A SYSTOLIC APPROACH TO LOOP PARTITIONING AND MAPPING INTO FIXED SIZE DISTRIBUTED MEMORY ARCHITECTURES

Transcription

1 A SYSOLIC APPROACH O LOOP PARIIONING AND MAPPING INO FIXED SIZE DISRIBUED MEMORY ARCHIECURES Ioanns Drosts, Nektaros Kozrs, George Papakonstantnou and Panayots sanakas Natonal echncal Unversty of Athens Department of Electrcal and Computer Engneerng Dvson of Computer Scence Zografou Campus, Zografou, Athens, Greece. {jdros, nkozrs}@cslab.ece.ntua.gr SUBJEC hs paper presents a new method for the problem of mappng of nested FOR-loops wth unform dependences, nto mesh-connected parallel archtectures. hs method s based on loop mappng for systolc arrays. he vrtual array of cells s derved from the ndex space, by applyng a lnear transformaton. hs array s dvded (cut) nto a fxed number of clusters, equal to the number of avalable real processors. he basc dea of our method s that the cuttng s performed along properly selected boundary drectons, so as to mnmze nter-cluster communcaton and equlbrate the number of vrtual cells for every cluster. Each cluster s then assgned to a dfferent processor, whch performs n a roughly ndependent manner, as the communcaton requrements are now mnmzed. hs mappng cuts down overall communcaton delays, whle usng a fxed number of processors from a (n-)-dmensonal meshconnected dstrbuted archtecture.. INRODUCION he prmary task n parallelzng a FOR-loop, s fndng a schedule of computatons nto tme, whle preservng the data dependences of the ntal lexcographc loop order. Most methods are based on fndng a lnear tme transformaton, snce lnear tme schedulng dffers only at a constant from the optmal schedule for fat domans, as Darte proved n [3]. Lnear schedulng was ntroduced by Lamport n [8] wth the hyperplane method, whch organzes ndced computatons nto well-defned dstnct groups called hyperplanes. Many methods were proposed n lterature to fnd the optmal hyperplane; some of them are based on the soluton of dophantne equatons [8, 0] and others on the use of nteger programmng [3, 3] or even lnear programmng n subspaces [3]. When ndex spaces wth unform dependence vectors are concerned, a polynomal complexty schedulng algorthm s presented n [7]. Once optmal parallel executon s found, an effcent method of mappng the concurrent groups of computatons (hyperplanes) nto the parallel archtecture should be appled. A systematc methodology for mappng nto fxed sze systolc arrays was presented n [8]. Snce the target archtecture s synchronously operatng, there s no need for communcaton eff- For further studyng on ths method, see [] and [3].

2 cent mappng and the man crteron for optmalty s now the total number of processors. Other methods dealt wth the same problem of mappng, whle reducng not only the sze but also the resultng dmenson of the systolc array (see [4, 6, ]). Researchers are tryng to mnmze the nter-processor communcaton, by dvdng the ndex space nto as much as possble ndependent groups of computatons. Shang and Fortes [4], and Per [], have presented methods for dvdng the ndex space nto ndependent sets of computatons, whch are assgned to dfferent processors, thus zerong the communcaton cost. Unfortunately, the ndependent parttonng of the ndex space s not always feasble. Sheu and a have presented a systematc method of parttonng the ndex space nto groups and assgn them nto dfferent processng elements [5]. her method frst projects the n- dmensonal ndex space onto the zero hyperplane. he resultng projected space s dvded nto groups of computatons, whch preserve the ntal optmal tme schedule. However, ther method does not reduce overall communcatons, snce t gnores the maxmzaton of ntragroup references. A smlar technque was presented by Kng n [5]. hs method produces better results, snce t groups together chans of related computatons, but requres a greedy (exhaustve) search to defne the group, where each computaton belongs. me schedulng s not explctly defned. hey only dscuss computatonal structures on two dmensonal ndex spaces. However, they ntroduced the dea of groupng related computatons together. In addton to ths, ther target archtecture has an unlmted number of processors, whch s not realstc for most of actual large ndex spaces. In ths paper we propose a new method, based on mappng nto an unbounded array of systolc cells. he ntal ndex space J n s lnearly transformed nto another Jï n, whch s used for ntal systolc mappng. ransformaton matrx s dvded nto the hyperplane transforma- WLRQDQGWKHVSDFHWUDQVIRUPDWLRQ6ZKLFKDFWXDOO\PDSVWKHLQLWLDOLQGH[VSDFHJ n to a (n- )-dmensonal projected vrtual space. hs space represents a vrtual array of cells, whch should be further dvded to ft the sze of the avalable hardware. Our method analytcally derves a parttonng, whch mnmzes communcaton requrements and delays. hs (n-)- dmensonal array s parttoned nto blocks of neghborng vrtual nodes, where each block s assgned to a dfferent physcal processor. We establsh the noton of a sngle cut of the projected ndex space, along a boundary drecton. All possble sngle cuts along these drectons are evaluated, by a formula calculatng the approxmate total communcaton requrements between the separate parts of each cut. Once the optmal cut (the one wth the mnmum communcaton requrements) has been selected, we perform the same procedure along the rest n- dmensons. he resultng set of cuts, called a mappng, dvdes the vrtual space nto clusters; all ponts of each cluster are assgned to the same physcal processor. he nter-processor communcaton s reduced, as neghborng vrtual processors are merged together nto a physcal one. In addton, by cuttng the vrtual space nto equal sze tles (except boundary effects), overall computatonal load s balanced. hus, the resultng space mappng s effcent, n terms of processor utlzaton and communcaton delays. Basc termnology and notaton used throughout ths paper s gven n Secton. Sectons 3 presents the propertes and algorthms used for parttonng; the vrtual array of systolc cells and the proposed mappng s elaborated n Secton 4. An example of our method s presented n Secton 5 and the obtaned results are dscussed n Secton 6.. PRELIMINARY CONCEPS AND DEFINIIONS We focus on algorthms, whch have the form of a nested FOR-loop structure, wth unform data dependences []. he algorthmc model s formally descrbed as follows:

3 for 0 = l 0 to u 0 do for n- = l n- to u n- do S S k end for end for where, 0 n-, are the loop varables, l and u are nteger-valued constants that represent the lower and upper lmts for loop varables, and S j, j k, are k assgnment statements. For the nested loop gven above, the ndex vector s defned as the vector of dmenson n: = [ 0,,, n- ]. Furthermore, each statement S j s an assgnment of the form: v 0 = E(v, v,, v p ), where v 0 s an output varable, v j, j p, are nput varables and E s an arbtrary expresson of the nput varables. It should be noted that all varables n the loop statements, may be ndexed by the ndex vector. Let J n be the set of ndces: n Τ J = {[ 0,,..., n ] : Z l u, where 0 n }, where Z s the set of nteger numbers. J n s an n-dmensonal nteger space. Each pont n ths n-dmensonal nteger space, s a dstnct nstantaton of the loop body. he nstance of statement S j s represented by S j (). Notce that the ponts of J n are ordered lexcographcally; the usual symbol `<` s used to denote ths (lnear) orderng. Dependences may exst between varables appearng n nstances of assgnment statements. A (drected) dependence between two nstances S j ( ) and S j ( ), s characterzed by the dependence vector d = -, as n [0]. he dependence matrx D contans as columns all exstng dependence vectors. A dependence vector s denoted by d, m. Such an algorthm wll be defned by ts ndex space J n and ts dependence matrx D and wll be represented by A(J n, D). 3. PARIIONING HE INDEX SPACE In ths secton, we descrbe our method of parttonng and mappng an n-dmensonal ndex space onto a mesh-connected MIMD archtecture. hs method s based on systolc loop mappng. When bounded number of cells or processors s gven, a parttonng methodology should also be appled to ft the vrtual array (resultng from mappng phase) nto the real one. We follow GPLS method (Globally Parallel Locally Sequental) 3. hs means, that snce the number of avalable cells (physcal processors) s less than the number of vrtual nodes, the vrtual array s parttoned nto blocks, whose number equals the sze of the real (physcal) array of processors. All vrtual nodes nsde the same block are sequentally executed -by the same processor. he cut of the vrtual array nto the fxed number of blocks s done along drectons, whch reduces the overall communcaton requrements and dvdes the array nto equal sze parttons. Snce everythng n each block s executed by the same physcal processor there are no communcaton requrements. he only communcaton overhead s mposed by nter-block message passng, whch s unavodable, snce all blocks are executed n parallel (Globally Parallel). Of course, there are now more local memory requrements, but the sze of memory n dstrbuted archtectures s large enough, and accesses to local memory for the same block are fast. Hereunder, the detals of our approach are further analyzed. In the followng lnes we wll use the notaton of [0]. 3 For the nverse approach LPGS see [4].

4 3. ransformaton of the Index Space J n We assume that a lnear transformaton matrx has already been selected. s gven as two VXEPDWULFHVDQG6DVLQ>@ = S. he frst row of the above matrx s the hyperplane vector whch determnes the tme executon orderng for the transformed ndex space -ï n : -ï n =.J n. he submatrx S, s the space transformaton, whch maps onto a (n-)-dmensonal array of systolc cells S.J n, as n [0]. For the needs of our method, we consder that was optmally selected. hs means that: gves the optmal hyperplane wth respect to the least makespan and S produces the optmal space mappng, n terms of the total number of cells and communcaton lnks, as shown n [7]. By applyng the transformaton matrx to the orgnal ndex space J, the latter has been transformed to a new ndex space Jï. By gnorng the frst dmenson of the transformed ndex space Jï, we obtan a projected (n-)-dmensonal ndex space, whch wll be parttoned and eventually mapped to a (n-)-dmensonal mesh of processng cells, as n [0]. 3. Cuts and ermnology In the rest of the paper, the prefx `h-` wll be used extensvely. It s read as hyper and stands for n-dmensonal. he termnology after the prefx s taken from the 3-dmensonal space. hat s: - h-space: the n-dmensonal space - h-plane: a plane n the n-dmensonal space, that s a lnear subspace of dmenson n- - h-lne: a lne n the n-dmensonal space, that s a lnear subspace of dmenson n- - h-mesh: a mesh n the (n-)-dmensonal space, that s an array of cells connected n a mesh topology. Wth all ths n mnd, the orgnal and the transformed ndex spaces J n and -ï n, are h-spaces. he projected transformed ndex space S.J n (the vrtual array of systolc cells) s an h-plane, and ts bound forms an h-polygon. he h-sdes of the h-polygon defne a number of h-lnes, whch we call bndng h-lnes: - bndng h-lne: an h-lne satsfyng the followng two propertes: at least n- transformed ndex ponts le on t (lne defnton); all remanng transformed ndex ponts are located n the same h-sem-plane. Some more termnology follows: - cut: an h-lne that s parallel to a par of bndng h-lnes; a cut separates the projected transformed ndex space nto two h-sem-planes - multple cut: a non-empty set of cuts, whose h-lnes are parallel and equdstant; ths mples the balanced parttonng of the computatonal load - mappng: a set of multple cuts, n whch there are no two dfferent multple cuts wth parallel h-lnes. 4. HE MAPPING MEHOD In the course of our method, one must keep n mnd the followng ssues: 4 matrx has been prevously selected so as, to best meet the needs of our problem, the transformed ndex space s an h-polygon wth edges that are parallel n pars, the real processor space, onto whch our transformed space s to be mapped, s an 4 As well as the assumptons found n [3].

5 h-mesh and that the optmal cut wll be gven n terms of contnuous space measures (not dscrete). 4.. Communcaton Cost Functon for Alternatve Cuts he cost value of each mappng s computed as the sum of the cost values of ts ndvdual cuts. he cost of a cut s defned to be the number of transformed dependence vectors that traverse the cut's h-lne. he formula that computes the cost value of a sngle cut s thus the followng (cost functon ): m p d c = l!, () p d = where: m s the number of dstnct dependence vectors, p s the vector that s perpendcular to the cut, d s a sngle transformed dependence vector, u s the Eucldean norm of vector u, l s the h-length of the segment of the h-lne that corresponds to the cut and s wthn the boundsriwkhwudqviruphgkvsdfhdqg!lvwkhghqvlw\riyhfwruv d along an h-lne perpendcular to d vectors. he fracton, wthn the sum of equaton, s equal to the cosne of the angle between the two vectors p and d and expresses the mplct noton that cuts that are as parallel as possble to the transformed dependence vectors, are traversed by fewer such vectors, and thus gve better (lower) cost values. he coeffcent p d s there to denote the depth of the calculaton for each dependence vector n the drecton of p. heorem Parameter! s equal to d RU! d. Proof: (A smple geometrc proof for the case that n = 3, s gven. A smlar proof can be easly derved, for greater dmenson.) Consder the dependence vector [ a,b] = d shown n Fgure, as the length segment AC. 0 dï b φ / dï φ a Fgure. Computng parameter! for the -dmensonal case (n = 3). Assume also that 0 s the lne perpendcular to vector d 7KHGHQVLW\!RIGHSHQGHQFHYHFWRUV d along the lne 0, equals to the number w of d vectors that traverse the CE segment, dvded by the length / of CE,.e.:! =. w /

6 In order to compute the number w, of the dependence vectors that traverse CE, one should frst notce that these vectors must have ther startng pont wthn the parallelogram ACEF. he area of ACEF s equal to the area of ABCD and thus s a.b. Snce the startng pont of each dependence vector has nteger coordnates and there are no holes n the projected transformed ndex space, we conclude that the number of vectors wth ther startng pont wthn ACEF, s equal to the area of ACEF. hus w = a.b. Also the length of the segment CE s computed by the formula: AD a ab w ab / = b cos(φ ) = b = b = thus, densty! equals to:! = = = d. (3) AC d d ab / d Cost functon s therefore smplfed to (cost functon ): c = l p d m = p. (5) he cost functon that s defned above, gves a heurstc measure, of how good a mappng s. Moreover, when the projected transformed ndex space s an h-parallelogram, applyng the cost functon to all dfferent mappngs, leads to the optmal soluton wth optmal processor utlzaton. 4.. Determnng Possble Cut Drectons Algorthm. Calculatng the bndng h-lnes Step.: Defne matrx V of dmenson nx n, contanng as columns all the permutatons ( n ) of the coordnates of the ndex space boundary ponts: l0 l0 u0 u0 l l u u V =. ln ln un un l n un ln un Step.&DOFXODWHWKHWUDQVIRUPDWLRQRI99ï 7.V. Step.3: Ignore the frst row of V, whch represents the tme coordnate, and construct matrx W of dmenson (n-)x n, contanng the other rows. W represents the coordnates of boundary ponts n the projected transformed ndex space. Step.4: Calculate the convex hull of all ponts contaned n W. he result s an h-polygon. he h-sdes of the convex hull are the bndng h-lnes. Step.5: Snce the h-sdes of the convex hull come n pars of parallel h-lnes 5, they can be represented n pars by nequaltes of the form: β α α α x + x + + n x n + α n γ. Let the number of such pars be b Mappng Algorthm In order to fnd all possble mappngs, we compute all dfferent ways n whch the projected transformed h-space can be allocated to the processor space. We assume that the processor 5 hs can be easly proved, snce the orgnal ndex space s an h-cube and the convex hull s the projecton of ts lnear transformaton.

7 space s denoted by a vector Œ, of n- elements. Each element shows how many processng cells are avalable on each drecton of the h-mesh, where the transformed h-space s to be mapped 6. hs s equal to the number of cuts that should be made along ths drecton plus one. Our mappng algorthm s dvded n two phases: pre-calculaton, where constant coeffcents are calculated once and for all, and calculaton, where the cost of all possble mappngs s calculated. In the followng, wll denote the number of dfferent non-trval values n vector Œ and % j wll denote these values, for j. Algorthm. Pre-calculaton of cost for multple cuts Step.: For all pars of bndng h-lnes: β α x + α x + + α x + α, b.,,, n n, n γ Step..: Let p be the perpendcular vector: p = [ α,, α,,, α, n ]. Step..: Calculate the constant coeffcent for cuts n the drecton of p : m depcost = p d j. p j= Step..3: For all dfferent numbers % j n the vector of processors, j : Step..3.: Calculate the constant coeffcent for multple cuts n the drecton of everyp, cuttng the projected transformed ndex space n % j segments: mccost γ β p, ψ j ψ j =, j depcost cutarea, p,..., p b, k = where cutarea() s a functon that returns the h-length of the cut s h-lne, accordng to Algorthm 4. Algorthm 3. Calculaton of the mnmal mappng Step 3.: For all vald mappngs: m = [ m, m,, mb ] : Step 3..: cost := 0. Step 3..: For all pars j of bndng h-lnes, j b: Step 3...: If m > then cost := cost + mccost,j, j, such that % j = m. Step 3..3: Keep track of the lowest cost. Algorthm 4. Calculaton of the h-length for a cut Parameters: - All vectors p perpendcular to the bndng h-lnes and p = [ p, p,, pn ] the one that s perpendcular to the cut. - he constant coordnate for the cut: p n. herefore, the equaton of the cut s h-lne s: p x p x p x = p. n n Step 4.: Let P the set of ponts that defne the cut's h-lne segment, ntally empty: P =. Step 4.: For all combnatons of (n-) bndng h-lnes not perpendcular to p (for all p p): Step 4..: Add the gven h-lne. Step 4..: Solve the lnear system of dmenson (n-)x(n-) to compute the pont of ntersecton of the n- h-lnes. Step 4..3: If the soluton satsfes all the remanng bndng h-lnes, then add t to P, else dscard t. Step 4.3: Calculate the h-length of the h-lne segment defned by the ponts n P. n 6 If the actual h-mesh has smaller dmenson than n-, some of ts elements wll have to be taken equal to.

8 Snce all ponts n P le on the gven h-lne, they defne an h-lne segment 7. H-lne segments are really fnte subspaces of dmenson n-, whose h-length must be calculated. he calculaton of the h-length of an h-lne segment that s defned by a set of k ponts, where k n-, can be reduced to the same calculaton, but for a set of n- ponts. he problem does not apply for the case of n = 3, but t s easy to see that n the case of n = 4 the h-lne segment s a polygon and we have to trangulate t, n order to calculate ts area. hus, t suffces to defne the h-length of an h-lne segment that s defned by n- ponts, and ths can be done nductvely: Algorthm 5. Inductve defnton of h-length Step 5.: Base case, for n = 3, use Eucldean dstance. Step 5.: Inductve case, for n > 3 do the followng: Step 5.3: Exclude one pont arbtrarly: u. Step 5.4: Use the same algorthm to calculate the h-length lï of the h-lne segment that s defned by the remanng n- ponts, n an h-space of dmenson n-3. Step 5.5: Fnd the projecton u of u on the h-plane defned by the remanng n- ponts Step 5.6: Calculate the Eucldean dstance d between u and u. Step 5.7: he result s the product of l and d. 5. EXAMPLE Consder the followng FOR-loop: for 0 = to 6 do for = to 4 do for = to 3 do α,, ) = α(,, ) + α(,, ) + α(,, ) ( In ths loop, every ndex pont α( 0,, ) s computed relatvely to the values of three other ponts of the ndex space. he dependence matrx s thus, as long as the transformaton assumed: 0 0 D = 0 0, = he transformaton matrx has been selected, so as to best demonstrate the proposed method. At the end of the example, an optmal transformaton wll be presented. he transformed ndex vector and data dependence matrx Dï are: = = = 0 + and + D = D = 0. 0 he frst lne of Dï denotes the tme dependence, of the transformed algorthm and s gnored, whle the remanng matrx has as columns the space dependences n the transformed ndex space. In our example, the transformed dependences are: d = [,], d = [,0], d = [0,. 0 ] In Fgure we can see the projected transformed ndex space, together wth the transformed dependences. In order to calculate the bounds of the projected transformed ndex space, we apply Algorthm. he matrces that are calculated n the frst three steps of the algorthm are, respectvely: 7 In the case that n = 3, we are certan that only two ponts wll be n P, snce all other ponts wll have been dscarded. However, for n > 3, the number of ponts n P can be larger.

9 V = , ï V = W = he convex hull of the ponts, whose coordnates are gven n matrx W, s a polygon whose vertces are the ponts: (, ), (7, ), (0, 5), (0, 7), (5, 7) and (, 4). d ï d 0 ï d ï j ï jï Fgure. he projected transformed ndex space. he nequaltes that descrbe the nteror of ths polygon, n the form of pars of parallel lnes, are the followng: Par : x 0, Par : y 7, Par 3: - z 5. P 3, P,3 P 3, P, P,3 j ï P, P, j ï j ï j ï Fgure 3. he mappng that was suggested for our example. Fgure 4. he real communcaton cost for the suggested mappng. For our example, we select Œ to be equal to [3, 3], whch means that the fnal -dmensonal mesh wll have 3 processors n the frst drecton and 3 processors n the other, gvng a total processor space of 9 cells. Next, we apply Algorthm, n order to pre-calculate the constant coeffcents for the mappng costs. he results of the algorthm are the followng: depcost = ( ) =, depcost = ( ) =, depcost 3 = ( ) =.44, mccost, = ( ) = ,

10 mccost, = ( ) = , mccost 3, =.44 ( ) = 0, where the length of the cuts have been calculated by usng Algorthms 4 and 5. It s not dffcult to see that the best mappng ndcated by Algorthm 3 conssts of two multple cuts, one n the drecton of par and one n the drecton of par 3. hs mappng s shown n Fgure 3. Its total cost s: cost = mccost + mccost , 3, = hs does not sgnfcantly devate from the real cost, as can be seen n Fgure 4. We can see that the real cost of the multple-cut n the drecton of par,.e. the number of dependence vector traversng the vertcal lnes, s equal to 0, whereas the estmated cost was Smlarly, the real cost of the multple cut n the drecton of par 3, s equal to, nstead of the estmated 0. We should note here, that f the projected transformed ndex space s not an h-parallelogram, there s nevtably a loss of processng cells. hs s the case n our -dmensonal example here, where the underused processors can be seen n Fgure 3. If we apply the method proposed n [9], for fndng an optmal transformaton when mappng to systolc cells, the followng results wll be produced, among many others, for our example: = 0, = 0, 3 = 0, 4 = hese matrces result n systolc arrays of 4, 4, and cells respectvely. hey all preserve the optmal hyper-plane [,, ]. he optmal transformaton for the needs of our problem s 4. hs matrx maps nto the least possble number of cells, as shown n [], and s better than 3, because t produces an array requrng only two external communcaton lnks (two external transformed dependence vectors). In the begnnng of the example, we have selected for presentaton, because the resultng structure s bg enough to demonstrate the merts of our method. 6. CONCLUSION In ths paper we have presented a new method for the parttonng and mappng of nested loops onto fxed sze, dstrbuted, mesh-connected archtectures. hs method s based on transformng the ntal n-dmensonal ndex space J n nto an equvalent -ï n, usng a transformaton (matrx), and then dvde the projected vrtual (n-)-dmensonal space, through S, nto blocks whch are assgned to dfferent processors. Interprocessor communcaton s consderably reduced, by choosng the optmal cut along each dmenson. As t was shown, the proposed method s formally presented and easly programmable. Future work ncludes affne by statement parttonng of the ndex space, to further reduce the redundant nterprocessor communcaton lnks. 7. REFERENCES []. Andronkos, N. Kozrs, G. Papakonstantnou, P. sanakas, "Optmal Schedulng for UE/UE-UC Generalzed N-Dmensonal Grd ask Graphs," to appear n Journal of Parallel and Dstrbuted Computng, he devaton s due to the fact that contnuous space propertes are used. Devatons are qute large n our example, because of ts small sze. hey become nsgnfcant when the sze of the ndex space s larger.

11 [] A. Darte and Y. Robert, Constructve Methods for Schedulng Unform Loop Nests, IEEE ransactons on Parallel and Dstrbuted Systems, vol. 5, pp. 84-8, August 994. [3] A. Darte, L. Khachyan, and Y. Robert, Lnear Schedulng s Nearly Optmal, Parallel Processng Letters, vol.., pp. 73-8, 99. [4] A. Darte and Y. Robert, Mappng Unform Loop Nests onto Dstrbuted Memory Archtectures, Parallel Computng, vol. 0, pp , 994. [5] C.-. Kng, W.-H. Chou, and L. N, Ppelned Data-Parallel Algorthms: Part II Desgn, IEEE ransactons on Parallel and Dstrbuted Systems, vol., pp , October 990. [6] N. Kozrs, G. Papakonstantnou and P. sanakas, Mappng Nested Loops onto Dstrbuted Memory Multprocessors n Proceedngs of the 997 IEEE Internatonal Conference on Parallel and Dstrbuted Systems (ICPADS 97), IEEE Press, Seoul, Korea. [7] N. Kozrs, G. Papakonstantnou, P. sanakas, Automatc Mappng and Parttonng nto Systolc Archtectures, Proceedngs of the 5th Panhelenc Conference on Informatcs, pp , Dec. 995, Athens. [8] L. Lamport, he Parallel Executon of DO Loops, Communcatons of the ACM, vol. 7, pp , February 974. [9] P.-Z. Lee and Z. M. Kedem, Mappng Nested Loop Algorthms nto Multdmensonal Systolc Arrays, IEEE ransactons on Parallel and Dstrbuted Systems, vol., pp , January 990. [0] D. I. Moldovan and J. A. B. Fortes, Parttonng and Mappng Algorthms Into Fxed Sze Systolc Arrays, IEEE ransactons on Computers, vol. C-35, pp. -, January 986. [] D. I. Moldovan, Parallel Processng: From Applcatons to Systems, Morgan Kaufmann Publshers, 993. [] J. K. Per and R. Cytron, Mnmum Dstance: A Method for Parttonng Recurrences for Multprocessors, IEEE ransactons on Computers, vol. 38, pp. 03-, August 989. [3] W. Shang and J. A. B. Fortes, me Optmal Lnear Schedules for Algorthms wth Unform Dependences, IEEE ransactons on Computers, vol. 40, pp , June 99. [4] W. Shang and J. A. B. Fortes, Independent Parttonng of Algorthms wth Unform Dependences, IEEE ransactons on Computers, vol. 4, pp , February 99. [5] J.-P. Sheu and.-h. a, Parttonng and Mappng Nested Loops on Multprocessor Systems, IEEE ransactons on Parallel and Dstrbuted Systems, vol., pp , October 99.