prossors. 8th Dsin Automtion Conrn (DAC), B. Wss n M. Gotshlih. Optiml DSP Mmory Lyout Gnrtion s Qurti Assinmnt Problm. In Int. Symp. on Ciruits n Systms (ISCAS), B. Wss, T. Zitlhor. Optimum Arss pointr Assinmnt or Diitl Sinl Prossors. ICASSP, IEEE X. Zhun, C. Lu, n S. Pn. Stor Assinmnt Optimiztions Throuh Vribl Colsn or Emb Prossors. In Proins o th ACM SIG- PLAN Conrn on Lnu, Compilr, n Tool Support or Emb Systms, pp. 0-, 00.