Optimization of Multistage Interconnection Networks

Transcription

1 IEEE TRNSCTIONS ON COMPUTERS, VOL. C-34, NO. 3, MRCH 985 Dynmi ssiility Tstin n Pt Lnt Optimiztion o Multist Intronntion Ntworks 55 DHRM P. GRWL, SENIOR MEMBER, IEEE, ND J-SONG LEU, STUDENT MEMBER, IEEE strt -T mrn o multipl prossor systms s sn t inrs us o multist intronntion ntworks (MIN's), uilt wit svrl sts o -input -output switin lmnts (SE's). T onntivity n ult tolrn o ts ntworks r importnt prolms s MIN's r xpt to t rt o ts systms. Tis ppr mploys vrstil rp mol o n SE tt oul rprsnt ll possil stuk typ trminl ults t t ontrol lins n input/output t lins. Tis tniqu ls to rp mol o ivn MIN, mnl to tstin o its ynmi ull ss (DF) pility. T si strty o mployin jny n rility mtris nls tstin unr vrious omintions o multipl ults. Simultion o vrious ntworks is rri out to vlut t vr pt lnts wi illustrts t t o onntion pttrn on t ntwork prormn. sin mtooloy or implmntin lss o V-input V-output ntworks wit m sts (m < n) o x MIN's is lso outlin so tt DF pility n t mximum vilility oul nsur. Optimlity o su ntwork unr t prsn o ults is vrii y t simultion rsults tt sow nliil inrs in t vr pt lnt. Inx Trms -jny mtrix, vr istn, onntivity, ynmi ull ss pility, rp mol, multist intronntion ntworks, rility mtrix, stuk-t ults. I. INTRODUCTION R ECENT vns in VLSI tnoloy v nour t us o multiprossor n multiomputr systms wit lr numr o prossin lmnts (PE's) n mmory mouls (MM's). In su systms, vrious tniqus r utiliz to support rstruturl t pts twn t PE's n MM's. Tus, t intrommunition is omin n inrsinly omplx ut invitl issu. Svrl sin tniqus or t intrprossor ommunition v n rviw [] n som o tm v n onstrut. T urrnt trn is to mploy multist intronntion ntworks (MIN's) wi rquirs smnttion o t ntwork into svrl sts, wit st prtilly stisyin t input-output onntion rquirmnts. Vrious sin issus o MIN's v n ovr in t litrtur. T min mpsis s n in inin tir quivln n nonquivln n omprin tir prmuttion pilitis []-[8], sinin or onlit-r pr- Mnusript riv Jnury 5, 984; rvis July 7, 984. n rlir vrsion o tis work ws prsnt t t Fourt Intrntionl Conrn on Distriut Computin Systms, Sn Frniso, C, My 4-8, 984. T utors r wit t Dprtmnt o Eltril n Computr Eninrin, Nort Crolin Stt Univrsity, Rli, NC muttions [9], [], loritmi ptility rtristis [], n unrstnin rltiv vnts n isvnts o MIN's []. But not mu ttntion s n pi to t spt o ult tolrn wi is ruil to t sussul oprtion o multipl prossor systm. It s om xtrmly importnt, s MIN's r onsir t rt o prlll systms. Tis ppr is onrn wit t ulttolrnt pility o MIN's wn mploy to provi intrprossor ommunition in multiprossor nvironmnt. T ntwork soul implmnt in su wy tt nontstropi ult my not or omplt sut-o n t systm soul ontinu workin wit ru pity. Tis rul rtion rtristi is not only pnnt on t prossors ilur, ut lso on t MIN's. Similrily, t pt lnt rquir to provi loil link twn two prossors inits t tim ly involv in trnsmittin t inormtion rom sour to ny sir stintion. Tus, vr pt lnts in ot MIN's n sinl-st intronntion ntworks (SSIN's) [3] sm to oo rprsnttiv o tir ommunition lys. T typ o multipl prossor systms w r onrn wit is sown in Fi. [], [4]. In tis systm mol, t PE's wit tir privt mmory mouls provi sir prlllism wil PE-to-PE trnsr is iv trou t MIN. Tus, ll PE's r onnt to ot sis o t ntwork so PE n trnsmit t vi t ntwork input si wil t output is usul in rivin t rom notr PE. ri introution o t MIN's n n ovrviw o xistin tstin tniqus is ovr in Stion II. Our ult mol inluin multipl stuk-t-ults t input-output lins n ontrol lins is sri in Stion III. nrliz tst prour n t us o jny n rility mtris r illustrt in Stion IV. Stion V outlins prour or omputin t stti vr pt lnt unr vrious ulty onitions. Stion VI provis n insit to t optimum sin mtooloy o MIN's so tt lrr numr o ults n tolrt. Finlly, onluin rmrks r inlu in Stion VII. II. MULTISTGE INTERCONNECTION NETWORKS ND EXISTING TESTING TECHNIQUES Svrl o t propos MIN's v n sin usin x SE s si uilin lok. simpl rprsnttion o su n SE is sown in Fi. () n its two possil stts 8-934/85/3-55$. 985 IEEE

2 56 H E-4 H M N ~6 N m CC m r P P * PE P I. I L I - B 9 -G TH Fi.. Intronntion Ntwork Multipl prossor systm orniztion. = o X o- I Y, xl X () y () y () = =- xlo / Y X,O ---( Fi.. () Switin lmnt. () Prlll onntion wit C. () Cross onntion wit C =. () Bslin ntwork or N = 8 (n = 3). r illustrt in Fi. () n (). T prlll n ross onntion o n SE is trmin y t loil lvl ppli t its ontrol lin. n N-input N-output MIN (or simpliity, N is ssum to som powr o ; sy N = C) is onstrut in svrl sts wit st onsistin o N/ SE's. T links twn sussiv sts o t ntwork r ssin in su wy tt input oul onnt to s mny outputs s possil. n n-st ntwork n provi pt twn ny o t N-inputs n to on o t N-outputs n su MIN s n rprou rom [] in Fi. (). T ontrol lins r not sown or lrity o t irm. T prolm o ommunition n t trnsr rt tns to inrsinly ritil wn itr t ntwork lo is vy, or wn som o t lins ppn to ulty. link twn ny input n n ritrry output nnot stlis i onlit ours or rquir lin ppns to ulty. In su situtions, i t r to trnsrr twn ny two PE's o Fi., t t my v to pss to on () EEE TRNSCTIONS ON COMPUTERS, VOL. -34, NO. 3, MRCH 985 or mor intrmit PE's or rin t stintion PE. Ts onsirtions nour us to look into MIN's wit m sts (m n). - T ult inosis or lss o switin ntworks s n sri y Oprmn n Tso-Wu [5]. Ty utiliz squn o tsts to nsur t orrt oprtion o t two possil stts o SE. n unltrl stt o n SE mns tt t ontrol lins r prmnntly stuk t or. Sn n Hys [ 6] v us rp mol or monstrtin t t o ults t t ontrol lins. In tir ult mol, ty onsir t output si to k to t orrsponin input lins, s rroutin o t trou t PE is possil. Ty v mol n SE y no n intronntion link is rprsnt y irt onntin t nos. ulty SE is init y no prtition into two stions. Tir min onrn is to tst wtr, unr ivn ult onition, input o t ntwork oul loilly onnt to ny on o t ntwork outputs in init numr o psss. Ty in tis proprty s t ynmi ull ss (DF) pility. T ontrol lin ults r si to ritil i ty stroy t DF rtristis. T sm mol s n us [7] in nlyzin t ult tolrn o vrious runnt MIN's. T mjor sortomin is to ssum t prsn o ult(s) only t t ontrol input(s). In Stion III, w propos rp mol wi ovroms tis limittion. Nrrwy n So [8] v onsir inosis mol or nrl switin ntwork onstrut wit k-input k-output swits (k > ). Tir si strty is to us known oo onntin t pts in intiyin oo swits n us tis prorssivly in inin ulty onntin pt. Ty o not onsir ults t t ontrol lins. In notr rnt ppr, Wu n Fn [9] v sri simpl tstin tniqu or MIN's. In tir ppr, ty v illustrt 6 irnt possil stts o n SE n ty onsir only two stts (prlll or ross onnt) s ultr situtions n ll otrs r intrprt s ults in t SE's. In tir novl sm, ty rquir only our squns or tstin ny MIN. In t, only two omplmntry squns r n n r rpt or two irnt ontrol sttins o t ntwork, on or prlll-onnt n notr or ross-onnt mos. E squn is slt in su wy tt on-out-o-two o is us in sp. Tis mns tt only on input (n n on output) o SE is m on wil t otr rmins zro. It is wll known tt t proility o loil ults insi n IC ip is quit smll n most o t ults our t t pins []. Tus, it is mor importnt to onsir stuk-t ults t inputs, outputs, n ontrol lins o t SE's. Wu n Fn's loritm n still tst ults t t input n output lin(s) o SE(s). Furtr utiliztion o t sm squns in tstin ontrol lin ult s n ovr in []. notr mol or t ontrol lin ults in Om [] n otr ntworks s rntly n onsir in [3]. Tir optimum -tst squn llows lotion o sinl-st multipl ults n is lpul in rmovin t ulty SE's, i sinl IC ip onstituts st. In notr rnt work [4], sin prours or (n - ) st Bns typ ntwork [5] tt oul tolrt sinl n multipl ults t t ontrol lins v n outlin. omprnsiv rviw o

3 GRWL ND LEU: MULTISTGE INTERCONNECTION NETWORKS 57 tstin tniqus s ppr in rnt pulition [5]. omprison o vrious runnt ntworks s n ivn in [6]. In tis ppr, w vlut t DF pilitis o n-, sinl, n m-st (m < n) MIN's unr stuk-t ults t t ontrol lins s wll s t t inputs n outputs o t SE's. III. FULT MODEL T mol or -input -output SE o Fi. () is sown in Fi. 3() [5], [6]. Tis rp mol is s on t onntivity proprty twn t input lins X n X n output lins Y n o t SE. E on o tm is ssin no. s X (X) o ts [Fi. ()] n onnt to itr Y, or ( or Y), ts r sown y t irt s in t rp mol o Fi. 3(). It is wort mntionin tt t rp mol o Fi. 3() [6] rsmls t rulr SW Bnyn ntwork [7] in pprn. But, inst o usin t SE o Fi. (), t Bnyn is sin wit iniviully ontroll pts n its rp sows t tul link pttrn. Similrly, t rp mol o ross-r swits [8] utilizs on or ross-point n sinl swit ult rmovs n in t rp. In tis wy, our mol o -input -output SE is ltotr irnt rom t Bnyn ntwork. Consirin t SE o Fi. () in, it oul in itr o t two oprtin mos, i no ult is prsnt. But wn t ontrol lin is stuk-t-zro (s--), t rp o Fi. 3() is ru to s sown in Fi. 3(). similr moiition is sown in Fi. 3() wnvr t ontrol lin is s--. T stuk-t ults t t input n output lins o n SE v to trt in irnt wy. I lin is ulty (s--, E, ), tn t lin nnot us to trnsmit ny t. Tis is rlt in t rp mol y rmovin t orrsponin no n n limintin ll t s onntin t no. Tis is sown in Fi. 4() or s- ult t on o t input lins wil Fi. 4() illustrts t mol wn ult is prsnt t on o t output lins. Multipl ults r rprsnt y,,b, n y (, /3, y E, ). Mols or t two possil oul ults r sown in Fi. 4() n (). Fults t t two inputs n outputs l to t mol sown in Fi. 4(), wil ults t ll t input n output lins o n SE limint ll t nos rom t mol. I t ontrol lin is stuk-t- or -, n on o t input or output lins is lso s--, tn t rp mols r ru s sown in Fi. 4() n (). Two possil omintions o ults wit t ontrol lin s-- n two o t input/output lins s-- r ivn in Fi. 4() n (i). In tis wy, Fis. 3() n () n 4()-(i) rprsnt ru rp mol unr ll sinl n multipl ults in n SE. T mol o Fi. 3() oul lso us [5], [6] to rprsnt t uppr n lowr rost us in Om ntwork []. In t s o uppr rost, t uppr input X is snt to ot t outputs Y n wil t lowr rost provis Y = = X. T rp mols o Fis. 3() n () n 4 rmin vli or vrious ults in t SE or Om ntwork. T itionl ulty situtions n t orrsponin ru rps or SE wi lowr n uppr rost, r sown in Fi. 5. Y. '- - Xz~~~~~: x X x () () y X, () Fi. 3. () Grp mol o t SE o Fi. (). () Grp mol o t SE wn C S--O. () Grp mol o t SE wn C s--i. y X -%( * xi () x Y.. X y () () x./ () Y y x s- ' () x Y. * x X () X _l *X () () (i) Fi. 4. () Grp mol o t SE wn X s--. () Grp mol o t SE wn s--. () Grp mol o t SE wn X s-- n s--,b. () Grp mol ot SE wnx, s-- nx s--,8. () Grp mol o t SE wn X s--, Y, s--,l, n s---y. () Grp mol o t SE wn ontrol s-- n X s--. () Grp mol wn ontrol s--o. n Y s--. () Grp mol wn ontrol s--, Xl s--, n s--,l. (i) Grp mol wn ontrol s--, Y, s--l, n s--y. In tis wy, on t SE's v n ppropritly mol, t nlysis is similr or SE's wit n witout rost pility. For onisnss o t txt, w will onsirin t MIN's implmnt wit SE's vin only two vli stts, i.., witout vin ny rost ility. IV. x I GRPH MODEL OF MIN ND ITS DF CPBILITY Bor w o ny urtr, lt us in two mtris, t jny mtrix n t rility mtrix, otin rom t rp mol w illustrt rlir. T jny mtrix o rp is t N x N mtrix [ij] wit ij = i tr is onntin link rom no i to no j in t rp;

4 58 IEEE TRNSCTIONS ON COMPUTERS, VOL. -34, NO. 3, MRCH 985 X - - _ V YI yl yl X () X () XI * v XI I () () ().- X X I Fi. 6. () jny mtrix o t SE o Fi. (). () jny mtrix o t SE o Fi. (). () jny mtrix o t SE o Fi. (). () Fi. 5. () Grp mol o n SE or Om ntwork wn s- uppr rost. () Grp mol wn s- lowr rost. () Grp mol wn s- uppr rost n X s--y. () Grp mol wn s- lowr rost n s--y. otrwis, ij =. Fi. 6() sows t jny mtrix o t iprtit irt rp [9] o * SE o Fi. (). Wn t ontrol lin is s-- or s--i, t mtris r s sown in Fi. 6() n (), rsptivly. T rility mtrix R o rp is in s n N*N mtrix [rj] wit rij = i noj is rl rom no i, n rij = otrwis. Hr, t inormtion w n rom t rility mtrix is wtr n input port o MIN n r n output port or not, n trtr, it is not nssry to rtin ny onntivity inormtion rom t input nos to t intrmit nos [.., no numrs -6 in Fi. 7()]. T rp mol o * SE ivn in Fi. 3() n us to otin t rp mol or t slin ntwork o Fi. () n is sown in Fi. 7(). T jny mtris o o t tr sts r sown in Fi. 7(), (), n (), rsptivly. Corollry : In n mr-st MIN, lt i t jny mtrix o t iprtit irt rp o t it st rprsntin its onntivity, tn R, t rility mtrix rom input nos to output nos o t MIN, oul ivn y R x m i. i= Proo: T proo is ovious. Now w n omput t rility mtrix R y multiplyin t jny mtris I,, n 3 n t rsultin R mtrix is sown in Fi. 7(). In similr wy, t mol oul otin or MIN wit ny numr o sts n wit ny numr o ritrrily lot ults n t R -mtrix oul us to tst or t DF pility. For xmpl, onsir t multipl sinl ults o Fi. 8(). s sown in t orrsponin rp mol o Fi. 8(), no n 8 s v n limint us o t ults. T jny mtris or st,, 3 r ivn in Fi. 8(), (), n (), rsptivly. T rility - mtrix, R = * 3 ivn in Fi. 8() sows tt not ll input nos n r ll t output nos. I multipl psss r llow (i.., output n k to t input si), tn itionl nos n ss in sussiv psss. Corollry : Lt R t rility mtrix o MIN, () C D o o o ' O O () B C D E O O () 8 I F G H () t O B IE O O O O O O O O () B C D E I. () F F G H I Fi. 7. () Grp mol or t slin ntwork o Fi. (). (),, irst-st jny mtrix or MIN o Fi. 7(). (), son-st jny mtrix or MIN o Fi. 7(). () 3, tir-st jny mtrix or MIN o Fi. 7(). () R, rility mtrix or MIN o Fi. 7().

5 GRWL ND LEU: MULTISTGE INTERCONNECTION NETWORKS 59 s-- B C D E F G H () () I () () O O O O BFGH () B C D E F G H O O () () Fi. 8. () Bslin ntwork wit ults. () Grp mol o t slin ntwork o Fi. 8(). () jny mtrix in irst st. () jny mtrix in son st. () jny mtrix in tir st. () R, t rility mtrix. () R or Fi. 8(). () R3 or Fi. 8() i B C D E F G H O I O O o O O O O O O O O O - B C D E F GH O O O > I I ' O O O O O O J () B C D E F GH. () BCDEFGH tn t rility in K psss (n n, its DF) oul ivn y Rk = Rk. Proo: T proo is sl-xplntory. DF is in s proprty tt provis input o t ntwork to onnt to ny on o its outputs in init numr o psss (n n, ny PE to ny otr PE). T R -mtrix o Fi. 8() sows tt t ntwork o Fi. 8() os not llow ll input nos to onnt to on o t output nos. But y multiplyin tr tims, it oul osrv [Fi. 8()] tt tr psss trou t ntwork o Fi. 8() r oo nou to provi ommunition pts rom ny input to ny on o t output nos. Hn, t DF proprty is stisi. It my not tt its DF oul not rtin or otr omintions o ults. For xmpl, n itionl s-- t t ontrol input o SE II in t ntwork o Fi. 8() ls to rp mol o Fi. 9(). Tis rp is lrly ivi into two unonnt surps, on onsistin o t input nos n n t output nos, B, n C, n t otr ontinin t rst o t input n output nos. Tis oul lso sn rom t rility mtrix o Fi. 9() wi oul prtition s two smllr nonzro sumtris s sown in t iur. Tis will tru or ny su s i R oul prtition s [9]. R =[~R' wr R' n R" r t two nonzro sumtris. Tis osrvtion oul sily vrii y otinin t rility mtris in two, tr, n our psss o t ntwork n is sown in Fi. 9(), (), n (), rsptivly. s R6 = R5 = R4, it is lr tt no mttr ow mny tims w multiply, w will nvr l to t ny ttr rsult. Tis mns t ntwork no lonr posssss t DF pility. Torm : In n N-input N-output MIN, t DF rtristi is srtin or sinl or multipl ults t t ontrol lin n/or input n output lins o on or mor SE's - i tr xists rility mtrix Rk ( S k N) in

6 6 IEEE TRNSCTIONS ON COMPUTERS, VOL. -34, NO. 3, MRCH 985 R C B C D E F G H o o II?II I o o o o o I n o o! O O OWrO - ''i i li Ij ' I R O O O -I O% O II I O () B C D X O O O O () E F G H I () C C B C D E F 'O O O O O O O O O O O () B C D E O O O () F GH F G H. * Fi. 9. () Grp mol o t slin ntwork o Fi. 8() wit t ontrol input o SE-II s--. () R, t rility mtrix. () RorFi. 9(). () R3 or Fi. 9(). () R' R3 or Fi. 9(). < - k psss, su tt rk = or ll i,j; i,j N. Proo: I MIN s t DF proprty in minimum o k psss tn Rj = Rj_ j S k. Tis mns tt ny input no o t ntwork oul r t lst on output no (irnt tn itsl s k is ssum) tr t irst pss, n trtr it will l to r t lst on itionl output no in sussiv psss. Tus, input no soul l to r ll N output nos in t most N psss i t ntwork s t DF pility. Tis mns tt n Rk ( k 6 N) must xist su tt rk = or ll i,j, S i,j 6 N. Wn tis is stisi, w will l to onnt ny input no to ny on o t output nos in t most k psss. Q.E.D. Corollry 3: I tr r ny stuk typ ults t t input si or t output si o MIN (rintr ll primry B C D () B C D E F E F G H I o: ( < () B C D X o 3 3. OD X X o Xo (C) G H w E F G I Fi.. () Distn mtrix or Fi. 7(). () Distn mtrix or Fi. 8(). () Distn mtrix or Fi. 9(). inputs n outputs, rsptivly), tn t DF proprty nnot provi. Proo: I tr is ult t ny on o its output lins, tn it is ovious tt notin n trnsmitt on tt lin, n n t orrsponin PE nnot riv ny t. In trms o t rp mol, t output lin nnot ss y ny on o t inputs. Similrly, i tr is ult t ny on o t input lins, tn it nnot ommunit to t output lins, n n t orrsponin PE nnot sn ny t. H Q.E.D. Torm : T ults in t MIN my stroy its DF pility i n only i tr xist t lst two rility mtris R' n R + or < N, su tt rz = r+t' or ll i,j; > i,j % N n t lst on rt = or ny i,j;. i,j -N Proo: I som ults in t MIN us it to los t DF proprty, tn tr rmins t lst on pir o input n output nos i n j tt nnot onnt in init numr o stps n is init y n ntry o rk =. Morovr, i R'+ = R' tn ll sussiv powrs o R (i.., R'+ n so on) will rmin qul to R'. Tis mns tt t

7 GRWL ND LEU: MULTISTGE INTERCONNECTION NETWORKS 6 () () Fi.. () 6-input 6-output Om typ SSIN. () Moii vrsion o SSIN. onntion rom i to j n nvr provi or ny numr o psss. Q.E. D. Corollry 4: In n N-input N-output MIN, RP+', t rility mtrix witin p + psss is qul to RP or ll p N. Proo: s pr Torm, i tr xists pt rom n input to n output no, tn t mximum numr o psss rquir is qul to N. Hn, tr N psss, w out to v t orrsponin $ lmnt o RN s n ny urtr pss nnot moiy t rility lmnts, n n ij lmnts. Hn, RN+ out to rmin t sm s RN. Q.E.D. V. DISTNCE MTRIX ND VERGE PTH LENGTH OF THE MIN's T istn mtrix D [9] o MIN is in s n N x N mtrix [ij] wit t ntris wn i j jj =t lst I (i ny) su tt rtj = in RI; < I < N oo otrwis. T ntris init t numr o t psss n or rqust to r t stintion. Fi. (), (), n () sows t istn mtris o t ntworks o Fis. 7(), 8(), n 9(), rsptivly. Corollry 5: I no onlit in pt lnt is ssum or t rnom rqusts, tn t vr o t minimum pt lnt (ll t stti vr n rprsnt y SV), n omput rom t istn mtrix D s ollows: SV =i N i=l j=l wr 6 rprsnts t ly tim n in pss. tr w v t ntris or t rility mtris o MIN, it is sy to st t istn mtrix n lult t vr pt lnts. On w v ll ts rsults, w my xmin t qustion o wtr t ntwork is t st rom t SV viwpoint or not. In otr wors; oul w v ttr SV y nin t link onntion pttrns or oin somtin ls? Tis spt oul sily xmin t lst or som o ntworks lik sinl-st intronntion ntwork (SSIN) [4], [3]. T minimiztion o pt lnts in n SSIN s n ovr in [3] n w ruls v n sust or inin t onntion pttrn in n SSIN. But t t o ults on t SV s nvr n onsir. simultion prorm is implmnt to osrv t prormn o 6 input 6 output SSIN (on st o t Om ntwork [] oniurtion) n t moii vrsion s sown in Fi. () n (). Fi. inits t t o nin t onntion pttrn wn no ults r prsnt. T impt o sinl ult on t SV is sown in Fi. 3 n t moii vrsion is sn to provi ttr pr- N N

8 6 IEEE TRNSCTIONS ON COMPUTERS, VOL. -34, NO. 3, MRCH 985 Ur) 4.' tol n I), Oi {L n - C, 9-4., ) C- IL) IN typc om r-moiii tin ( om typ to- -moi4i 9 7 I o m 6 Fi.. T t o nin onntion pttrns wn no ult is prsnt in 6-input 6-output SSIN. I I. 8 I l o m 6; Fi. 5. T t o nin onntion pttrn wn sinl ult ours in t ontrol lin t t irst st o -st 6-input 6-output ntwork. m I, 4) v ILD.,,W. 4.) -' L ti, qin.e. CL '4- to 'WI L )--_-. om typ moii om m i 4 i typ ' 7 3 B ]lo i 6C Fi. 3. T t o nin onntion pttrn wn sinl ult ours in ontrol lin o 6-input 6-output SSIN. 7 I. L lo 6 Fi. 6. T t o nin onntion pttrn wn sinl ult ours in t ontrol lin t t son st o -st 6-input 6-output ntwork. C. 'F C. 4., C, _-4 om -': o/ ^ molii. typ - M 7 3 l lo J Fi. 4. T t o nin onntion pttrn wn no ult ours in t -st 6-input 6-output ntwork. ; I. om mo i 4 i typ 4 7 o 6 Fi. 7. T t o nin onntion pttrn wn sinl ult ours in t link twn sts o -st 6-input ntwork.

9 GRWL ND LEU: MULTISTGE INTERCONNECTION NETWORKS ormn. T nxt qustion to rss is wtr su onlusions r vli or nrl MIN wit svrl sts. Som onlusiv simultion rsults r otin rom t omputr prorm. Unr no ults, Fi. 4 sows t t o nin onntion pttrn rom -st 6-input 6-output Om-typ ntwork to moii vrsion. Fis. 5, 6, n 7 sow t prormns o ts -st ntworks wn sinl ult ours in ontrol lin t t irst st, son st, or t link onntin t two sts, rsptivly. 63 B VI. OPrIMuM DESIGN OF MIN FOR DF CPBILITY T m-st MIN sin wit m = n s n wily ovr in t litrtur []-[4], [7]-[]. Ts ntworks r sin in su wy tt tr is on-to-on orrsponn twn input n output nos, i.., rom input lin, tr is uniqu pt to on o t output lins, n ull onntivity rquirmnts r stisi. Systmti wys o sinin ts ntworks v lso n sri [6]. For m > n, ltrnt pts twn input-output pir provis runny n rnt work [6] provis til ount o tir ult-tolrnt pilitis. Our min onrn is to sri sin mtooloy or lss o MIN's, wit m < n, so tt t vilility n rul rtion o t prlll omputin systm oul optimiz. In otr wors, t ntwork oul sin su tt t DF proprty oul rtin or s mny ults s possil. ltou it my possil to vis otr sms too, t propos mtooloy os provi rtin r o optimlity rom t DF viw point. T sin prour is s on st tory. T two stps r s ollows. ) Prtition t N (= ') inputs n outputs into N/(m) sts wit st onsistin o m inputs n " outputs. B) Dsin t mr-sts o t ntwork su tt m-inputs o on st oul onnt to m-outputs o notr st. Tus, st will onsist o m sts, wit st orm wit '- SE's. In tis wy, o t suntworks oms ully onnt ntwork o siz m inputs n ' outputs. On su xmpl or m = n n = 4 is sown in Fi. 8, wrin 6 inputs n 6 outputs r ivi into our susts wit roup onsistin o our lmnts. T input susts r (,,, ), (,,, ), (i,j, k, ), n (m, n, o, p), wil t outputs r ivi into (, B, C, D), (E,F,G,H), (I,J,K,L), n (M,N,O,P) susts. Tis stisis prt ) o t sin prour. Prt B) is ssur y ssinin output nos in su wy tt tr is no ommon lpt twn t inputs n outputs o sust. In otr wors, t outputs rom t input no sust (,,, ) r not onnt to t output no sust (, B, C, D). It must kmmr tt or t multipl prossor systm o Fi., ot nos n r loilly t sm, s t PE works s link twn t input-output pir n. Hn, no vnt is in y onntin nos n trou t MIN. Su sin prour or n ritrry vlu o m < n is sown in Fi. 9. Bor w o y6 ~ \\ Fi. 8. p~~~6 Two-st MIN wit sinl stuk typ ults t vrious SE's (only on roup o 4-SE's, V, VI, VIII, n XIV, not ulty). onsir t optimlity o our sin prour, tr lmms r in orr. Lmm : In t ult-r prtition o mr m SE's onntin m inputs n orrsponin outputs, ny input n ss ny on o its outputs in just on pss. Proo: T sin mtooloy sri rlir mks o t prtition ntworks MIN wit m inputs-m outputs. Morovr, t R-mtrix o prtition oul sn to ontin ll on lmnts n n DF is stisi in only on pss. Hn, ny o its inputs n ss ll o its output lins in on pss. Q.E.D. Lmm : In prtition roup onsistin o m * m' SE's n onntin m inputs n t orrsponin output lins, i som or ll SE's v sinl ults (xpt t t primry input n output lins), tn primry input lin n onnt to t lst on o t primry output lins. Proo: From Fi. 3() n (), it is ovious tt ult t t ontrol lin o n SE llows input to onnt to on output. sinl ult t on o t inputs o t SE llows t otr input to onnt to ot t outputs [Fi. 4()] n sinl ult t t output si o t SE prmits ot inputs to onnt to t nonulty output [Fi. 4()]. It my not rom Fis. 3 n 4 tt ult t n input (output) lin is rlt s ult t t orrsponin output (input) lin. Hn, simultnous ults t n input n n output lin o t sm SE r onsir multipl ult. s t primry input n output lins o t MIN r

10 64 IEEE TRNSCTIONS ON COMPUTERS, VOL. -34, 3, NO. MRCH 985 ~ Inputs * outputs k pt trou t orrsponinpe tks us to no o t son prtition. T son pss tks us to on m Inputs o t outputs outputs o tir prtition n so on. In t worst s, op - m Iti will tk (K ) psss or w r t Kt prtition. t s tis roup ontins ll lty SE's n t orrsponin / io outputs R mtrix ontins ll "" lmnts, t nxt pss n tk us ttoll t output nos o t Kt prtition. Now, t - m /*p 3,; * - minputs j* ss to ll ' inputs o t irst prtition is possil. k pt troupe's tks us k to t irst prtition n E4,v/,* T nxt pss provis ss to ll ' outputs o t Inputs * t m outputs son prtition. I tis pross is ontinu, totl o (K - + K) = (K-) psss is rquir to ss ny on o t output nos. Q.E.D. Fi. 9. Optimum sin o n m-st MIN. Lmm 4: T mximum numr o tolrl ontrol lin ults, unr t onitions ivn in Torm 3, is ssum to ult r, on input n onnt to t m (n--m-). lst on output lin. Tis n lso prov usin t Proo: For ivnmr-st ninput-' output MIN, t jny mtris or st n y usin t rsultnt numr o SE's in on st = n-, so t totl numr o on-pss rility mtrix. Q.E.D. SE's = m *. T numr o prtitions ="- n t Lmm 3: spil s o Lmm riss wn t numr o SE's in on prtition =m* m'. orin to ontrol lins o ll t swits r stuk-t-zro or -on; tn t sttmnt in Torm 3, only on prtition is ssum to t rp mol or prtition ntwork will ontin m ult r wil ontrol lins in ll otr prtitions my unonnt surps, wit on irt rom on ulty. Tror, t mximum numr o tolrl ontrol primry input to only on o t primry outputs. lin ults =mr (n - m-). Q.E.D. Proo: From Fi. 3() n (), wnvr t ontrol Lmm 5: T mximum numr o tolrl link lin lin o n SE is ulty, tr is on-to-on onntion. Tis ults unr t onitions ivn in Torm 3 is mns only on input is onnt to on n only on o t (m - ) * (Q -"- ). outputs, n n t jny mtrix or st will Proo: For ivnm-st n-input-n output MIN, v only on nonzro lmnt or row n ol- tr r (m - ) intrmit onntions twn t umn. Hn, t ovrll R-mtrix will lso v only on " " sts n " link lins or st. Hn, tr is totl ntry in row n olumn. Tis woul l to n o (m- )n intrmit links (xpt t primry inputs ovrll on-to-on onntion, wit m unonnt su- n outputs). E prtition will onsist o (m - )' intrrps or roup. Q.E.D. mit links. s only prtition is ssum to ult T optimlity o t sin in trms o DF pility is r n sinl link ilur pr SE is llow in ll otr monstrt y t ollowin torm. prtitions, t totl numr o tolrl link ults = Torm 3: Ininput-'outputMINisimplmnty /(m - ) (n - m) = (m - )(n- m-) Q.E. D. mr-sts (m < n) orin to t sin prours ) n Sin t numr o ults tolrl in our sin is irly B), tn DF pility is nsur or multipl sinl ults los to t numr o SE's in t ntwork, w oul lim provi ttm r m` SE's onstitutin on prtition o m tt our mtooloy provis los to optiml solution. s inputs n m outputs r ssum ult r. T uppr n xmpl, t rp mol o t MIN sown in Fi. 8 oun or t numr o psss rquir to provi t DF wi ontins svrl sinl ults is provi in Fi.. is (`.i) wr K = -. T rility mtrix in t irst pss is ivn in Fi.. Proo: T rp mols or possil sinl ults v Unr t rnom rqust loin n los-to-inis rin ivn in Fis. 3() n () n 4() n (). T on- trtion [3], t omputr simultion mntion rlir prontivity onsirtion (n t jny mtrix), is im- vis t prormns o t MIN o Fi. 8 n is sown portnt or t DF pility. typil m-st MIN is in Fi.. T vr pt lnts r omput or no sown in Fi. 9. Tn t worst s ult oul si to ult s n wn sinl ult is prsnt t itr t prsnt wn t SE's o ll t (K - ) prtitions r ontrol lin o itr st or t link onntin t two ulty. For simpliity (n witout losin t nrlity) lt sts. T rsultin urvs init tt sinl ult s us ssum tt t kt prtition is lty. T rility vry mrinl inrs on t vr tim ly n oul mtrix R or prtition oul otin n t onsir to vry vlul simultion rsult to sup- R-mtrix or t irst (K - ) prtitions woul stisy port our lim tt our sin is oo n los to optiml Lmm wil Lmm is pplil to t lst prtition. rom ult-tolrn viwpoint> It s not n possil to s intrprt rlir, in t irst (K - ) prtitions, on ompr our sin mtooloy to otrs, s, to t st o pss woul llow ny input to onnt to t lst on o our knowl, tr os not xist ny su tniqu in t t outputs; wil t Kt prtition woul llow ny input to litrtur. onnt to ny on o its outputs in on pss. Corollry 6: T rstrition impos y Torm 3 is not nssry onition or t DF proprty. Lt us ssum tt w strt ss rom on o t inputs o t irst prtition. Unr t ssum ults, t irst pss Proo: Torm 3 is suiint or nsurin DF, ut will llow ss to t lst on o its output nos n t not nssry. T ults my su tt t R-mtrix l-

11 J IC ~ ~~~~~~~~ L i I- M, N * ~ ~~~~~ k * _ ~~~~~~~~~~~~C,+ i P m n l - 4 B jv --.~'. u - ost SE s--x B.-S--- st SE --x link --x 7 L L Fi.. lo w 6 T t o t sinl ults tt our in -st 6-input 6-output ntwork. GRWL ND LEU: MULTISTGE INTERCONNECTION NETWORKS 65 F G 6 -> C' K p 66 Fi.. rp mol o Fi. 8. D B B C D E F G IIJ K L M N O p O () i o (o o j k m n D E F p Fi.. R, t rility mtrix or Fi.. mnts my ontin ritrry l's n t multipl pss (n, multiplition o t R mtrix to itsl) my l to n R' mtrix wit ll "" ntris. Q.E.D. On su xption is illustrt in Fi. 3, wi n si to possss i r o ult tolrn. Tus, our sin prour is vry usul in implmntin MIN wit t DF pility in t prsn o ults. Torm 3 intiis t st o sinl ults t t SE's so tt it is possil to srtin t DF rtristi vn witout otinin rp mol n witout prormin lot o onntivity n rility omputtion. _ - () Fi. 3. () On-st MIN wit 4 ults. () Grp mol or Fi. 3(). VII. -- _ CONCLUDING REMRKS T ult mol o n SE is us to mol MIN's, n n jny mtrix n rility mtris r mploy to provi systmti prour o tstin t ntwork's DF pility. T vrstility o t ult mol nls us to tst t ntwork unr multipl stuk typ ults ot t t ontrol lins n t input-output lins o t SE's. In ition,

12 IEEE TRNSCTIONS ON COMPUTERS, VOL. -34, NO. 3, MRCH t R-mtrix in sussiv psss is lso usul in omputin t vr pt lnt. T ntwork sin prour nls t MIN to posss mximum ult tolrn, n n in turn optimizs t vilility o t systm. T sltion o onntion pttrn is sn to ky issu in minimizin t pt lnts in ot t SSIN's n MIN's wi oul lso us s n inx or t prormn. I w onsir t ult tolrn s wll s t minimiztion o t pt lnts simultnously, tn t optimiztion prolm o t MIN oms xtrmly omplx n w op to prsnt itionl rsults in t nr utur. REFERENCES [] D. P. rwl n T. Y. Fn, " stuy o ommunition prossor systms," Rom ir Dvl. Cntr, T. Rp. RDC-TR-79-3, D [] C. L. Wu n T. Y. Fn, "On lss o multist intronntion ntworks," IEEE Trns. Comput., vol. C-9, pp , u. 98. [3] H. J. Sil, "T tory unrlyin t prtitionin o prmuttion ntworks," IEEE Trns. Comput., vol. C-9, pp. 79-8, Spt. 98. [4] D. K. Prn n K. L. Konpni, " uniorm rprsnttion o sinl- n multi-st intronntion ntworks us in SIMD mins," IEEE Trns. Comput., vol. C-9, pp , Spt. 98. [5] D. P. rwl, "On rp torti ppro to n- n (n - I)-st intronntion ntworks," in Pro. 9t nnu. llrton Con. Commun. Contr. Comput., Spt. 3-Ot., 98, pp "Grp torti nlysis n sin o multist intronntion [6] ntworks," IEEE Trns. Comput., vol. C-3, pp , July 983. [7] L. N. Buyn n D. P. rwl, "Dsin n prormn o nrl lss o intronntion ntworks," in Pro. 98 Int. Con. Prlll Prossin, u. 4-7, 98, pp. -9; lso in IEEE Trns. Comput., vol. C-3, pp. 8-9, D [8] D. P. rwl n S.C. Kim, "On non-quivlnt multist intronntion ntworks," in Pro. t Int. Con. Prlll Prossin, u. 5-8, 98, pp [9] M.. ii n D. P. rwl, "On onlit-r prmuttions in multist intronntion ntwork," J. Diitl Syst., vol. V, no., pp. 5-34, Summr 98., "Two sinl pss prmuttions in multist intronntion nt[] works," in Pro. 98 Con. Inorm. Si. Syst., Mr. 6-8, 98, pp [] D.. Pu, D. J. Kuk, n D. H. Lwri, "Hi-sp multiprossors n ompiltion tniqus," IEEE Trns. Comput., vol. C-9, pp , Spt. 98. [] C. L. Wu n T. Y. Fn, "T rvrs-xn intronntion ntwork," IEEE Trns. Comput., vol. 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Nkmur, " smplr o iruit switin ntworks," IEEE Computr, vol., pp. 3-48, Jun 979. [9] F. Hrry, Grp Tory. Rin, M: ison-wsly, 97. [3] P. Y. Cn, P. C. Yw, n D. Lwri, "Prormn o pkt switin in ur sinl-st sul-xn ntworks," in Pro. 3r Int. Con. Distri. Comput. Syst., Ot. 8-, 98, pp Drm P. rwl (M'74-SM'79) ws orn in Blo, Ini, on pril, 945. H riv t B.E. r in ltril ninrin rom t Rvisnkr Univrsity, Ripur, Ini, in 966, t M.E. (ons.) r in ltronis n ommunition ninrin rom t Univrsity o Roork, Roork, Ini, in 968, n t D. S. T. r rom t Frl Institut o Tnoloy, Lusnn, Switzrln, in 975. H s n Mmr o t Fulty in t M.N. E + X Rionl Eninrin Coll, l, Ini, t Univrsity o Roork, t Frl Institut o Tnoloy, Soutrn Mtoist Univrsity, Dlls, TX, n Wyn Stt Univrsity, Dtroit, MI. Currntly, is wit t Nort Crolin Stt Univrsity, Rli, s Prossor in t Dprtmnt o Eltril n Computr Eninrin. His rsr intrsts inlu prlll/istriut prossin, omputr rittur, ult tolrn, n inormtion rtrivl. Dr. rwl s srv s Rr or vrious rput journls n intrntionl onrns. H ws Mmr o Prorm Committs or t COMPCON Fll o 979, t Sixt IEEE Symposium on Computr ritmti, n Svnt Symposium on Computr ritmti. Durin t yrs srv s Mmr n t Srtry o t IEEE Computr Soity Pulitions Bor, n s n wr t Soity's "Crtiit o pprition" or is srvis. Currntly, is t Cirmn o t Ruls o Prti Committ o t Pulitions Bor. H ws t Trsurr o t IEEE-CS Tnil Committ on Computr rittur n s t Prorm Cirmn or t Tirtnt Intrntionl Symposium on Computr rittur l in Jun 984. H s n Co-Gust Eitor o t IEEE Trnstions on Computrs Spil Issu on Computr ritmti n is n Eitor o t nw Journl on Prlll n Distriut Computin pulis y mi Prss. H is lso Distinuis Visitor o t IEEE Computr Soity. H is list in Wo's Wo in t Miwst, t 98 Outstnin Youn Mn o mri, n in t Dirtory o Worl Rsrrs 98's sujts pulis y t Intrntionl Tnil Inormtion Institut, Tokyo, Jpn. H is mmr o t CM, SIM, n Sim Xi. J-Son Lu (S'84) ws orn in Yun-Lin, Tiwn, on Otor 5, 957. H riv t B E. r in EECS in 98 rom Cun-Yun Coll, Tiwn,. i -55u. > t M.S. r in omputr stuis rom Nort Crolin Stt Univrsity, Rli, in 983, n is now Rsr ssistnt workin towrs t P.D. r in t Dprtmnt o Eltril n Computr Eninrin, Nort Crolin Stt Univrsity. His urrnt intrsts inlu prlll/istriut prossin n omputr ommunition.