An Efficient Algorithm for the Physical Mapping of Clustered Task Graphs onto Multiprocessor Architectures

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1 An Effiient Algorithm for the Physil Mpping of Clustere Tsk Grphs onto Multiproessor Arhitetures Netrios Koziris Pnyiotis Tsnks Mihel Romesis George Ppkonstntinou Ntionl Tehnil University of Athens Dept. of Eletril n Computer Engineering Computer Siene Division Computing Systems Lortory Zogrfou Cmpus, Zogrfou 15773, Greee e-mil: {nkoziris, mromes}@sl.ee.ntu.gr Astrt The most importnt issue in sequentil progrm prllelistion is the effiient ssignment of omputtions into ifferent proessing elements. In the pst, too mny pprohes were evote in effiient progrm prlleliztion onsiering vrious moels for the prllel progrms n the trget rhitetures. The most wiely use prllelism esription moel is the tsk grph moel with preeene onstrints. Nevertheless, s fr s physil mpping of tsks onto prllel rhitetures is onerne, little reserh hs given prtil results. It is well known tht the physil mpping prolem is NP-hr in the strong sense, thus llowing only for heuristi pprohes. Most reserhers or tool progrmmers use exhustive lgorithms, or the lssil metho of simulte nneling. This pper presents n lterntive pproh onto the mpping prolem. Given the grph of lustere tsks, n the grph of the trget istriute rhiteture, our heuristi fins mpping y first pling the highly ommunitive tsks on jent noes of the proessor network. One these «kone» tsks re mppe, there is no ktrking, thus hieving low omplexity. Therefore, the remining tsks re ple eginning from those lose to the «kone» tsks. The pper onlues with performne n omprison results whih revel the metho s effiieny. 1. Introution In the lst yers the evolution in the fiels of VLSI tehnology n omputer networking hs given rise to the utiliztion of istriute omputing systems. Distriute omputers re ttrtive to mny emning pplitions sine they provie the user with moulrity, slility n low ost eentrlize proessing power. Nevertheless istriute omputing hs soun rwk whih isourges pplition evelopers to use it. The existene of mny prllel proessors is not fully exploite euse of the interproessor ommunition overhe. The egrtion of optiml speeup when the numer of proessors inreses is use y the exessive sometimes mount of messges etween non-neighoring ells. Reserhers hve lrey fouse on suh lowperformne prolems n propose vrious methoologies to the optiml prlleliztion of the given progrms. Experiene hs shown tht n effetive solution to the generl tsk grph sheuling prolem onto given rhiteture is multistep pproh whih inlues the tsk grph esription of the prolem, effiient sheuling of the tsks onto virtul fully onnete prllel mhine, merging tsks into lrger lusters n finlly ssigning lusters into the physil proessor topology. As fr s the tsk grph sheuling with preeene onstrints is onerne, the most wiely use moel is the irete yli grph with eges n noes hving vrious weights. Noe weights represent omputtion time for the orresponing tsks, while ege weights re ommunition requirements etween tsks. Eh ege iretion gives set of preeene onstrints whih shoul e preserve. The generl tsk grph sheuling prolem with ommunition elys is NP-COMPLETE s presente in [9]. Srkr [10], Gersoulis n Yng [3] presente goo heuristis for solving the tsk grph sheuling prolem with ritrry ommunition osts. All previous reserhers were supposing unoune or oune numer of proessors with CLIQUE topology.

2 This hypothetil topology is fr wy from the relisti se. Moern istriute memory prllel systems re orgnize in ifferent topologies, inluing hyperue, mesh or ring rhitetures. There is strong nee for effiient plement of lustere tsks into the rel proessor network. The mpping of virtul into rel topologies is lle the physil mpping prolem or the tsk llotion prolem. The tsk llotion prolem n e optimlly solve in speil ses suh s two-proessor istriute systems, or liner rry of ny numer of proessors. If the trget rhiteture ontins two proessors, then the tsk llotion prolem is stte s mximum flow minimum ut prolem [11] whih n e polynomilly solve using, for exmple, the For-Fulkerson lgorithm. There lso exists heuristi presente [7], whih resses the generl m-proessor prolem using the 2-wy min ut lgorithm m times. Most of the theoretil work on mppings onsiers struture grphs like gris, hyperues, trees, et [8]. An inresing numer of pplitions emn methos eling with irregulr grphs. The generl mpping prolem is unfortuntely NP-omplete, thus llowing only for effiient heuristis. Our pper els with the intrtle prolem of luster llotion onto prllel rhitetures, y proposing n lterntive pproh to the ove mentione methoologies. We moel ny trget prllel rhiteture y non-irete grph with unitry ege weights only. This representtion retins the meningful only etils of the trget. Next, the grph of tsks is epite s nonirete grph with vrious ege osts n noe weights resulting from lustering heuristis s the one propose y Gersoulis in [3]. The tsk ssignment proeure strts y tring the lusters whih re most likely to e ple in neighoring ples, where the mximum numer of links is ville. This step ens y pling these kone lusters into these ples. For eh of the kone lusters, the lgorithm retes sets of neighoring lusters whih re nite for plement into the jent ells of every pressigne kone luster. The lgorithm ens when ll lusters re ssigne. In the reminer of this pper the following re presente n further elorte: Setion II reviews the multistep pproh of tsk grph sheuling with preeene onstrints using speifi topology istriute rhiteture. Setion III presents the propose grph moel for the prllel rhiteture n the utilize ost funtion for evluting ifferent physil mppings. In setion IV we outline the steps of the propose lgorithm. Finlly setion V the propose lgorithm is ompre to other pprohes in terms of effiieny n the outperforming results re shown. 2. The Multistep Approh The generl sheuling prolem of n ritrry tsk grph with ommunition elys onto fixe size n onnetion pttern istriute rhiteture is NP- COMPLETE. El-Rewini et H. Ali in [1], [2] prove this NP ompleteness y representing the prolem of tsk llotion onto istriute system with split grph. The tsk llotion is therefore equivlent to weighte lique grph prtitioning whih is NP-COMPLETE, thus proving inherent intrtility. In orer to fin effiient methos, reserhers hve followe multistep pproh, where eh step resses limite instne of the generl prolem. The suessive steps re outline s follows: tsk lustering, luster merging n physil mpping Tsk Clustering - Sheuling tsk grph with ommunition elys onto oune/unoune CLIQUE of proessors. First the tsk grph with omputtion/ommunition osts n preeene onstrints is sheule onto fully onnete network of proessors. The lssil CLIQUE rhiteture is therefore use s trget hving limite or unlimite numer of proessors [4]. Reserhers, in this first step, propose lgorithms whih minimize the mximum mkespn, isregring the tul proessor s topology. Even when the mkespn metri is to e minimize, the sheuling prolem remins NP- COMPLETE in the mjority of generl ses. Ppimitriou et Ynnkkis in [9] hve prove tht the lssil sheuling prolem of tsk grph with ritrry ommunition n omputtion times is NP- COMPLETE. They propose 2-optiml pproximtion lgorithm. In ition to this, Srkr in [10], Gersoulis in [3] propose fster heuristis with eptle performne. All these lgorithms perform the sme initil step: Cluster the tsks into lrge noes, so tht the grin of the prllelism is inrese n the use of istriute proessors is minimizing the tsk grph mkespn. The resulting topology, y pplying this initil step is grph of lusters. Insie every luster there exist severl tsks whih re time sheule in the sme proessor n hve zero intrommunition overhe. As fr s lusters interommunition is onerne, this is the summtion over ll tsks interommunition overhe for ll lusters. This orgnises the set of tsks into lusters with the following property: Eh luster ontins ll tsks whih re to e exeute on the sme proessor. This step is lle tsk lustering.

3 2.2. Cluster Merging into p physil lusters In this step, the set of lustere tsks is mppe onto lique of oune numer of proessors. Sine the set of lustere tsks is lrger thn the numer of ville proessors, this step ssigns two or more lusters to the sme proessor. Srkr in [10] hs propose sheuling heuristi with O( V ( V + E )) omplexity, where V stns for the numer of noes n E for the numer of eges. A lower omplexity heuristi whih is use in PYRROS [12] is the work profiling metho. It merges lusters whih hve pproximtely the sme rithmeti lo Physil Mpping of p physil lusters onto p network onnete proessors. While so muh effort hs een one onerning the first steps, the finl step of luster llotion onto the physil proessor topology hs not given onsierle ttention. Few reserhers suh s Bokhri hve presente some heuristis. Mny sheuling tools whih suh s OREGAMI or PYRROS or PARALLAX use heuristis or pproximtion lgorithms whih re sometimes effiient. For exmple Gersoulis et Yng in Pyrros [12] use Bokhri s heuristi whih is se onto simulte nneling. This lgorithm strts from n initil ssignment, then performs series of pirwise interhnges y reuing the ost funtion n stops fter O(p 3 ) steps. In Oregmi s [6] MAPPER, tool for tsk llotion onto istriute rhitetures, greey heuristi is use, lle the NN-Eme. It urrently supports only mesh n hyperue proessor networks thus limiting its potentil use. Even in the se of those two rhitetures, it uses rther simple metho, y listing ll eges in sening orer of their weights n ssigning them to the proessor s network eges The NN-Eme Algorithm Given tsk grph, it first onstruts list of ll the eges in the grph, sorte y weight. The heuristi then trverses this list in liner time n for eh ege, ssign enpoints s follows: If oth noes hve lrey een ssigne, o nothing If only one noe hs een ssigne, then ssign the other noe to the losest free proessor If neither noe hs een ssigne, rnomly hoose free proessor n ssign one noe to it n the other to its losest free neighor. The sorting step nees O( E log E ) time, n the rest is O( E ). It is fst heuristi, ut it is limite to hyperue n mesh topologies. Oviously, it hs poor performne in mny ses euse it is se on single-ege jeny, n it oes not tke into ount ny set of jent noes. In Exmple 5, we pply oth our metho n the NN-Eme, n the outperforming results of the propose heuristi re shown. In PARALLAX [5], ll tsk llotion heuristis ssume fully onnete networks of proessors. Only the mpping heuristi onsiers ritrry proessor interonnetion topology. It uses moifie list sheuling tehnique with priorities, y tking into ount the ommunition elys. The noe s priority in the list is its level. In [8], Monien n Suorough reviewe results on mpping speifi tsk grphs into the most populr prllel rhitetures. 3. Grph Moels Our prolem onsiers two grphs: The grph of virtul proessors, or equivlently the grph of lustere tsks, efine s G (V, E ) n the grph of physil proessors, efine s G p (V p, E p ). Definition 3.1: Consier two grphs n G w (V w, E w ) where V is the set of noes n E is the set of eges with V w = V p, we efine the following funtion F m :V w V p s the physil mpping funtion : v w,v w V w with (v w,v w ) E w, (F m (v w ),F m (v w )) E p Let us efine some topologil prmeters for the proessor grph/network: Hop is the unit istne etween ny two iretly onnete proessors in the network Distne etween two proessor noes u n v is lle the numer of hops in the shortest pth onneting u n v. Distne is represente y ist(u,v). The following formul efines the ost funtion use to evlute the ifferent mppings: Cost Funtion for mpping F m : CF(F m ) = ist(f m (v w ), F m (u w ) )xomm(v w, u w ) Y X ( Z Z Z where: ist( F m (v w ), F m (u w ) ) is the shortest pth in the proessor grph etween the 2 proessors-noes where the tsks v w n u w re mppe. omm(v w, u w ) is the totl ommunition ost etween the tsks v w,u w The following properties hrterize the effiieny of mpping from G w to G p : Ege-iltion is the mximum istne in G p n ege of G w hs to e route. Formlly:

4 Ege-ongestion is the mximum numer of eges from G w route vi n ritrry ege (or noe) of G p In most theoretil works eling with the physil mpping prolem, minimizing ege iltion n ege ongestion re the primry gols. Our ojetive is therefore to fin the mpping F m whih minimizes CF m, or formlly: opt Definition 3.2: A mpping F m with respet to Cost Funtion CF() is lle optiml if: CF(F opt m ) = min { CF(F m ) F m MAP}, where MAP is the set of ll possile mppings 3.1. Exmple Consier the G (V,E ) with ij osts shown in figure 1, n the proessor grph G p (V p,e p ): Figure 4: Mpping F M1 CF M1 = 2 + 4X = Figure 5: Mpping F M2 G C Figure 1. The Cluster Grph 1 2 CF M2 = 4 + 2X X2 = 21 Oviously, F m1 is etter thn F m2 in terms of totl ompletion time, sine it imposes less ommunition overhe thn F m2. The lest ommunition is hieve y mpping F m3 shown elow: G P 3 4 Figure 2 : The trget rhiteture 1 2 Figure 6: Optiml Mpping F M3 CF m3 = x3 = Figure 3: CLIQUE rhiteture Oviously there exist 4! lterntive mppings whih n e foun y exhustive serh. For exmple, onsier the following lterntive mppings n evlute the orresponing ost funtion CF m (): 4. The Physil Mpping Algorithm (PMAP) This heuristi tries to fin the most ommunition intensive tsk-noes n mp them n their neighors into neighoring proessing noes on the proessor grph. At the eginning, the proessor grph is nlyze n, for eh noe, the numer of jent noes is lulte. Susequently, the noes of the tsk grph re sorte y sening orer of their totl ommunition weight n numer of neighors. The heuristi then ples the most emning tsk-noes in terms of totl ommunition links n ost to the respetive noes of the proessor grph. One these ore tsk noes re ple, there is no ktrking. Next, the heuristi ples the jent of the

5 ore noes to jent ells of the proessor grph, y mking lolly est-fit omprisons. The lgorithm performs the following steps: 4.1. First Phse: 1 st step: Ajust the mximum numer of neighors in the tsk-grph to the mximum numer in the proessor grph, y removing the less ommunition ost eges for eh tsk-noe. 2 n step: Sort the noes of the proessor grph y sening orer of their neighoring links n the noes in the luster grph G y the numer of ommunition links they nee. 3 r step: Assign the most emning tsk-noe to the est suitle noe of the proessor grph n his neighoring noes to the est suitle neighoring noes of the proessor grph Seon Phse: 4 th step: Ple k the eges remove t the first step. 5 th step: From the proessors not llote yet fin the one who neighors with the most proessors lrey ssigne. 6 th step: Fin tsk tht is istne y t most i (t the eginning i=1) from ll the tsks tht re mppe to the neighoring proessors of the proessor foun on the previous step. If foun ssign this tsk to the proessor of the 5 th step. 7 th step: Repet steps 5-6 until no more ssignments n e me. If there re more proessors to e llote, inrese i y one n go k to the 5 th step Detils of the lgorithm 1 st Phse: Ajust_neighors(G ); Aj(v) = {u: (u,v) E } /* Plement of the most ommunition ostful noe */ fin v G : rnk(v ) = mx rnk(v) fin v p G p : rnk(v p) = mx rnk(v) F m(v ) = v p; /* Plement of neigors to neigoring ells */ while Aj(v p) egin fin v : rnk(v ) = mx rnk (v ) Aj(v ) = Aj(v )-{v } fin v p : rnk(v p) = mx rnk (v p) F m(v ) = v p PROCS_ALLOC = PROCS_ALLOC {v p} PROCS_LEFT = PROCS_LEFT-{v p} TASKS_LEFT = TASKS_LEFT-{v } en 2 n phse: /* Plement of ll other noes */ Ple_k_eges; i=1 while PROCS_LEFT egin ssigning=flse for ll u PROCS_LEFT egin fin u Aj(u) PROCS_ALLOC = mx NEIGH = { v i: F m (v i) (Aj(u) PROCS_ALLOC)} /*set of tsks whih hve lrey een ssigne to neighoring pros*/ Aj(1,v i)=aj(v i) Aj(i,v i)=aj(i-1, v i) {u:(u,w) E C,w Aj(i-1,v i)} let CAND= Aj(i,v i) TASKS_LEFT if CAND egin fin v n : rnk(v n) = mx rnk (v), v CAND F m(u)=v n PROCS_LEFT=PROCS_LEFT-{u}; PROCS_ALLOC = PROCS_ALLOC {u}; TASKS_LEFT = TASK_LEFT-{v n} ssigning=true en en /* of for*/ if ssigning=flse then i=i+1 /* i inreses when uring the previous yle there hs een no ssignment */ en Ajust_neighors routine: neigh_r=mx Aj(v), v G p For eh v G : Aj(v) > neigh_r repet fin u Aj(v): omm(v, u) = min omm(v, w) w Aj(v) Aj(v)=Aj(v)-u until Aj(v) = neigh_r en of for; 5. Exmple Consier the G (V, E ) with ij osts shown in figure 7, n the proessor grph G p (V p,e p ) in figure 8: g Figure 7: The Cluster Grph for Exmple e 4 h 5 2 f

6 Figure 8: The trget rhiteture for Exmple 5.1.First Phse Sine # of Neighors of > mx Neigh we just the tsk grph y eleting eges : (,h) = 1, (, e) = 2. Tsk # of Neighors Totl rnking omm e f g h Tle 1: Sttsistis for the tsk grph V p =1,V =f F m (f)=1 Aj(f)={, e, h } F m ()=2 F m (e)=3 F m (h)=5 TASKS_LEFT={,,,g} 1f e h Figure 9: Assignments fter the 1 st phse 5.2 Seon Phse: insert k eges : (,h)=1,(,e)=2 1 st yle i=1 u 1 =4, Aj(4) PROCS_ALLOC = 2 u 2 =6, Aj(6) PROCS_ALLOC = 2 u 3 =7, Aj(7) PROCS_ALLOC = 2 u 4 =8, Aj(8) PROCS_ALLOC = 0 Let u = 4: NEIGH = {,e} CAND=Aj() Aj(e) TASKS_LEFT= Let u = 6: NEIGH={,h} CAND=Aj() Aj(h) TASKS_LEFT={g} F m (g)=6 2 n yle i=1 u 1 =4, Aj(4) PROCS_ALLOC =2 u 2 =7, Aj(7) PROCS_ALLOC =2 u 3 =8, Aj(8) PROCS_ALLOC =1 Let u = 4: NEIGH = {,e} CAND =Aj() Aj(e) TASKS_LEFT= Let u = 7: NEIGH = {h,e} CAND=Aj(h) Aj(e) TASKS_LEFT= Let u = 8: NEIGH={g} CAND=Aj(g) TASKS_LEFT= No ssignment uring this yle: i inreses y one 3 r yle i=2 u 1 =4, Aj(4) PROCS_ALLOC =2 u 2 =7, Aj(7) PROCS_ALLOC =2 u 3 =8, Aj(8) PROCS_ALLOC =1 Let u = 4: NEIGH = {,e} CAND = Aj(2,) Aj(2,e) TASKS_LEFT = {,} F m ()=4 4 th yle i=2 u 1 =7, Aj(7) PROCS_ALLOC =2 u 2 =8, Aj(8) PROCS_ALLOC =2 Let u = 7: NEIGH = {h,e} CAND=Aj(2,h) Aj(2,e) TASKS_LEFT = Let u = 8: NEIGH= {,g} CAND=Aj(2,) Aj(2,g) TASKS_LEFT = No ssignment uring this yle: i inreses y one 5 th yle i=3 u 1 =7, Aj(7) PROCS_ALLOC =2 u 2 =8, Aj(8) PROCS_ALLOC =2

7 Let u = 7: NEIGH={h, e} CAND=Aj(3,h) Aj(3,e) TASKS_LEFT={,} F m ()=7 F m ()=8 5h 6g 1f 3e Figure 10: Assignments fter the 2 n phse PMAP result 5.3. NN-Eme lgorithm 8 6. Performne Results We report our experiments on the PMAP lgorithm. For the ske of omprisons we hve rete progrm whih, given the tsk n the proessor grphs, genertes ll possile mppings n lultes the totl ommunition time se on the metri of setion III. We hve lso rete rnom tsk grph genertor with vrious ommunition-ege weights n numer of tsksnoes n grph genertor se on fixe topologies. We hve lso pplie the PMAP heuristi n the NN_Eme heuristi on the sme grph pirs. The results otine re shown in the tle elow. We n see tht the PMAP lgorithm gives etter results thn the NN_Eme lgorithm, espeilly when the tsk n proessor grphs hve fixe topologies. But even when the grphs hve een rnomly rete PMAP outperforms NN-Eme y more thn 10%. Sorting of Eges in esening orer of their weight: Noes Tsk topology Proessor topology PMAP vs. NN_Eme optiml vs. NN_Eme (e,f) 5 (,) 2 (,) 4 (,e) 2 (,f) 4 (g,h) 2 (,) 3 (h,f) 2 (g,) 3 Tle 2: Weight of eges Assign (e,f) ege to n ritrry ege, e.g: F m (e)=1, F m (f)=5 Assign (,) ege to n ritrry ege, e.g: F m ()=2, F m ()=6 Assign (,f) ege: Sine f is lrey ssigne, ssign to the losest free neighor: F m ()=7 All noes of (,) re ssigne. Assign (g,) ege: Sine is lrey ssigne, ssign g to the losest free neighor: F m (g)=3 Assign (,) ege: Sine is lrey ssigne, ssign to the losest free neighor: F m ()=8 All noes of (e,) re ssigne. Assign (g,h) ege: Sine g is lrey ssigne, ssign h to the losest free neighor: F m (h)=4 All noes of (h,f) re ssigne. The following tle ompres the performne of the lgorithms for this exmple: Algorithm Cost NN_eme 39 PMAP 34 Optiml Rnom Rnom 13% 26% Rnom Rnom 12% Rnom Rnom 11% - 32 Ring Mesh(8x4) 29% - 32 Mesh(8x4) Ring 28% - 32 Mesh(8x4) Mesh(8x4) 39% - 32 Mesh(8x4) Hyperue 10% - 32 Hyperue Mesh(8x4) 21% - 10 Ring Mesh(5x2) 32% 42% 10 Ring Binry Tree 10% 20% 10 Mesh(5x2) Ring 14% 24% 10 Mesh(5x2) Mesh(5x2) 31% 42% 10 Binry Tree Ring 20% 30% 10 Binry Tree Mesh(5x2) 21% 33% 7. Relte Work Tle 3: Performne results There re not mny systems whih tkle with the prolem of tsk llotion on speifi trget rhitetures, thus solving the physil mpping prolem. PYRROS, evelope y Yng n Gersoulis, uses the heuristi of Bokhri whih is se on simultion nneling. In OREGAMI [6] tsk llotion is performe in three steps: ontrtion of the tsk grph to smller grph, ssignment of the ontrte lusters of tsks to proessors n finlly routing of messges through the interonnetion network to minimize ontention. In OREGAMI, there exist nne lgorithms for typil

8 interonnetion ptterns onerning the trget prllel rhitetures, whih hve een evelope priori. If the interonnetion pttern is not in the ove lirry, the NN- Eme greey heuristi is use. PARALLAX [5], is nother sheuling tool whih ontins tsk llotion heuristi ssuming speifi interonnetion topology for the trget rhiteture. 8. Conlusion In this pper new pproh ws presente for the physil mpping of tsk grphs into prllel rhitetures hving ritrry interonnetion topologies. It is ovious the generl prolem is intrtle thus llowing only for effiient heuristis. The propose lgorithm hs low omplexity n gives very goo results for grphs with equilirte ommunition los etween jent tsks. This new lgorithm ws implemente n teste, n the outperforming results were reporte. Future work inlues the introution of itionl riteri whih woul inrese the heuristi s performne s fr s nonounterlne grphs re onerne. 9. Aknowlegements Journl of Prllel Progrmming, vol. 20, no. 3, 1991, pp [7] V. Lo. Heuristi Algorithms for Tsk Assignment in Distriute Systems. IEEE Trns. Comput., vol. C- 37, no. 11, pp , Nov [8] B. Monien An H. Suorough. Emeing one Interonnetion Network in Another. Computtionl Grph Theory, Springer-Verlg, Computing Supplement 7, pp , [9] C. H. Ppimitriou n M. Ynnkkis. Towr n Arhiteture-Inepenent Anlysis of Prllel Algorithms. SIAM J. Comput., vol. 19, pp , [10] V. Srkr. Prtitioning n Sheuling Prllel Progrms for Exeution on Multiproessors. Cmrige, MA: MIT Press, [11] H. Stone. Multiproessor Sheuling with the Ai of Network Flow Algorithms. IEEE Trns. Soft. Engin., vol. SE-3, no. 1, pp , Jul [12] T. Yng n A. Gersoulis. PYRROS: Stti Tsk Sheuling n Coe Genertion for Messge Pssing Multiproessors. Pro 6 th Int l Conf. Superomputing (ICS92), ACM Press, New York, N. Y., 1992, pp This reserh ws supporte in prt y the Greek Seretrit of Reserh n Tehnology (GSRT) uner PENED 99/308 Projet. Referenes [1] H. Ali et H. El-Rewini. Tsk Allotion in Distriute Systems: A Split Grph Moel. Journl of Comintoril Mthemtis n Comintoril Computing, vol. 14, pp , Otoer [2] H. El-Rewini, T. G. Lewis n H. Ali. Tsk Sheuling in Prllel n Distriute Systems. Prentie Hll, [3] A. Gersoulis n T. Yng. On the Grnulrity n Clustering of Direte Ayli Tsk Grphs. IEEE Trns. Prllel Distri. Syst., vol. 4, no. 6, pp , Jn [4] N. Koziris, G. Ppkonstntinou n P. Tsnks. Optiml Time n Effiient Spe Free Sheuling for Neste Loops. The Computer Journl, vol. 39, no 5, pp , [5] T. Lewis n Heshm El-Rewini. Prllx: A Tool for Prllel Progrm Sheuling. IEEE Prllel & Distriute Tehnology, vol. 1, no. 2, My 1993, pp [6] V. Lo, S. Rjophye, S. Gupt, D. Kelsen, M. Mohme, B. Nitzerg, J. Telle n X. Zhong. OREGAMI: Tools for Mpping Prllel Computtions to Prllel Arhitetures. Int l