APPLICATION OF STRASSEN S ALGORITHM IN RHOTRIX ROW-COLUMN MULTIPLICATION

Transcription

1 Informtion Tehnology for People-entre Development (ITePED ) APPLIATION OF STRASSEN S ALGORITHM IN RHOTRIX ROW-OLUMN MULTIPLIATION Ezugwu E. Aslom 1, Aullhi M., Sni B. 3 n Juniu B. Shlu 4 Deprtment of Mthemtis, Ahmu Bello University, Zri-Nigeri ABSTRAT This pper presents stuy of Strssen s lgorithm n its pplition to rhotrix row-olumn multiplition. The speil se of o-size rhotries ws onsiere silly y mens of oth ping n peeling pproh. A sequentil reursive implementtion of the lgorithm using jv progrm ws one for rhotries of reltively meium sizes n lso prllel MPI frmework implementtion ws likewise moelle n presente. 1.. Keywors: Strssen s lgorithm, Rhotrix multiplition, Prllel omputing 1. INTRODUTION Rhotrix is new mthemtil prigm for mtrix theory. The stuy of rhotrix is onerne with the representtion of mthemtil rrys of rel numers in rhomoil form like strutures, s n extension of ies on mtrix-tertions n mtrix noitrets propose y (Atnssov n Shnnon, 1998). We represent the struture of n-imensionl rhotrix s follow. R n tt- t1t t-1t-1 tt1 t-t t-1t- t-1t-1 t-t-1 tt (1.1) where:, 1,..., tt enotes the min entries n, 1,..., t-1t-1 in (1.1) enotes the hert entries of the rhotrix (Sni, 4). A rhotrix woul lwys hve n o imension. It is importnt to note tht for ny n- imensionl rhotrix enote y R n will hve R 1 ( n n 1), entries (Ajie, 3) n n Z 1, thus we reiterte tht ll rhotries re of o imension. Aoringly, the 5 th n 7 th imension re given s: NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

2 R n R onsequently we n efine the rows n olumns of rhotrix min entries s: t,1 t1,1 1,1... t, t1, 1,... t,3 t1,3 1, t, t t1, t 1, t n 44 1,1 1, t1 1, t,1, t1, t 3,1 3, t1 3, t t,1 t, t1 t, t while the rows of the hert entries re similrly efine (Sni, 4). Given ny two rhotrix sy Rn n Qn the set of entries ij for R n n ij for Q n multiply eh other n entries lk for R n n lk for Q n lso o likewise. Thus, the multiplition of ny two rhotries is thus efine s: ( ) ( ) = Rn Q n= i1 j1, l1k1 i j, lk t t 1 ( ), ( ) ij 1 ijij 1 lk 1 lklk 1 (1.) The multiplition proess of (1.) is similr to tht foun in mtrix, whih is given y. l 1 ij ikkj (1.3) k. BAKGROUND FROM GROUP THEOREM TO OMPLEX NUMBER The exponent of mtrix multiplition hs prove so elusive tht it hs een given nme, whih in essene is esrie s the smllest numer suh tht ( n ) multiplitions suffie for ll. Similrly, sine ll n must e prt of ny omputtion, is lerly t lest. The ro to the urrent est upper oun on ws uilt on lever ies n inresingly omplex omintoril tehniques. Mny reserhers long elieve tht the stnr, o( n 3 ) lgorithm ws the est possile. Then, in 1969, Volker Strssen stunne the reserh worl with his.81 o ( n ) lgorithm for multiplying mtries. Still sometimes use in prtie, Strssen s lgorithm is reminisent of shortut, first oserve y Guss, for multiplying omplex numers. Though fining the prout ( + i)( + i) = + ( + )i seems to require four multiplitions, Guss oserve tht it n tully e one with three multiplition,, n ( + )( + ), euse + = ( + )( + ). Let A, B [ i]/( i 1), where A ( i) n B ( i ), then AB ( ) ( ) i. This proess uses 4 multiplitions. How possile is it optimlity? ertinly y efinition: m1 ( )( ) ( ) m m3 NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

3 AB ( m1 m m3 ) ( m m3 ) i Thus, omplex multiplition is omplishe with only 3 multiplitions over..1 A Review of Strssen s Algorithm In 1969, Strssen introue n lgorithm to ompute mtrix multiplitions. This lgorithm reues the numer of multiplitions require for omputing the A A1 B B1 A, B, A1 A B1 B M 1 := (A 1,1 + A, )(B 1,1 + B, ) M := (A,1 + A, )B 1,1 M 3 := A 1,1 (B 1, - B, ) M 4 := A, (B,1 - B 1,1 ) M 5 := (A 1,1 + A 1, )B, M 6 := (A,1 - A 1,1 )(B 1,1 + B 1, ) M 7 := (A 1, - A, )(B,1 + B, ) 1,1 = M1 +M4 - M5 +M7 1, = M3 +M5,1 = M +M4, = M1 - M +M3 +M6 In essene, inste of using the norml 8 multiplitions of trivil pproh, Strssen s lgorithm only uses 7. We n ontinue to pply Strssen s lgorithm reursively to hieve the time omplexity log7.87 of n = n. onsiering A i, j, B i, j, n i, j s mtries inste of slrs N llows mtrix multiplition over n size mtries to e performe using only 7 N = 7 log n = n log 7 = O(n.81 ). Whih mens it will run fster thn the onventionl lgorithm for suffiiently lrge mtries. Unless the mtrix hs imension of k you nnot pply Strssen s lgorithm iretly. A metho lle ynmi peeling is le to solve this prolem effetively (Spene n Knoi). Dynmi peeling mkes mtrix imensions even y stripping off n extr row or olumn s neee, n putting their ontriutions k to the finl result lter. This interesting feture motivte us to inorporte this lgorithm into the present reserh re. From our erlier efinition of 1 1 prout of two x mtries A n B from eight to seven, y expressing the entries of AB s liner omintions of prouts of liner omintions of the entries of A n B. This trik, pplie to four loks of n x n mtries, reues the prolem to seven multiplitions of n-1 x n-1 mtries. Let A, B, (.1) rhotries, we relte its row-olumn multiplition to e similr in every spet with tht of mtries multiplition n hene the importne of the present stuy. We elieve tht the Strssen s lgorithm n lso e pplie to the row-olumn multiplition of rhotries even though; rhotrix is lwys of o imension. Huss-Leermn pioneere the esign of n effiient n portle seril implementtion of Strssen s lgorithm lle DGEFMM (Huss-Leermn n Joson, 1996). DGEFMM is esigne to reple DGEMM n otine etter performne for ll mtrix sizes while minimizing the temporry storge. DGEMM, op( A) op( B) (IBM Engineering n Sientifi Suroutine Lirry). There re severl prllel implementttion of Strssen s lgorithm on istriute memory rhitetures. The methos typilly elong to three lsses. The first lss is to use the onventionl lgorithm t the top level (ross proessors) n Strssen s lgorithm t the ottom level (within proessor).one si pplition of this is tht of the Fox s Brost-Multiply- Roll (BMR) methos (Fox et l., 1987). Another similr implementtion pproh is foun in (Ohtki et l., 4) ut is more fouse on istriution sheme prtiulrly for heterogeneous lusters. The seon lss is to use Strssen s lgorithm t oth the top n ottom level. hou (hou et l., NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

4 1995) eomposes the mtrix A into x loks of sumtries, then further eomposes eh sumtrix into four x loks (i.e., 4 x 4 loks). This wy he is le to ientify 49 multiplitions n uses 7 or 49 proessors to perform multiplition onurrently. Tle.1: Strssen s Algorithm Phse 1 T1 = A + A T6 = B + B T = A1 + A T7 = B1 B T3 = A + A1 T8 = B1 B T4 = A1 A T9 = B + B1 T5 = A1 A T1 = B1 + B Phse Q1 = T1 T6 Q5 = T3 B Q = T B Q6 = T4 T9 Q3 = A T7 Q7 = T5 T1 Q4 = A T8 Phse 3 T1 = Q1 + Q4 T3 = Q3 + Q1 T = Q5 Q7 T4 = Q Q6 Phse 4 = T1 T 1 = Q3 + Q5 1 = Q + Q4 = T3 T4 We eqully elieve tht if suh tsk n e hieve y portioning mtries into suloks of x, we n eqully exten this metho to rhotries, y wy of prtitioning the hert entries into x su-loks n then pplying the o imension tehniques of solving the Strssen s lgorithm s isusse in (Huss-Leermn et l., 1996). The lst lss is using Strssen s lgorithm t the top level n the onventionl one t the extr mtrix itions impose y Strssen s lgorithm eomes less ompre to the sve mtrix multiplition ost. Therefore Strssen s lgorithm is etter use ross proessors on the top level. Finlly, for mtries with o imensions, some tehnique must e pplie to mke the imensions even, pply Strssen's lgorithm to the ltere mtrix, n then orret the results. Originlly, Strssen suggeste ping the input mtries with extr rows n olumns of zeros, so tht the imensions of ll the mtries enountere uring the reursive lls re even. After the prout hs een ompute, the extr rows n olumns re remove to otin the esire result. We ll this pproh stti ping, sine ping ours efore ny reursive lls to Strssen's lgorithm. Alterntively, eh time Strssen's lgorithm is lle reursively, n extr row of zeros n e e to eh input with n o rowimension n n extr olumn of zeros n e e for eh input with n o olumnimension. This pproh to ping is lle ynmi ping sine ping ours throughout the exeution of Strssen's lgorithm. A version of ynmi ping is use in (Huss-Leermn n Joson, 1996). Let AB e the esire mtrix prout, n let e mtrix with n even imension, where enotes n unspeifie vlue. Then Strssen s lgorithm n e use to ompute The prout AB ppers in the upper-left lok inepenent of the vlues sustitute for the s in A n B. The sme woul e true if the zeros woul hve een ple in the ottom row of B inste of the lst olumn of A. In the ove igrm one extr row n olumn hs een inserte; however, if esire, itionl rows n olumns n e inserte so long s the resulting mtrix hs even imension. Furthermore, the extr rows n olumns oul hve een inserte in ritrry lotions so long s the olumn inex in the first mtrix mthes up with the row inex in the seon mtrix. Typilly the extr rows n olumns ontin zeros n re inserte so tht A n B re in the top-left or ottom-right loks (Dougls et l, 1994). NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

5 Another pproh, lle ynmi peeling, els with o imensions y stripping off the extr rows n/or olumns s neee, n ing their ontriutions to the finl result in lter roun of fixup work. The fixup is require to inlue the ontriution of the rows n olumns tht were elete. More speifilly, let A e n m x k mtrix n B e k x n mtrix. Assuming tht m, k, n n re ll o, A n B re prtitione into the lok mtries where A hs even imensions, 1 is olumn vetor, 1is row vetor, n is single element. Similrly, where B hs even imensions, 1 is olumn vetor, 1 is row vetor, n is single element. Given this seven su-mtrix prouts, we n ompute W s W = S 5 + S 6 + S 4 S = (A + D)(E + H) + (B - D)(G + H) + D(G - E) (A + B)H = AE + DE + AH + DH + BG DG + BH DH + DG DE AH BH = AE + BG Similrly we n ompute X s X = S 1 + S = A(F - H) + (A + B)H = AF AH + AH + BH = AF + BH Agin we n ompute Y s Y = S 3 + S 4 where AB is ompute using Strssen's lgorithm, n the other omputtions onstitute the fixup work. In Setion 3, we isusse how we tully pplie similr Strssen s tehnique to esign our own rhotrix multiplition lgorithm n oe its omputtions. 3. VERIFIATION OF STRAS- SEN S ALGORITHM BY DIVIDE-AND-ONQUER TEHNIQUE Interestingly, Strssen s lgorithm orgnizes rithmeti involving the surrys A through H so tht we n ompute I, J, K, n L using just seven reursive mtrix multiplitions. We n esily verify tht they work orretly. Let us ssume tht n power of two n let us prtition P, Q, n R eh into four n n mtries, so tht we n write R = PQ s W X A B E F then, Y Z DG H Let s efines seven su-mtrix prouts: S 1 = A(F - H) S = (A + B)H S 3 = ( + D)E S 4 = D(G - E) S 5 = (A + D)(E + H) S 6 = (B - D)(G + H) S 7 = (A - )(E + F) = ( + D)E + D(G - E) = E + DE + DG DE = E + DG Finlly, we n ompute Z s Z = S 1 S 7 S 3 + S 5 = A(F - H) (A - )(E + F) ( + D)E + (A + D)(E + H) = AF AH AE + E AF + F E DE + AE + DE + AH + DH + F + DH Thus, we n ompute R = PQ using seven reursive multiplitions of mtries of size n n. Similrly, if we let funtion t (n) enote the running time of the lgorithm on n input of size n, n hrterize t (n) using n eqution tht NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

6 reltes t (n) to vlues of the funtion t for prolem sizes smller thn n. We get reursion eqution if n t( n) (3.1) n 7t( ) kn f n, If we use the itertive susttion metho n we ssume tht the mtrix size is firly lrge enough, then y sustituting (3.1) of the reurrene for eh ourrene of the funtion t on the right-hn sie, we n hrterize the running time t(n) s n t( n) 7t( ) kn, for some onstnt k >. Hene y the mster theorem, we hve the following: Theorem 1: we n multiply two n n 7 mtries in O( n log ) time. Thus, with fir it of itionl omputtion, we n perform the multiplition for n n mtries in time.88 O ( n ), whih is o ( n 3 ) time. 3.1 Strssen s Algorithm n Rhotrix Row-olumn Multiplition In orne with the Winogr s vrint of Strssen s lgorithm, whih uses seven mtrix multiplitions n 15 mtrix itions we exten n implement similr lgorithmi omputtion on rhotrix multiplition. It is well-known tht this is the minimum numer of multiplitions n itions possile for ny reursive mtrix multiplition lgorithm se on ivision into qurnts. The ivision of the mtries into qurnts follows eqution (.1). Our mjor iffiulty is on prtitioning rhotrix elements into x loks of qurnts knowing tht rhotrix hs n o imension. In tul sense this might seem omplite, ut we n still omine oth Strssen s lgorithm with the stnr multiplition tehnique of mtrix to hieve our gol. In this setion we o not isuss the theoretil proof ehin this lgorithm s they re overe elsewhere. The following steps illustrte the splitting proess of rhotries into min n hert entries. Definition: Sifting is the proess of seprting min entries or hert entries form the entire given entries of the rhotrix. It is emote y the symol with the min entries on the left n the hert entries on the right. The ove efinition n e illustrte y the following exmple. R Note tht the following reltions is true from the ove efinition where n Where enotes rhotrix emement n enotes rhotrix sifting. Similrly, NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

7 Q 5 where n It is importnt to oserve tht the hert entries of R n n Q n from the ove extrtion re not themselves rhotries, ut represents omputtionl step mong group of steps require in performing multiplition tsks. One we re le to split this, then the next step is to fin wy of prtitioning these entries into loks of x, s require y the esire lgorithm. R n n 1 1 If the rhotries R n, Q n re not of type n x n, we hve two options, first option is to either fill the missing rows n olumns with zeros ( metho referre to s ping pproh) n prtition R n n Q n into eqully size lok mtries or we elete the unprtitione rows n olumns ( metho referre to s peeling pproh) n then lter inorporte them into the omputtion proess. But first let s onsier the ping pproh s shown elow. R n nq n NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

8 A 1 1, A1 3, A 3 n A 1 B 1 1, A1 3, A 3 n A 1 n 1 1, 1 3, 1 3 n with A i, j, B i, j n i, j n 1 1 n then 1,1 = A 1,1 B 1,1 + A 1, B,1 1, = A 1,1 B 1, + A 1, B,,1 = A,1 B 1,1 + A, B,1, = A,1 B 1, + A, B, With this onstrution we hve not reue the numer of multiplitions. We still nee 8 multiplitions to lulte the i,j mtries, the sme numer of multiplitions we nee when using stnr mtrix multiplition. Now omes the importnt prt. We efine new mtries M 1 := (A 1,1 + A, )(B 1,1 + B, ) M := (A,1 + A, )B 1,1 M 3 := A 1,1 (B 1, - B, ) M 4 := A, (B,1 - B 1,1 ) M 5 := (A 1,1 + A 1, )B, M 6 := (A,1 - A 1,1 )(B 1,1 + B 1, ) M 7 := (A 1, - A, )(B,1 + B, ) whih re then use to express the i,j in terms of M k. Beuse of our efinition of the M k we n eliminte one mtrix multiplition n reue the numer of multiplitions to 7 (one multiplition for eh M k ) n express the i,j s 1,1 = M1 +M4 - M5 +M7 1, = M3 +M5,1 = M +M4, = M1 - M +M3 +M6 We iterte this ivision proess n times until the su mtries egenerte into numers (elements of the ring ). At the en of the reursive omputtion we now o wy with the zeros e to ugments the missing rows n olumns to get k our rhotrix. Alterntively we onsier one of the key pprohes to eling with mtries with o imension, the peeling metho isusse in (Huss-Leermn et l., 1996). The peeling pprohe isusse n e use to pply Strssen s lgorithm to smller rhotries otine y eleting rows n olumns. In this se, fter pplying the Strssen s lgorithm, still we nee to o some itionl work, referre to s fixup. The fixup hs to e tken into onsiertion in other to inlue the ontriution of the initil rhotrix s rows n olumns tht were elete erlier on. The peeling pproh orrespons to loking methos foun in nnon s lgorithm n is stritly pplie to only the rhotrix min entries whose imension is usully o, n not to the hert entries whose imension woul lwys e even. The input rhotries re prtitione into loks, possily of ifferent sizes, suh tht the rhotrix prouts is omptile with the imensions of the loks, tht is, if A is the ij -lok of the rhotrix R (A) n B jk is the jk -loks of the rhotrix Q (B), then the olumn imension of Aij must e equl to the row imension of B jk for ll i, j, n k. If the imensions of the loks re omptile in this sense, then ik, the ik- ij NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

9 lok of AB, is equl to i, k n k1 k, j where j runs over the loks in the i-th olumns of the lok rhotrix otine y prtitioning A. After prtitioning the input rhotries into loks, Strssen s lgorithm n then e pplie to ompute the lok prouts of rhotries whose imensions re even. The fixup work inlues the lok prouts tht re not ompute with Strssen s lgorithm n the work require to omine the lok prouts. One prtitioning sheme hooses the upper left-hn lok to e the lrgest surhotrix, with even imensions, ontining the element in the upper left-hn orner. In this sheme there n e t most four lok, whih ours when oth the row n olumn imension re o. In this se R( A) 1 3 if we ssume 1 A then where Ais n (m - 1) x (k - 1) mtrix, 1 is (m - 1) x 1 mtrix, 1 is 1 x (k - 1) mtrix, is 1 x 1 mtrix 3 A 3 NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

10 R( B) if we ssume,, 3 1 n, 3 1 then, B B 3 n B is n (k - 1) x (n - 1) mtrix, 1 is (k - 1) x 1 mtrix, 1is 1 x (n- 1) mtrix, is 1 x 1 mtrix. The prout R AB is ompute s r 1 R r r 1 B 1 1 A B A 1 1 The ove peeling pproh for whih we pplie to solve for the rhotrix min entries s stte efore o not pply to the hert elements with even imension, hene the goo news is tht we n iretly pply the loks: R R 1 R R 1 where, 1, 1, n re (m - 1) x (k-1) mtries n D, D 1, D1 n D re (k - 1) x (n - 1) mtries respetively. 3. Strssen s Algorithm omputtion Approh The omputtion proess n e ivie into two mjor prts; the first prt hnles the reursive omputtion of x lokle mtries with even imensions while the ltter prts hnles the fixup ( i.e., multiplition resulting from the reursive omputtion with the peele off row n 1 1 D 1 D D Strssen s lgorithm for the omputtion of the hert entries s shown elow. The lgorithm silly esries how to perform single level reursion on D 1 olumn) se resulting from the erlier peeling proesses rrie out. For the first prt, the omputtion n proees s shown in Tle 3.1 for oth rhotrix min entries n hert entries. Figure 3.1 n 3.1 shows the omputtion grph for the Winogr vrint of Strssen s lgorithm. The su-loks rhotries A n B, n n D re the initil inputs, shown lelling the verties t the top of the two grphs. All eges re irete from inputs, higher verties to lower ones elow. Tle 3.1: Strssen s Algorithm for Rhotrix Multiplition NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

11 omputtion for rhotrix min entries omputtion for rhotrix hert entries Phse 1 n Phse 1 n S1 = A1 + A T1 = B1 B S1 = 1 + A T1 = D1 D S = S1 + A T = B T1 S = S1 + T = D T1 S3 = A A1 T3 = B B1 S3 = 1 T3 = D D1 S4 = A1 S T4 = B1 T S4 = 1 S T4 = D1 T Phse 3 Phse 3 P1 = A B P5 = S3 T3 P1 = D P5 = S3 T3 P = A1 B1 P6 = S4 B P = 1 D1 P6 = S4 D P3 = S1 T1 P7 = A T4 P3 = S1 T1 P7 = T4 P4 = S T P4 = S T Phse 4 Phse 4 R = U1 = P1 + P U = P1 + P4 U3 = U + P5 R1 = U4 = U3 + P7 R = U4 = U3 + P7 U6 = U + P3 R1 = U7 = U6 + P6 R = U1 = P1 + P U = P1 + P4 U3 = U + P5 R1 = U4 = U3 + P7 R = U4 = U3 + P7 U6 = U + P3 R1 = U7 = U6 + P6 Thus, for exmple, in figure 3.1, A n A1 re initil inputs, whih re omine to proue S3. S3 is susequently omine with T3 to proue P5, n so on, until the finl results, su-loks of R, re proue t the ottom level. A A1 A1 A B B1 B1 B S3 S1 T1 T3 S T S4 T4 P1 P P3 P4 P5 P6 P7 U U6 U3 R R1 R1 R Figure 3.1: Depeneny grph for Winogr s vrint of Strssen s Algorithm [Huss-Leermn] 3.3 Strssen s Algorithm Fixup se NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

12 The seon prt of our omputtion els with the ynmi peeling pproh expline erlier on, se on the previous nlysis, it hs een shown tht the ynmi peeling oul e useful metho, ut it hs not een previously teste through tul implementtion for rhotries. We wrote jv progrm to implement the essentilly t-riven omputtion n presente frmework moel for the prllel MPI implementtion. After omining opertions, there re potentilly three fixup steps: Step1: R 1 1 R, Step: A r 1, 1 r1 1 Step3: r 1 r 1 1 The three steps n e ompute using the stnr rhotrix multiplition metho. It shoul e note tht, An B re the result otine from our initil reursive omputtion for even imensione lokle rhotries n 1, 1,, 1, 1 n re the peele off row n olumn elements. 4. PROGRAM TASK GRAPH FOR STRASSEN S ALGORITHM It is strightforwr to represent Strssen s lgorithm in tsk grph. A tsk grph is irete grph whose eges represent epenene reltionship n noes represent tsks (either sequentil or prllel progrms). The noe lote t the r til shoul e exeute efore the noe t the r he. Figure 3. isplys the orresponing tsk grph of one-level reursion of Strssen s lgorithm. Winogr s vrint of Strssen s lgorithm uses the require 7 multiplitions n miniml numer of 15 itions/sutrtions. This is simplifie into the following 4 stges of omputtions. Stges (1) n () B 1 r 1 r. S1 = A1 + A, S = S1 A = A1 + A A, S3 = A A1, S4 = A1 S = A1 A1 - A + A, T1 = B1 B, T = B T1 = B B1 + B, T3 = B B1, T4 = B1 T = B1 B + B1 B. Stges (3) ompute the seven prouts P1 = AB, P = A1B1, P3 = S1T1, P4 = ST, P5 = S3T3, P6 = S4B, P7 = AT4, An stge (4) omputes U1 = P1 + P, U = P1 + P4, U3 = U + P5, U4 = U3 + P7, U5 = U3 + P3, U6 = U + P3, U7 = U6 + P6. Hene we n esily verify tht R= U1, R1 = U7, R1 = U4, n R = U5. We n now pply the ove tsk nottion to proue simplifie tsk grph s shown elow. S3 S1 Strt S S4 T1 T3 T T4 P1 P P3 P4 P5 P6 P7 R R1 R1 R Stop Figure 3.: Tsk grph for Winogr s vrint of Strssen s Algorithm NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

13 5. A FRAMEWORK OF TASK- PARALLEL APPROAH USING MPI We n use strightforwr metho to prllelize Strssen s lgorithm. Bse on the lgorithm, there exist seven mtrix multiplitions (i.e., P1-P7). If we employ seven proesses to ompute the multiplitions, the progrm esign eomes very simple. The si ie is tht we ivie the tsk grph in Figure 3. into seven prts n eh prt is hnle y proess. Figure 3.3 epits the tsk prtitioning ross the seven proesses. Noes with the sme olour elong to single proess. It is importnt to note tht the positions of noes in Figure 3.3 re ientil to those in Figure 3. were omputtions re rrie out for step 1 n of S n T, step 3 of U n step 4 of R. Sine Figure 4 iretly reflets our progrm esign moel, it is strightforwr to write n MPI progrm se on the grph. For instne, proess P3 performs omputtions of S1, T1, U4 n R1 whih orrespon to the rown noes on the right. Str P4 P P3 P5 P P4 P3 P6 P P1 P P3 P4 P5 P6 P P P3 P4 P Figure 3.3: A moel of MPI tsk proesses grph to implement Strssen s lgorithm. 6. ONLUSION We hve esrie theoretil omputtionl moel for rhotrix multiplition using the Strssen s lgorithm, in whih se we were le to implement sequentil version of the lgorithm n t the sme time presente prllel frmework for the MPI implementtion on istriute memory systems. This metho we elieve is slle n portle to other pltforms euse most of the esire lle suroutines lrey exist in stnr lirry. Hene it is worthwhile to put more effort into this metho so tht the si opertion of rhotrix multiplition oul e improves further. 7. REFERENES Ajie A.O. (3). The onept of Rhotrix in Mthemtil Enrihment, Interntionl Journl of Mthemtil Eution in Siene n Tehnology, Atnssov K.T. n Shnnon A.G. (1998). Interntionl Journl of Mthemtis Eution in Siene n Tehnology, 9, hou., Deng Y., Li G. n Wng Y. (1995). Prllelizing Strssen s Metho for Mtrix Multiplition on Distriute-Memory MIMD Arhitetures, omputers for Mthemtis with Applitions, 3():49. Desprez F. n Suter.Mixe F. (1). Prllel Implementtion of the Top Level Step of Strssen n Winogr Mtrix Multiplition Algorithms, Proeeings of the 15 th Interntionl Prllel n Distriute Proessing Symposium (IPDPS 1), Sn Frniso. Dougls., M. Heroux, G. Slishmn n R. M. Smith. (1994). A portle level 3 BLAS Winogr vrint of Strssen s mtrix-mtrix multiply lgorithm. GEMMW: Journl of omputtionl Physis, :1-1. Fisher P.. n Proert R.L. (1974). Effiient Proeures for using Mtrix Algorithms in Automtion Lnguges n Progrmming, Numer 14 in NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,

14 Leture Notes in omputer Siene, Pge Springer-Verlg. Fox G.., Hey A.I n Otto S. (1987). Mtrix Algorithms on the Hyperue I: Mtrix Multiplition, Prllel omputing 4:17-. Huss-Leermn S. n Joson E. (1996). Implementtion of Strssen s Algorithm for Mtrix Multiplition, Proeeings of the 1996 AM/IEEE onferene on Superomputing, Pittsurgh. IBM Engineering n Sientifi Suroutine Lirry Version 3 Relese 3 Guie n Referene. Doument Numer SA- 77-4, IBM. Joson S.E.M., Johnson J.R., Tso A. n Turnull T. (1996). Strssen's lgorithm for mtrix multiplition: Moeling, nlysis, n implementtion. Tehnil report, enter for omputing Sienes. Tehnil Report S-TR Ohtki Y., Tkhshi D., Boku T. n Sto M. (4). Prllel Implementtion of Strssen s Mtrix Multiplition Algorithm for Heterogeneous lusters, Proeeings of the 18th Interntionl Prllel n Distriute Proessing Symposium (IPDPS 4), Snt Fe. Sni B. (4). The Row-olumn Multiplition of High Dimensionl Rhotries, Interntionl Journl of Mthemtil Eution in Siene n Tehnology, Spene M. n Knoi V. Effiient Implementtion of Strssen s Algorithm for Mtrix Multiplition, Tehnil Report, Rie University. Strssen V. (1969). Gussin Elimintion is not Optim Numer. Mt: NIGERIA OMPUTER SOIETY (NS): 1 TH INTERNATIONAL ONFERENE JULY 5-9,