Graph Theoretical Analysis and Design of Multistage Interconnection Networks

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1 637 I TRNSTIONS ON OMPUTRS, VOL. -32, NO. 7, JULY 1983 [39].. svnt,.. jski, n. J. Kuck, "utomtic sign wit pnnc grps," in Proc. 17t s. utomt. on, I omput. Soc. TMSI, 1980, pp [40] M.. Mcrln, " orml inition o ISPS or proving proprtis o rwr scriptions," s. Rs. n., rngi-mllon Univ., Tc. Rp., to pulis. Micl. Mcrln ws orn in oston, M, on Octor 4, rciv t.. gr in pysics rom ornll Univrsity, Itc, NY, in 1969, n t M.S. n P.. grs in lctricl nginring rom rngi-mllon Univrsity, Pittsurg, P, in 1978 n 1981, rspctivly. is currntly tcing computr scinc t oston ollg, stnut ill, M, wil stuying tology t Wston Scool o Tology,. mrig, M. s work s consultnt in sign utomtion n vriiction t rngi-mllon Univrsity, t Univrsity o Soutrn liorni n ll Ls, Murry ill, NJ. is rsrc intrsts inclu sign utomtion, sign vriiction, orml smntics n rwr scription lngugs. lic. Prkr (S'78-M'75) rciv t M.S... gr rom Stnor Univrsity, Stnor,, in 1971 n t P.. gr rom Nort rolin Stt Univrsity, Rlig, in urrntly, s is n ssistnt Prossor in t prtmnt o lctricl nginring-systms t t Univrsity o Soutrn liorni. Prior to tt s ws n ssistnt Prossor o lctricl t 00<y:;:i l. 0X: nginring t rngi-mllon Univrsity, Pittsurg, P. r rsrc intrsts inclu utomtic syntsis n optimiztion o igitl rwr, rwr-scriptiv lngugs, n microprogrmming. rp Torticl nlysis n sign o Multistg Intrconnction Ntworks RM P. RWL, SNIOR strct-tis ppr introucs two grp tortic mols tt provi uniorm procur or nlyzing 2 g-input/2 -output Multistg Intrconnction Ntworks (MIN's), implmnt wit 2-input/2-output Switcing lmnts (S's) n stisying crctristics cll t "uy proprty." Ts mols sow tt ll suc n-stg MIN's r topologiclly quivlnt n nc prov tt on MIN cn implmnt rom intgrt circuits sign or notr MIN. T propos tcniqus lso llow inticl moling n comprison o prmuttion cpilitis o n-stg MIN's n otr link-controll ntworks lik ugmnt t mnipultor n SW nyn Ntwork n nc, llows comprison o tir prmuttion. In t cs o ny conlict in t MIN, n uppr oun or t rquir numr o psss s n otin. In prlll systm, prticulr prmuttion my sirl mor rquntly tn t otrs. Wit tis qustion in min, two simpl mtoologis or signing MIN's r scri wic coul pss givn prmuttion witout ny conlict. It is sown urtr tt i t output stg o on MIN n t input stg o notr MIN mutully stisy t "uy proprty," tn ts two stgs coul mrg togtr to rsult in ntwork topologiclly quivlnt to t ns ntwork. MMR, I prticulr cs o spcil intrst is wn t scon MIN is t rvrs o t irst MIN. Tus, ny MIN cn simult ns ntwork in just two psss i t r snt in t orwr irction in t irst pss n in ckwr irction in t scon pss. Tis sows tt lowst possil oun o (2n - 1) psss r rquir to civ ny ritrry prmuttion in som spciic singl-stg ntworks. Inx Trms-ns ntwork, uy proprty, conlict-r prmuttions, grp moling, multistg intrconnction ntworks, numr o psss, prmutility, singl-stg ntwork, topologicl quivlnc. I. INTROUTION Svrl suprsystms scri in t litrtur [ 1]- [4] r sown to provi noug computtionl powr to solv complx prolms on rl-tim sis. In ll ts systms, t computtionl prlllism is otin rom multipl procssors,. wil t switcs provi t rconigurtion cpilitis in t systm. T intrconnction ntworks (IN's) r cpl o llowing simultnous communiction twn t mmory locks (M's) n t procssing lmnts (P's) n nc, Mnuscript rciv rury 11, 1982; rvis Sptmr 10, Tis r consir s t rt o t prlll systms [5]. T work is s upon mtril prtilly support y t Ntionl Scinc systm prormnc is grtly ct y t intrprocssor ountion unr rnt IST T utor ws wit t prtmnt o lctricl n omputr ngi- communiction [5]- [7]. typicl xmpl o n IN is n (m nring, Wyn Stt Univrsity, troit, MI is now wit t prtmnt o lctricl n omputr nginring, Nort rolin Stt X m) cross-r switc mploy in t xisting.mmp computr [8] n in propos urrougs scintiic procssor [3]. Univrsity, Rlig, N /83/ $ I

2 638 Suc m2 switcs provi progrmml t pts twn t P's n t M's. In suprcomputr nvironmnt, tousns o P's r to us, try orcing m to consirly lrg, wic in turn mks t us o cross-r switcs unconomicl. n ltrntiv is to ivi IN into svrl stgs; n suc sgmnt ntwork wit c stg stisying prtil connction rquirmnts yils to Multistg IN (MIN). Svrl xisting n propos systm signs [9] - [10] r s on t MIN's. T MIN's llow ny input to connct to ny on o t outputs n possss vry simpl control lgoritms. nc, ty r spcilly usul or suprcomputr wit lrg numr o P's n M's. Vrious MIN's scri in t litrtur utiliz 2-input/2-output switcing lmnts (S) s uiling lock. Svrl MIN's v rcntly n compr y Tompson [11], Sigl [12], n Wu n ng [13], v us topology scriing ruls to sow t topologicl quivlnc twn slin, moii t mnipultor, lip, omg, inirct inry n-cu n rgulr SW nyn (S = = 2) MIN's, ll implmnt wit 2 X 2 S's. ii n grwl [14], [15] us tis procur in prtitioning prmuttion into two-singl pss conlict-r prmuttions. T xisting nlyticl procurs or MIN r xtrmly complx; ty r suitl or only vry w MIN's wit 2 X 2 S's. Tis ppr utilizs simpl grp tortic pproc or nlyzing n signing t MIN's. rp tory s n osrv to vry usul nlyticl tool or gnrliz connctors, concntrtors, n connctor ntworks [16]. Msson us t connctivity concpt to otin grp mol o cross-r switcs n vlop clss o ntworks cll inominl switcing ntworks. Suc grp mol s n illustrt in ig. 1. In notr novl work, ok n Lipovski [17] introuc nyn ntwork wic utilizs group o sic nyn mouls. T nyns r scri y symmtric irct grps n r siclly link-controll ntworks. r w propos two irnt grp mols o 2 X 2 S sown in ig. 2 n tn xtn tm in otining grp mols o ny MIN. Ts grp mols.r sown to vry lpul in compring t prmuttion cpilitis o vrious MIN's. Ty provi n sy wy o rsolving t conlicts n or signing suitl MIN. In t irst prt o tis ppr, w r concrn wit clss o N-input/N-output (N 2n) MIN's, uniormly structur in n stgs suc tt c pir o S's rom t it stg is connct wit only on pir o S's longing to t (i + 1 )t stg. Tis is known s "uy" proprty [18] wit t pir o S's in t it stg cll "output uis" ( I nj2) o ig. 3. T pir t (i + I)t stg is nm s "input uis" (j n]4) o ig. 3. Two input-uy pirs lso constitut two pirs o output uis. Tis is illustrt in ig. 3 wr input uy pirs (i - j2) n (3 -j4) lso orm output uis (i-13) n (2-14) T strict uy proprty orcs t ntwork to moulr in ntur n nc cilitts xpnility wic mks it suitl or VLSI implmnttion. Tis, in turn, mks ntwork xtrmly usul or suprsystm. It is intrsting to not tt lrg clss o MIN's ll into t uy typ ctgory n w o not consir otr typs o ntworks lik I TRNSTIONS ON OMPUTRS, VOL. c-32, NO. 7, JULY 1983 Outputs X 4 c= ig X 2 cross-r switcs. irct grp mol o ig. l. ig. 2. ig. 3. =O Xi Yi 'Xi -o Yi Xi Y1 X2 Y2 X2 0 Y2 X2 Y2 (c) Switcing lmnt. Prlll connction wit = 0. (c) ross connction wit = 1. Output i-t uis (+1) t i-t (i+l)t stg stg stg stg Output u Input uis S Output uis uy proprty o S's. Intrstg uy proprty in uniorm MIN. nonrstrict lt ntworks [18]-[19] n nonuniorm nyn ntworks [17]. T nonuy typ ntworks v n sprtly covr in notr ppr [20]. In t ltr prt o tis ppr, w r concrn wit t sign o ns-lik [21] ntwork rom on or mor uy-typ MIN's, so tt ny prmuttion coul provi in conlict-r mnnr. Tn, t lowst possil numr o psss or civing ny ritrry prmuttion in som spciic singl-stg ntwork s n riv. T two grp mols o S's r prsnt in Sction II, n Sction III provis til trtmnt o t irst mol in scriing topologicl quivlnc o ll MIN's wit uy proprty. T usulnss o t scon mol is monstrt in Sction IV wr t prmuttion cpilitis o MIN's r compr wit t link-s ntworks lik t ugmnt t mnipultor [22], [23] n t rgulr SW-nyn Ntwork wit S = = 2 [17]. T nxt sction provis tcniqu to trmin t numr o psss rquir to civ sir prmuttion in givn ntwork conigurtion. Two gnrliz sign mtoologis or signing MIN, suc tt givn prmuttion coul civ witout ny conlict wil still mintining ull-connctivity, is covr in Sction VI. Sction VII introucs tcniqu to connct two MIN's so tt t prmutility o ns-typ ntwork coul ttin. Singl stg ntworks wit spciic intrconnctions v n invstigt to trmin t numr o psss rquir in civing ny prmuttion, n lowr oun s lso n otin. oncluing rmrks r inclu in Sction VIII.

3 RWL: MULTIST INTRONNTION NTWORKS 639 XIl yi X2 4 Y2 in :3 c r- _ o 9 - X2 -( 2 ig. 4. K2 Y2 K2 y2 K2 Y (c) rp mol o switcing lmnt. Switcing lmnt n its ltrntiv grp mol. (c) ltrnt positioning o input/output links. II. RP MOLS O 2 X 2 SWITIN LMNT (S) irst Mol grp mol is otin y rprsnting n S y no n irct gs, rwn rom t inputs s t sourc nos to t outputs s t sink nos. T rsulting grp o n S witout ny control lin is sown in ig. 4; n it is similr to t pysicl lyout o t links wit c S rplc y no. It my lso not tt intrcnging t pysicl positions o x1 n x2 n/ory1 n Y2 (ig. 4(c)) cngs t topology scriing rul [13] wil t grp mol o S rmins uncng. Scon Mol n ltrntiv pproc is to ssign nos to t links rtr tn to t S (ig. 4). s t control signl o t S llows t connction o inputs x - x2 to Y - Y2 or Y2.YI rspctivly, tis mns tt in tis mol, no must rc y only on no n long only on o t two gs wn spciic vlu o t control signl is us. Tus, tis grp mol is s on t connctivity n t pproc is similr to t grp mol o cross-r switcs [ 16]. It is wort mntioning tt t grp mols o ig. 4 n 4 cn s wll rprsnt t uppr (lowr) rocst (i.., t input xj(x2) is connct n trnsmitt to ot t outputs yi n Y2) typ o S introuc y Lwri or is Omg ntwork [24]. III. NTWORK MOLIN N TIR QUIVLN Initil riction cost o LSI n VLSI circuit is irly ig n t prouction cost is rltivly low. Tis is t min rson or voiing custom-m cips, n it is usully visl to concntrt on t us o o-t-sl componnts. Tus, i t I's or only on typ o MIN is commrcilly vill, tn its us n vrstility coul nnc i it coul mploy in implmnting ny sir MIN. T ig. 5. g v, _-. slin ntwork. rp mol o slin ntwork. -topologicl quivlnc o only w MIN's v n stlis in t litrtur [13], [23]. r, w prov suc quivlnc o ll possil uy-typ MIN's. Tis lso sows tt it is possil to us t I's sign or on typ o uy MIN to otin notr typ o uy MIN suitl or givn ppliction (s Sction VI). urtrmor, it provis t quivlnt mpping o vrious S's so tt t control lgoritm or ny MIN coul corrctly n convnintly ppli. T torm on quivlnc is lso usul in proving t vliity o t rsults otin in Sction VII. T grp mol o t S sown in ig. 4 coul sily us to otin grp mol o MIN, i t ollowing r ssum. ) no is ssign to c S. ) Only t primry inputs n t inl outputs r ssign to no. c) Nos r not ssign to ll t intrmit inputs n outputs s ty o not possss ny crctristics o t ntwork topology. ) Nos rprsnting S's o two stgs r connct y irct gs. Tis procur ls to grp mol o ntwork n suc mol or 8-input n 8-output slin MIN [13] o ig. 5 is sown in ig. 5. T strictly uy typ MI ntworks scri in t litrtur look irnt cus irnt suling strtgis v n utiliz or intrconncting vrious stgs o t ntwork. ut, mtmticl rltions v n riv [13], [20], [23] to sow tt som o t strictly uy typ ntworks r topologiclly quivlnt. Wit our grp mol ny uy typ MIN coul trnsorm into irct grp n tn t topologicl quivlnc is mtmticlly trnslt into grp isomorpism [20].

4 640 Torm 1: ll o t 2n-input 2n-output nonrunnt strictly uy-typ MIN's r topologiclly quivlnt to c otr provi ty stisy t ollowing conitions. ) c ntwork utilizs 2-input/2-output S s sic moul. ) c s n stgs. c) Tr r 2n-I S's pr stg. ) or ll stgs o t ntwork strict uy proprty is osrv in t cs o ot t input n t outputs. ) S's r connct in suc wy tt ny input link cn connct to ny on o t output links. Proo: T torm is prov using simplii rprsnttion o t grp mol s sown in ig. 6. T tils insi t rctngulr ox r not sown to prov t torm or ny gnrl vlu o n. It is impli tt t structur o t grp provis conncting pt rom ny input no to ny on o t output nos. To prov t torm, ssum tt it is tru or ig. 6 n w will sow its corrctnss or (2n+1)-input MIN. Suc MIN cn otin y ving two MIN's o siz 2" n tn ing on mor stg itr t input si or t t output si. On suc possil conigurtion wit ngmnttion o stg t t output si is sown in ig. 6. c pir o nos stisying input uy proprty r lso mrk t t nt stg o ig. 6. To provi ull connctivity, nos t t nt stg o two 2" ntworks r to connct togtr t t (n + I)t stg. Lt us ssum tt n y r connct to. Tn t uy proprty rquirs n y to connct to notr no; sy c. Tn t output uy rstrictions ncssitt tt wil constructing t (n + 1 )t stg, t rspctiv input uis o n -y (wic r : n () ougt to connct to two nos o t (n + 1 )t stg (sy n ) n tr is only on uniqu wy o oing tis. Tis procss coul continu or ll nos o nt stg, tus proviing only on typ o grp. s t nos (, 3) n (,y, 6) r ritrrily slct, it is possil to otin otr grps wic will isomorpic to ig. 6. Suc uy typ MIN is lso rrr to s nonrunnt MIN s tr is uniqu pt or c input-output pir. Q... IV. LTRNT RP TORTI PPRO N ITS USULNSS Vrious typs o MIN's v n propos n voct or tir us in multiprocssor systms. s rsult tr is n tmospr o conusion s to wic ntwork is suprior to t otrs. T prmuttion cpility o ntwork is xtrmly importnt or icint oprtion o suprsystm. MIN wic coul provi lrgr numr o connctions twn its inputs n outputs is prrr provi t complxity o t control is mintin t t sm lvl. Tr r ntworks lik t mnipultor [221 or ugmnt t mnipultor (M) [23] n nyn Ntwork [17] wic o not utiliz 2 X 2 S's s tir sic mouls n c link is controll iniviully. It soul point out tt Wu n ng [ 13] v moii t t mnipultor n t nyn ntworks so tt it is sil to implmnt tm using 2 X 2 S's n ty v sown topologicl quivlnc o ts moii vrsions. ut, tr os not xist ny stisctory tcniqu I TRNSTIONS ON OMPUTRS, VOL. -32, NO. 7, JULY L c - M cj I( Multistg ' Ntwrk c, =grp (n-stg P % 2n inputs) 2n 2n-1 2n- 2n Input nos nos Output nos 1st stg nt stg nos >T 2n~~~~~~~ - 2n 26+ 2n 2n no2 Otu + 1 Input nos nos (n1 Osu nos 1st stg nt stg ns Os ig. 6. Simplii rprsnttion o n-stg MIN grp mol. onstruction o (2n+l) input MIN grp rom two ntworks o siz 2n. wic comprs n quntiis prmutility o M, nyn n otr MIN's. Tis sction is vot to n ltrntiv grp tortic tcniqu tt provis uniqu mol o ll MIN's n nc, nls us to sow t rltionsips twn vrious ntworks. In orr to otin similr grp mols o two typs o MIN's, t grp mol o ig. 4 is utiliz rsulting in notr grp o slin ntwork, n is illustrt in ig. 7. In M, c cll is controll iniviully, n t t output rom cll x t stg i coms t t input to cll x' t stg i - 1 x' is in y itr o t ollowing tr possiilitis. ) Strigt: x' -x. ) Uppr rocst: x' x + 2i (mo N). c) Lowr rocst: x' x - 2i (mo N). Suc n M wit iniviul cll control is sown in ig. 8. T scon grp mol o ig. 4 lso rprsnts t link connctivity n c cll o M coul consir s n input/output no o S. Tis llows t irct us o t grp mol, sown in ig. 4 n mks t propos tcniqu prticulrly ttrctiv in otining grp mol o t M. Tis grp mol sown in ig. 8 contins som runnt gs to provi ltrnt pts in t ntwork. or xmpl, t input no "" cn connct to t output no "" using tr irnt pts o "-0-0-," "-0-2-," n "-4-2-." In tis wy, w will v t lst two itionl pts. T grp o notr link controll ntwork (nyn) is sown in ig. 8(c). 0 N

5 RWL: MULTIST INTRONNTION NTWORKS ~~~~~~ 5 INPUT OUTPUT ig. 7. n ltrnt grp mol o t slin MIN. Tus, t scon moli is usul in compring t prmuttion cpilitis o MIN lik slin ntwork wit t M or t nyn. s is clr rom igs. 7, 8 n 8(c), ts mols contin t sm numr o nos. Now, t rltionsip mong t slin, M, n nyn ntworks coul sily xmin. sugrp o t M, rwn s slin ntwork, is sown in ig. 9 n tis sows tt t M is topologiclly quivlnt to t slin. owvr, t M still contins som unus links sown wit t rokn lins in ig. 9. Tus, it is mor pproprit to quntiy tt t slin ntwork is compltly m witin t M n t lt-ovr runnt gs o t M sow its vrstility. Tis mns tt t M s mor prmuttion cpilitis compr to ny otr MIN s in y t Torm 1, try proviing n ltrnt proo o n rlir rport work on M [23]. In trms o grp tory, t slin, n ll otr uy typ MIN's in y Torm 1, coul viw s sugrps o t M, i.., t M ms t slin n otr MIN's. It is wort mpsizing tt it is iicult to prov tis rsult using t topology scriing tcniqu introuc in [13]. T grp mol o SW nyn [17] wit S = = 2, sown in ig. 8(c) provis vry intrsting rsult. ig. 8(c) lso sows t mpping o slin ntwork o ig. 7 n try sows tt SW nyn wit S = = 2 n slin ntworks r isomorpic rom grp tory viwpoint n nc, r si to topologiclly quivlnt. Otr rgulr SW nyn ntwork wit lrgr vlu o spr n n out will m t nyn wit S = = 2 n nc will v mor prmuttion cpility s compr to t slin or otr uy typ MIN's. It my not tt t link-controll ntworks mploy t lst two tims t numr o S's in MIN n nc, lrgr numr o control signls v to suppli or t link-controll ntwork. In tis wy, t grp tortic pproc provis unii wy o compring t prmuttion cpilitis o ll possil MIN's. V. NUMR O NTWORK PSSS OR ONLIT- R PRMUTTIONS s mntion rlir, connctivity o MIN is criticl wit rspct to t ovrll prormnc o lrg prlll systm. T gr o prlllism tt coul utiliz y t P's is trmin y t prlll ccssiility o t sir t. T connctivity o ntwork is crctriz y t numr o conlict-r simultnous connctions possil in ntwork. or 2n-input/2n-output MIN, tr r 2"! possil ST ig. 8. ugmnt t mnipultor. rp mol or t ugmnt t mnipultor (M) o ig. 8. (c) SW nyn wit S - =2. ig. 9. o ~~~~~~~~~~~~~~~ 9 00o 0 o () O ~~~~~~~~~~~~~ rp mol o M wit ming o t slinmn (unus gs r sown wit rokn lins). input/output prmuttions. It s rcntly n sown [14] tt crtin typs o ijctions r possil witout ny conlict n suc prmuttion sts r pnnt on t ntwork conigurtion. ut, or givn ntwork structur, not ll t prmuttions r conlict-r [14]-[15]. In ct, it is monstrt ltr in tis sction tt in t worst cs, on pss my provi connctions or only 2n/21 input-output pirs in conlict-r mnnr. In givn ntwork, it is importnt to invstigt wtr inputs-to-outputs mpping or ijction coul civ witout ny conlict or not. In cs o conlict

6 642 in prlll systms, it is usul to trmin t minimum numr o psss rquir to civ suc ijction. Tis is prorm y iviing t input-output ijctions into svrl groups in suc wy tt c group o ijction coul prorm in on pss. Tus, t numr o psss MIN rquirs or ny ritrry prmuttion cn consir s rsonl msur o its ctivnss or tt prticulr ijction. t tis point, t concpt o "utiliztion ctor" (U) is introuc. In givn grp, i n g is us or stlismnt o t ncssry link twn sourc n stintion nos, tn t link is utiliz onc or t U o tt g is ssum s 1. In tis wy, t U o ny link is givn y numricl vlu qul to t numr o tims tt link is us. T concpt o U is mploy to comput n prov lowr oun or t minimum numr o psss rquir to civ givn prmuttion. To o tis, ny on o t two grp mols coul utiliz. In t grp mol o ig. 4, only two gs o t grp mol r us or c stt o S, n t otr two gs r utiliz only in t complmntry oprting mo. T grp mol o ig. 4 utilizs ll t gs irrspctiv o t vlu o t control signl. Ts rgumnts l to t ollowing corollris n r givn witout ny proo. orollry 1: I MIN is mol using t irst mto, tr is no conlict i U o ny g is not mor tn 1. orollry 2: I scon mol is us, tn tr is no conlict i c no is rc y only on g, i.., t itiv U o ll gs irct to no, is not mor tn 1. T corollris 1 n 2 cn ppli to vlut t prmuttion cpility o MIN wit rspct to sir prmuttion. Using mols 1 or 2, t grps or MIN coul rwn n tn t U or givn ijction cn otin. Torm 2: In MIN, t lowr oun or t numr o rquir psss is givn y k, wr k is givn y t lrgst U (in mol 1) or t cumultiv lrgst U (in mol 2) to provi sir prmuttion. Proo: or ny input-output pir conction, tr is on-to-on corrsponnc, i.., uniqu pt is rquir or stlising t t pt. Tis ls to utiliztion o t corrsponing gs in t grp mol o t ntwork. I t U o ny g (or pir o gs in mol 2) is mor tn on, tn conlict xists n t only wy it cn rsolv is to llow svrl psss n to us t conlicting g only onc in c pss. Tis rquirs t lst s mny psss s t mximum numr o t U mong ll t gs (or pir o irct gs in mol 2). Q... T concpt o U in t irst mol cn si to similr to t ntris o t conlict-rsolution tl [1 3]. ut, U is lpul in rriving t orollry 3 n Torm 3. orollry 3: I t conlict is prsnt in only on stg, tn t numr o rquir psss is givn y t k-vlu o tt stg Ṗroo Tis is ovious rom n xmpl sown in ig. 10. T conlict is prsnt in link stg 2 (1) o ig. 10, ig. 10, n k-vlu is 2. nc, only two psss r rquir to civ t prmuttion. T irst pss will provi ig. 10. RWL: MULTIST INTRONNTION NTWORKS _ 1 9 c g I -½ 1 1, Utiliztion ctor or t grp mol o slin MIN: Prmuttion to civ cg Utiliztion ctor o t t/ ltrnt grp mol or t slin MIN: Prmuttion to civ (cg J c prmuttion wil t rst is provi in t scon pss. It is wort mntioning tt crtin spciic ijctions cn pss in 2-psss n ty v n intii y ii t l. [15] n tr is n to xtn tis procur or ny givn prmuttion. Torm 3: In n-stg MIN, t uppr oun or t numr o rquir psss to civ ny ritrry prmuttion is givn y 2[n/2J wil t minimum numr o inputsto-outputs mpping in c pss quls 2n/21. Proo: or proving t torm, lt us look t n xmpl sown in ig. 11. T rquir prmuttion is suc tt t mximum vlu o t U is 4 n only our gs r us in t cntr stg. In ct, tis is t worst cs n rquirs our psss to civ t prmuttion. T pt rom inputs "" to outputs "MNOP" r sown y rk lins. It is clr rom tis, tt t pts rom input nos "" mrg togtr t t no "y," us common g "yz" n rom no "z," ty gin strt mnting outwrs to "MNOP." Suc worst cs pts orm two ck to ck connct inry trs n ig. 12 illustrts tis or vrious vlus o n. Only prt o t grp is sown or clrity o t igrm n concisnss o t txt. It is clr rom tm tt ully lnc tr structur is otin rom t input nos until cntrl link ving mximum U is rc. Tis is lso tru or t output si n nc t mximum U o t cntrl link cn givn y 2Ln/2]. nc, ts mny psss r rquir to civ t worst cs prmuttions. lso, t numr o in- Q

7 RWL: MULTIST INTRONNTION NTWORKS 643 I ~~~~~~~~~~~~ :52 cgi jkimnop MNOP IJKL / Minimum no. o rquir psss Io kik L m n 0P p I p ig. 11. Utiliztion ctor or 16 X 16 slin MIN or t prmuttion ( cgijklmnop\ MNOPIJKL/ Minimum numr o rquir psss = 4. puts-to-outputs mpping in c pss cn otin i totl numr o inputs is ivi y t totl numr o psss n cn otin s 2rn/21. Q... VI. NRLIZ SIN MTOOLOY T mily o prmuttions tt coul pss in conlict-r mnnr vris rom on MIN to notr MIN [14]. I it is known priori tt, in suprsystm, prticulr prmuttion is utiliz mor rquntly tn otrs, it will usul i t slct MIN conigurtion provis suc prmuttion witout ny conlict. T otr ijctions my or my not civl in singl pss. Tis lps in stlising simultnous prlll connction or rquntly us prmuttion n nc nncs t ctivnss o t prlll systm. Our ojctiv in tis sction is to in unii sign mtoology so tt MIN cn sign tt will llow givn ijction in conlict-r mnnr wil mintining t ull connctivity o t ntwork. T ull connctivity rquirs tt t outgoing gs in t grp mol r lwys irct to nw nos so tt link twn ny input to ny on o t outputs, coul stlis. s ll MIN's utiliz som sort o sul n xcng, t prolm coul nvision s prorming propr sul in c stg tt will rsolv ny conlict' or givn input-output prmuttion. s t ntwork is to sign to pss crtin prmuttion n t irst mol provis irct loction o c S, it is, sir to consir t irst grp mol. T ollowing xiits t usulnss n univrslity o t propos grp tortic pproc. Two irnt procurs o signing MIN's r prsnt using t irst grp mol. Suppos t ollowing 8-input 8-output prmuttion is to pss conlict-r. (Input: c g Output: Mtoology 1: Rorring t prmuttion ccoring to t pysicl position o t output link ls to (Input: g c koutput: Onc ll t nos o t grp r mrk, t only qustion is ow to connct tm proprly. To o tis, t prmuttion is ivi into two groups. Tn c input no o irst group n t corrsponing no o t scon group, r connct N 12 2l 4 4 o, (c) () ig. 12. Prts o t grp illustrting worst U or vrious vlus o n. Prt o t grp illustrting worst cs U in MIN wit n = 5. Worst U pt or n = 6. (c) Worst U pt or n = 7. () Worst U pt or n = 8. to no o t nxt stg n tis no is mrk s comintion o two prvious nos [s ig. 13]. Tis is rpt or ll t nos o t two groups n v n sown in ig. 13. Trtr, s illustrt in ig. 13, c l is urtr ivi into two prts n t procss o t irst stp is continu until only on no is lt in group. Tis ls to grp s sown in ig. 14 n t corrsponing MIN is givn in ig. 14. It coul sily vrii tt t S's o t rsulting MIN stisy t strict-uy proprty t c stt o t ntwork. Torm 4: T ntwork otin rom ormntion procur mintins t connctivity proprty wil t sir prmuttion is pss in conlict-r mnnr. Proo: In t irst stp, t prmuttion is rrrng ccoring to t pysicl loctions o t output nos n tn group suc tt t two input nos r connct to t no o t scon lvl. T corrsponing output no numrs ir y N/2 (N 2n). T nxt stp mrgs two nos wos stintion nos ir y N/4 n so on. Tis compltly vois ny conlict n s t trvrsl rom ny input no to output no stisis t strict uy proprty, inry trs wit roots t ll t inputs (outputs) r orm. Tus ull connctivity is nsur. Q... Mtoology 2: In tis procur, t= grp is rwn strting rom t output no ckwrs to t input no. T givn prmuttion is ivi into two groups n on no rom c group is slct to connct to no o t prvious

8 644 I nput: g c Output: Input: I g c Output: ) : Input: I:9 Output: (c) ig. 13. No grouping in Mtoology 1. No grouping: irst stp. Scon stp o no grouping. (c) Tir stp o no grouping. irst Scon T ir ourt I TRNSTIONS ON OMPUTRS,. VOL. c-32, NO. 7, JULY 1983 Input c g Output I Input I c 9 Output! I Input : Ic : : 9: Output * j -.. t J' i t. t * tj (c) ig. 15. No grouping in Mtoology 2. No grouping: irst stp. Scon stp o no grouping. (c) Tir stp o no grouping. 4 I ourt Tir St SI Scon S4L -, irst Sp I, ig. 14. Mtoology I grp mol to implmnt conlict-r prmuttion o(input cg ooutput MIN or ig. 14. row. Tis procss is continu n is illustrt in igs. 15 n 16. Torm 5: T uy typ MIN otin in ig. 16 psss t givn prmuttion- witout ny conlict n ull connctivity is lso provi y t ntwork. Proo: T vliity o tis procur cn prov ollowing t rgumnts similr to Torm 4. Q... It is wort mntioning tt t mtoology 1 os not rquir suling o links t t output stg (similr to t Omg ntwork). No suling is n t t input stg wn Mtoology 2 is us. orollry 4: In t sign procur iscuss ov, i t corrsponnc twn ny two groups r not mintin, tn t MIN otin my nonuy typ [20]. ut, it will pss t givn prmuttion in conlict-r mnnr. Proo: I t orr o t nos is not mintin tn t uy proprty cn not ssur. ut, t ntwork will still ig. 16. < < \ Mtoology to 2 grp mol to implmnt conlict-r prmuttion o Input g Output MIN o ig. 16. v ll t input trs im into it n nc will provi t prmuttion in conlict-r mnnr. Q... It my not tt it woul xtrmly usul i ntwork coul implmnt to llow mily o prmuttions n not just only on prticulr ijction. Muc mor ort is n to intiy mtoology n t propos grptortic mtoology pprs to goo pproc in otining solution to tis gnrl prolm. VII. NS TYP N SINL-ST NTWORKS Two mtoologis or signing MIN's to pss givn prmuttion witout ny conlict, v n scri in t prvious sction. ut, wn MIN is us or systm wit lrg numr o procssing lmnts n s to utiliz or svrl irnt pplictions, it migt impossil to scrtin t us o on prticulr prmuttion mor r-

9 RWL: MULTIST INTRONNTION NTWORKS quntly tn t otrs. Tus, in gnrl, irnt typs o prmuttions my sirl n t ntwork my not l to stisy ll rquir ijctions. In suc css o conlicts, simpl solution is to us multipl inpnnt psss o Sction V. Tis is tim consuming n migt proiitiv or rl tim pplictions i lrg numr o psss r rquir. notr solution is to incrs t numr o stgs to (2n - 1) n v ns ntwork [211 wic posssss uniqu proprty o pssing ny input-output ijction. comintion o slin-rvrs slin [13] n Omgrvrs Omg [25] v n sown to quivlnt to t ns ntwork. r, it is sown ltr tt ntwork quivlnt to ns, cn implmnt y n pproprit comintion o two uy typ n-stg MIN's. Tis novl rsult is prticulrly ttrctiv or systm wit n-stg MIN, s on pss in t orwr n notr in t rvrs irction cn simult ns- ntwork n nc cn provi ny ritrry prmuttion only in 2-psss. urtrmor, control lgoritm or t ns ntwork is xtrmly complx wil t MlN's r it controll n t stintion rss itsl spciis [24] t vlu o t control signl. O cours, it is wll known tt it is iicult to trmin t intrmit lvl mpping n w r working on tis prolm to mk t st us o MIN's it-controll proprtis. ns ntwork or N = 8, is sown in ig. 17 n its grp using mol o ig. 4, is otin in ig. 17. It migt not tt t strict uy proprtis r stisi y t ns ntwork. T n-stg slin [13] MIN o ig. 5 lso utilizs rcursiv scm similr to t ns ntwork. It s uniqu crctristic tt t slin n t rvrs slin (i.., t slin ntwork wit input-output sis intrcng) r not only topologiclly quivlnt, ut r lso unctinlly quivlnt [12], [26]. Tis s n illustrt in ig. 18, wic sows t mpping o t rvrs slin MIN s slin o ig. 5. Tus, t two ntworks coul intrcng witout ltring t unctionlity o t ntwork. s mntion rlir, two stgs o t slin v n mtmticlly sown to simult t ns ntwork [261. Tis rsult coul sily prov y our grp mol. onncting t grp mols o ig. 5 n 18, it rsults in t grp o ig. 19. It is wort noting tt in t mil stg, t no pirs - n ' - ' r connct to on no in c prt o t grp n limintion or ovrlpping o tis sction will not cng t connctivity proprty o t grp n nc t prmuttion cpilitis o t ntwork. Tis ls to t grp o ig. 19 wic is xctly inticl to t grp mol o t ns ntwork o ig. 17. Torm 6: irct comintion o n or (n - I) stgs o t slin n (n - 1) or n stgs o t rvrs slin MIN's (rtr cll composit slin ntwork) cn pss ny o t 2n! possil prmuttions in conlict-r mnnr. Proo: s iscuss rlir, t lst stg o t slin cn mrg wit t irst stg o t rvrs slin n tis rsults in grp inticl to t ns ntwork n nc t rsultnt ntwork posssss t sm prmuttion cpilitis s tt o t ns ntwork. Q... c. " ig input-8 output ns ntwork. rp mol o t ns ntwork. ' ' c', ', l '4 9"4 '4 ig. 18. rvrs slin MIN mpp s slin MIN. rp mol o rvrs slin MIN. orollry 5: T slin or t rvrs slin MIN's cn us in itr or in ot t prts o ig. 19. Proo: s sown rlir, t slin n t rvrs slin MIN's r topologiclly s wll s unctionlly quivlnt to c otr. nc, ty cn sustitut or c otr. Q... Torm 7: Torm 6 is vli i t slin n t rvrs slin MIN's r rplc y ny otr two MIN's, provi tt ) ll t conitions o Torm 1 incluing t uy ' ' ' I' ' t ' 645

10 646 I TRNSTIONS ON OMPUTRS, VOL. c-32, NO. 7, JULY l ig. 19. R Rvrs slin omposit grp o slin n rvrs slin ntworks. Ruc composit grp o slin n rvrs slin ntworks. proprty r stisi y t MIN's, ) uy proprty twn t two ntworks is stisi, i.., t input uis o t lst stg in t irst MIN r mrg wit t output uy pirs in t irst stg o t scon MIN. Proo: Just lik t slin MIN, two MIN's cn put togtr si y si s is sown in ig. 20 wit t tils o t MIN's omitt or t gnrlity o t ntworks. s pr our Torm 1, t MIN's 1 n 2 r isomorpic n topologiclly quivlnt to t slin ntwork. Q... n xmpl is sown in ig. 21. orollry 6: Two psss, on orwr n t otr rvrs irction in ny n-stg MIN cn provi ny prmuttion witout ny conlict. Proo: ny n-stg MIN n its rvrs stisy t conitions o Torm 7 n t conctntion o MIN n its rvrs cn simult ns ntwork. ut, in plc o putting t MIN n its rvrs togtr, t sm ct is osrv i w v MIN n mov orwr in on pss wil going ckwr in t scon pss. Q... Tis concpt is xtrmly usul in suprsystm s ny prmuttion cn civ in n-stg MIN in just two psss. I t numr o procssing lmnts is consirly lrg, t implmnttion o n-stg MIN my irly complx, tn it migt wort mploying singl-stg ntwork riv rom spciic MIN's mploying similr intrstg connctions in ll t stgs. Lwri's Omg MIN [24] is suc n xmpl. Vry rcntly, svrl otr singlstg ntworks v n rport [27]. T ntwork riv rom Omg, is sown in ig. 22 Torm 8 sows tt tis on stg Sul-xcng-Rvrs Sul ntwork cn provi ny prmuttion in (2n - 1) psss; wic is lowr tn t xisting ouns o (3n - 1) psss or sul-x- 't 4 ct t t t c 4 9 ' RucMIRM ' Ruc M I Ntwrk Ntrk ' #1 S K' #2 T. ig. 20. omposition o two MIN. Ruc composition o two MIN. cng singl-stg ntwork [28], [29]. In tis ntwork, Sul is us in t irst n-psss y y ing ''c'''- 'g'' to input cg. T Rvrs Sul is us in t lttr (n - 1) psss y conncting outputs to ''''''''. n xmpl o prmuttion sown in ig. 10 rquirs two psss or t slin MIN. It cn sown tt to civ tis prmuttion, t Omg ntwork [24] lso rquirs mor tn on pss. Tis prmuttion cn civ y t on-stg ntwork o ig. 22 in (2n - 1) psss n t stt o t inputs n outputs in ll t psss r sown in ig. 22. Torm 8: In singl-stg sul-xcng-rvrs sul ntwork, only (2 1og2 N - 1) psss r rquir or L' ' ' ' i ' ' ' ' I' ' P l' '

11 RWL: MULTIST INTRONNTION NTWORKS 647 -M ntvwork - ' ' 4-Rvrs MIntwork *-Invrs M I ntwork- omposit grp o MIN n its rvrs. Ruc composit grp o MIN n its rvrs. (c) Ruc composit grp o Rvrs MIN n MIN. Rvrs sul s ul ' X R I M Q, I' M Q ~ II/o 'I ~~~~~~ l I l '~~~~~~~~~R K- l NlIntwr c./\ ~~~~~~~~~~~l ig. 21. c 9. Input irst Pss Sown Pss Tir Pss ourt Pss it Pss outut Rvrs Rvrs s sul Ss sul Ss sul Ss sul Ss sul Ss g-.g - c *c cx o -. <>(>< 9 g>< >< 9 <-* > c ><z'>, ><V -* g 9'c-c - < - - _ o - _ i ig. 22. On stg Sul-xcng-Rvrs Sul xcng ntwork (n psss o sul n (n - 1) psss o rvrs xcng rquir). Succssiv psss in on-stg ntwork o ig. 22 or prmuttion (g) \) ny ritrry prmuttion. Proo: s pr Torm 7 n orollry 6, composit ntwork otin rom ovrlpping Omg n Rvrs Omg MIN's coul simult ns ntwork n ntwork similr to t ns coul otin. In tis rsulting structur, t irst n-stgs will consist o sul n xcng oxs n t ltr (n - 1) stgs will v rvrs-sul n xcng. In t ntwork o ig. 22, t irst n-psss uss sul n xcng wil t ltr (n - 1) psss mploy rvrs-sul n xcng. nc, t ntwork o ig. 22

12 648 I TRNSTIONS ON OMPUTRS, VOL. c-32, NO. 7, JULY 1983 cn si to simult ns ntwork in (2n - 1) psss. Q... It my o intrst to t rrs tt svrl suc singl-stg ntworks v vry rcntly n rport [27] wic cn simult n-stg MIN in just n-psss. Tis woul mn tt n-orwr n (n - 1) rvrs psss troug ny on o ts singl-stg ntworks, coul provi ny ritrry prmuttion. IX. ONLUIN RMRKS Tis ppr introucs novl pprocs or nlyzing n signing MIN's. so tt ty coul ctivly utiliz in utur suprsystms. T grp mols r lso lpul in compring t prmutility o MIN's wit 2 X 2 S's wit t link controll MIN's. T grp mols provi sign mtoology o n-stg MIN suc tt givn prmuttion' coul pss conlict-r. T vrstility o t propos grp tortic tcniqu is lso rlct y t rsult sowing tt ny uy-typ MIN cn provi n ritrry prmuttion in just two psss n tis coul consir to on o t mjor rsults nncing t us o MIN's. urtrmor, or singl-stg ntwork, t propos tcniqu llows t us o som spciic singl-stg ntworks in simulting ns ntwork n nc civing conlict-r t trnsrs. T propos grp mols coul sily xtn or lrgr-siz S's [30], [31] n runnt ntworks, n tis woul ncourg t us o MIN's in suprsystms. T grp mol is lso usul in tsting o ts ntworks, n t ntwork prtitioning coul sily on or t multipl SIM/MIM [32] computtion mo. RRNS [1] S. 1. Krtsv n S. P. Krtsv, "Prolms o signing suprsystms wit ynmic rcitcturs," I Trns. omput., vol. -29, pp , c [2] K.. tcr, "it-sril prlll procssing systms," I Trns. omput., vol. -3 1, pp , My [31. K. Kuck n R.. Stoks, "T urrougs scintiic procssor (SP)," I Trns. omput., vol. -31, pp , My [41. P. grwl n R. Jin, " piplin psuoprlll systm rcitctur or rl tim ynmic scn nlysis," Spcil Issu, I Trns. omput., vol. -22, pp , Oct [5] T. Y. ng n. P. grwl, " stuy o communiction procssor systms," Rom ir vlopmnt ntr, Tc. Rp. TR , c. 1979, 179 pp. [6] L. uyn n. P. grwl, "Prormnc nlysis o T lgoritms on multiprocssor systms," I Trns. Sotwr ng., vol. S-9, July 1983, to pulis. [7]. Lint n T. grwl, "ommuniction issus in t sign n nlysis o prlll lgoritms," I Trns. Sotwr ng., vol. S-7, pp , Mr [8] W.. Wul n.. ll, ".mmp- multi- mini-procssor," in Proc. IPS on., vol. 41, prt II, J, 1972, pp [9] K.. tcr, "T lip ntwork in STRN," in Proc Int. on: Prlll Procss., ug. 1976, pp [10] J.. nnis, "T vritis o t low computrs," in Proc. Ist Int. on. istriut. omput. Syst., Oct. 1-5, 1979, pp [ 1].. Tompson, "nrliz connction ntworks or prlll Procssor intrcommuniction," I Trns. omput., vol. -27, pp , c [12]. J. Sigl, " Mol o SIM Mcins n omprison o Vrious Intrconnction Ntworks," I Trns. omput., vol. -28, pp , c [13]. L. Wu n T. Y. ng, "Routing Tcniqus or clss o multistg intrconnction ntworks," I Trns. omput., vol. -29, pp , ug [14] M.. ii n. P. grwl, "On conlict-r prmuttions in multistg intrconnction ntworks," J. igitl Syst., vol. V, pp , Summr [15] M.. ii,. P. grwl, n J. J. Mtznr, "Two singl pss prmuttions in multistg intrconnction ntworks," in Proc. 14t nnu. on Inorm. Sci. Syst., Princton, NJ, Mrc 26-28, 1980, pp [16]. M. Msson,.. inr, n S. Nkmur, " smplr o circuit switcing ntworks," omput., pp , Jun [17] L. R. ok n. J. Lipovski, "nyn ntworks or prtitioning multiprocssor systms," in Proc. nnu. Symp. omput. rc., c. 1973, pp [18]. M. is n J. R. Jump, "nlysis n simultion o ur lt ntworks," I Trns. omput., vol. -30, pp , pr [191 J.. Ptl, "Procssor-mmory intrconnction or multiprocssors," I Trns. omput., vol. -30, pp , Oct [20]. P. grwl n S.. Kim, "On Nonquivlnt multistg intrconnction ntworks," in Proc Int. on. Prlll Procss., ug , 1981, pp [21]. Oprmn n N. Tso-Wu, "On clss o rrrngl switcing ntworks," ll Syst. Tc. J., vol. 50, pp , My-Jun [22] T. Y. ng, "t mnipulting unctions in prlll procssors n tir implmnttions," I Trns. omput., vol. -23, pp , Mr [23]. J. Sigl n S.. Smit, "Stuy o multistg SIM intrconnction ntworks," in Proc. 5t nnu. Symp. omput. rc., pr. 1978, pp [24]. Lwri, "ccss n lignmnt o t in n rry procssor," I Trns. omput., vol. -24, pp , c [25] K. Y. L, "On t rrrngility o (2 log N - 1) stg prmuttion ntwork," in Proc Int. on Prlll Procss., ug , 1981, pp [26]. L. Wu n T. Y. ng, "T rvrs xcng intrconnction ntwork," I Trns. omput., vol. -29, pp , Spt [27] S.. Kim, "rp tortic pproc to multistg intrconnction ntworks," P.. issrttion, Wyn Stt Univ., c. 6, pp , July [28]. S. Prkr, "Nots on sul/xcng-typ switcing ntworks," I Trns. omput., vol. -29, pp , Mr [29]. L. Wu n T. Y. ng, "T univrslity o t sul-xcng ntworks," I Trns. omput., vol. -30, pp , My [30] L. uyn n. P. grwl, "VLSI prormnc o multistg intrconnction ntworks using 4 * 4 switcs," in Proc. 3r Int. on istriut. omput. Syst., Mimi, L, Oct , 1982, pp [31] L. uyn n. P. grwl, "sign n prormnc o gnrl clss o intrconnction ntworks," in Proc Int. on. Prlll Procss., llir, MI, ug , 1982, pp [32]. P. grwl, "Tsting n ult-tolrnc o multistg intrconnction ntworks," omput., vol. 15, pp , pr rm P. grwl (M'74-SM'79), or iogrpy n potogrp, s p. 330 o t pril 1983 issu o tis TRNSTIONS.

Functions. A is called domain of f, and B is called codomain of f. If f maps element a A to element b B, we write f (a) = b

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