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On Bene’ Rearrangeuble Networks By PK, TOWANG {tani ses 31, 192 VY. , Level considered a clas of malt-stage switching noble and rot hat f the linkage pater blcen tc saga 9 chason ina specific ‘oay, then the reeling maboors are roerasgeobs. We ofer simpler Dont ty pnting out the raaton betwee ReneS laa netcorks and the Steian-Puaid hen on thee tape Cine volves V.E. Boned comilered the class, donoted here tyr Bots, sucga)y of all eonnesing networks» nth the following properties {0 viatwo side, with X terminals on auch side where 4 = TT G5) oi Dull of an add umber a= 20} Tf ata <1, = arb 1 conected ea gponfod by permutations 1)" gern the notation of Bone, eh oo sada 25) fowls of Mn dential equate evitahas of ste n, (G8) fem Sane RE Toe LHeaed proaeded to prove a specific wey of chcoting « (Se0 page 118 of Ret 1) "Onder the itches ofeach sage to define «for ugiven 1 = take the first avitel of fray with 1 outlet and ny 0 divisor of N, land connect those outlets on to eas of tho Gosh switch of fas ¢ igri. tothe woud anita off, and eonel au eutlets one eo on FT the maety ewitehes of fy {when 2 the svitches off, ; have one link onthe inlet sc, wre aatn with th et swite; proce cyliealy in this way tilall the outlts off ae asigned.” ened alee specised — puis for b= {4 3, --- 2% Wo calla network + # Bl» me, oo tee) constructed in this manner a eyeic ‘Baneé netorork, 202 rae wan sey soca couaNhuy sasontNe ened proved that a ovule Bensl nelwur ‘+ rearangenb, The proof given By Bact in Ref is howeves, quite lengthy end invelvet din the pnesent work, me ofér an elterasts proof by pointing out the links between eyelie Bene netsarka and Khe Slepian-Duggid Theorem ‘on tnte-saze Clos nebwarks, Tis hoped Uhl the imapler prot will ead to nes Insights Into the problem of sonstrvting, arrange site, We show a way to comet # © Bl), may --~ Mon) fea a © ‘B(oaynsy +o oy) sh Guat’ earangeeba then, i reurengesble ‘We ematruot a three-stage network by having the fist stego and ‘ho thi singe enh oonsst of N/m, copies of my Xm, equare evita, Av" Mein ¢ C110" Choy, 8x, are the saeond tego comics of te opies af sus, yy, « Keck ewitoh in the Sst ota oad tho ‘hie stge i Thee Tne to ery Bin tho mand wage. (Tt does nok tatter which ale or ontet of By is Hnked to wily Ay or which Ce) his gives network » e-Bay, % y=» my). Now i i sin con strvetod in this manner snd 20 on dow to the fatee-stage nator, we then call» Deneé network, Lat a Denes network ia rateangeable will for from 1 multistage version othe Slesian Duguid ‘Thesrem fn thnweage Cl natura, Ho be compote, ofa sate thin multistage version a i woot Theorer 1: Let we Booey 1+ Mes) be @ Boned network, Phen via rearrange Proof: For ¢~ 0, we haven spodil ces of a ny Xm aquere swith ‘high i clay recrmngpah't ad un eantimation ie needed. Supe that ‘hoor Lia up for ¢ — 1-— 1 3 0, wo prove Theorem 1 for ree ‘A mmsimal ssigament between etch inlet terminal and each outet, terval i portation o ofthe et of numbers (2! 4'= 1,2 >= vere = [[ ster ellowing Dene, we nond ony conser sil eqameste) A given aia esiginen con ny be owed ts feta dling ati ta con each atch ite Ret lag ad toch avi in Whe rd sage ieee sob of x8 the ona Saga) Bey = yy sory yy hose tha Y= my Dy the th eves inthe Bink sty A= Lay $4 = 4, co" Nghe be thot site in the Wed stage, © = [0,2] = 1,2, ~~ Bale Coe sider pacticlar net otagyeviteh A, Spe ew nk teria ec ACRAELE SAWS an A ave origned by ¢ tn thin sage ale (urinals, Bo Dawte hy Yata e Che Uhindtege emites Uses conta Yu Adress cand Ib Se = Fe te 1, 2 ee nh Sina Js By he 2 tet of om, aletaenés vat Seo oath distinct clement has onl lions, there exe lnk dite? clements in Jig 8s Zor on Po 1,2, Wye luge the condition of. Hal's Tone! 01 lite epeeceuistionf srs atid werd te eit att 2 — 172 = Ddcoe Nel such tay Zye Gand Z ~ Chen te 4 = Ay Be can be chose ach that 2; # Ay esye Z,- Nowe we ety clomge ta zoule tas ns esl 2, through the fst acond-age sila By, hich Dy indies gph ia zearerzenbi. Tho preb em is then educed to tact ofa mistinal amigamert in netoolo?Rype fey — 1 teh iy resctily applying she sue srqameat, we olan. sab seclgariente or th Diesen 2: Howry sylin Mone acinul Ge « Monat valor, hence earanca “Proofs A ingle singe ele Dens nec ine 9. NY equero mais, ish iva ened notwork. Supore Theorem 2 it true for 2 — 1) Tage networt aioe ( 2 We prove = (20 1) stage eyelic Bene elm ion Banos neta Tay = tesfi poder: €28( yyy <> yn) Bea evaio Rene retwork, Wesull how thot the won pm tyes 0+ wacil fee tee Anco oops no 9 Where w+ Tuy hy 200 {2 uko a ean Berl rere Furt-cnaare nl at sus Gol ‘cul awit ft fata yoy. Toke fo cach of ve eras by the ctv ofa Monod retinas, By ote inductive hypothesis, 2 i 0 Doves network, This i enuugh ta proce that » ie w Bene neznor' Tt the notation {Afi}! dole a seb of ents Ae Sein te rrr aan, 1b B/S fy danoto a rh af ets 1 Fo n'y We deinen bin Bon the two we A end and we LASHES aC etch of is Ha co every eit of Kh nor Wor saw eye Bots vets — Evid. gears bearer ir ea mage bo une. Noca thaw the e, fur} 1,2, o°> (ia rear se dares Bp fateuod AYRGIUE ale) ER = Lo 201 ime nnn MME TRADE JOR, JANI Hn tee & afr, sd panes for F201 Denomrpore the eames P= faa °°* ou soe inl (oh my 2,1 my gurmists uf Chae wate 0 ail ag] Mleron R= tent ein fe nfl mom va fin /ra ws Tu teh a, il mille i eel lige ec a a oonriatant vith their ordernge in» Buopase & @ evitoh e fp) aad gf ite eo- linac in 9g Cy a he git Hn Flo). Then & « Fle) nd Tae the soit gi AF w2 wate gf uniauals a offs em ce ® (usage comin fy 24 igo bulge weet tana-n iis o Vion vema, if 4.66.6), 1,2! — 1, hae coordinate gas exprosed in equation (2), the aie aly a owe in R60 with eoondinate 2, 5 cpresced in equation () [Newt we show chat » is » eyelic Benet actor, iby Teb x — Het etn than og = 2,3 ++» ean be decribed by eimed fetes Mee? <DEL KA Least) B for eae k pamnavorauu SETRORR? 205 wnhere [= NY namie and N= Mie for oe te flee ma (Note that in oqution (2), the owe sate of si fooedinstes "of 2.) From equations (1) and (2, sane writen i mgt = fm = net | 9 Th o Let 9, ¢ £1.) be a eiteh ving coordinate =< (mod f}. Thea sree the hie expres ® °. o ssineo TEIZI ns divides fy frore omstions (1) and (there exits ov unious uO s [fic seh that H+" forwme a2 ro) eg non he omeontng pong 7 nat 0 fh epithe ate © shane fy = naff mad sone tm neta fim aleife Hoan =O, by © cuffs ve so-o its 6 1 cun bo emily verified thet U 3 2 <1, Equation (7) mays tal m saiteh a, © f(r) which has evontinate af = 2 mo has eaoninate g — (moa fi» [Net meson that the tv ste 1G / ff) an (Geo/e stl 200 Tam wax sxeTme wer OAL JoURAR, TAYEARY A fare ach tht {God ff AH9 IRE Bad ® od Fedo f/f. ao) hold, then Gi snd C.., ace coordinates of the wae set of awit, "Bjuation (9) len {Otto H Ine! = Dh k= 1,2, mutated = ole ifs +-1)= beet 2ovemsttatedh Equation (10} implies eaftesG} = jute A = fplefts, +44 en fla -2) tte aitnGdh the <fffes etn fs tet eee ees? Rut fom eguation (2,09 ~ w then its eerespending gs nnelfas@ meu ea—o ie "Therefore 1Ga/th fod} and 10, /%s(0)) ome elesly enodinatee ofthe men st of aches, ‘That enh stein 6) Hinkel to wh a by yn diet rom of (mod F/O) Ime — 1) Fak = 1 = wl seaoraae sank? a [nod ine $B = Ba ed ba ‘ince » = ffs +> pevax i eytamntic with rxpoot to ibs mile stagy ard yy€ Bovey ey so Mesh iy of = eps fork = C+ 1, 3 1 Ara finally, again by sx argument of symmetry, teh rita in fo sine a euch {Dy oy 2 sa fon Ioevetative uf Btwn’ Leadon Bath, Soe, 20 (18), yp.