THE BELL SYSTEM TECHNICAL JOURNAL Fotine a ‘Outer Noster On the Addressing Problem for Loop Switching y BL GRAHAM and HO. POLAK, “Phe ethan lo perform the ewig janet of the Bet Syst. hone een deve acer the fademeoael garam pers tha the hotig time of the cowed old bg compared to fe ie Need 1 2 99 the coll ns ccnnieni eta forma ef eomorenceaticn wk and amang Conyers the psbity its that Maseny, ihe deine at So Head vit rea asa though 2 caremaztion netork without the phat rentztion 0} annple delicate pack bere Feeney. Pe ch ssn aon propane vee fd uae valled “los i aubacrbers, perhaps ea! thovgit of a8 ceneputer trina ar oh tor wnrny hoip Phase “lad” loops ace ores by easious ine tra ancthe at llme tao se thw coined tone anther wal no a “nation” oop. I} @ messsie from ane tao it eatinad for a abucriter on nvatinr loop i proces tevvnd the urging toto a stable stching poet wer i may ‘hone tetera diferent Uap, thie proves omtsmaing wn the measane raves ite detivaton, Phe question rat nen, How the message Sto heh segue af toope to jlo, T would be diel or tie ach junction ob ble yo imple te the deat 2409 -Ta wn. SYSTEM TREN FOL, GTO 8 trees at the tnd of the message which would dlrine which beter the mettage shoul wake at that junction, Tn thi paper ce tapite atti of adescng the ope whi hae 1) HE peeoie an extremely simple routing arate tobe asd by the metanges in reaing their destinations (G0) By wen shia aoteny, 2 mecage wth aboays fake the share pesair ath beeen acy fo leat loops th tke sane tion. {G5 The mothed of eben applies to any eallection of lcpt, no matters epee interconnections {Phe aiveaniny mene 06 propose willbe appli? grimly a heal loca whare te muta ateenenaaiony wy be go mri. Tf eetesn ‘arent of hirarcieal erase tral us Be repaid waa srry, a anggeted by J. Ie, Pree” ii posable (» ahiow esinga which ave bath eon at que fet. ‘The muthods uted to verform tho willing foetion® of the Bell ‘Sram have bers developed unde ch adsense tha he Tg Lene of Uke com ele eal ile many) to ete eo eet tip the eal, Tee iy sense un hd oione af a ose bill the llvempl ie mle co tablish Tre cmuecon. Ip soasdaring exten Tar of eomanniention with ane tanang ooraptcers, an ell se te oneideration of may shoe fir Lime vison’ stitching, the fossiblt aries thu a moss, wih ite dertinaon a ile head, might Tia is a hgh onanunioaLinn newer Witton awatiog the ped reliaesion uf w curate dione palhbelore beginning om it ‘ne ach acon hae been propened by JR Pier, and may: bo called “up seiteing.” We iayine abserbers, perez bow tht fsa eumroulee teal or ler data gsuaraing devioes, ov onc-ns Toone. If a moartage ip deeined for a aubeeriber ox anoter oop i proceeds around se vigiatng ha ta aitable ssitehing poist where iurmay hone to unter dllerert leap wl eentizue the procass ti iron eesti on ‘The question aow comes up, Low che eveugy so knoe ahich so vases of ooze ta follom, A muffeiontiy complicated menury in the orgiating loop might of ours, lok up xn apopata rou, and then Altea seine » onplete pach; bul thie ve old sed peeps in woo emrrermec Po opproprate solution, Th would be more convenient and sometimes pioferable ifthe equipment al eh junelinn eal spp” somo simple fest tothe destination nddnss av che hesd ofthe mestige wie would determine whieh choivs the mentge shonid male at tht junction Tn the nation-wide Inap svilhing sate sa conesived by Pierey wo ean anvaage lea lps eon lope, anda nstional loop, The implest lmaginable structures nein which eseh Ines np has an interchange fly with tareginnal lop al with ru other lo soy veh regions Toup iuterchanges with the national loop and otherwse only with et Tel loge, Hoar does it work? Suppo tht u mosage origina oe Top X. snd basis dordnation ia lol loop T, where X ane Y my ot ay not be ideuteal, When the meege oomen to the intorchang breton X'and 3 gional loop, i exit onto the regional To i an tly i? YX. Ie later exte onto the mation Toop 17 Vere Aiden dsm A's rgion; others tho mowing stays om X's ein Top uri teaches. ‘erefore what shoud addcese Lok Tike? We see thas if prton of the loop eddzees epaosets the egonel fop, wn tether portion the lol oop, siting devisions wll ho made on tho basis of Hltity ow wadontty a nr portions of the sending nd he rvsiving dre “Ta loup eantgurtion jue ceribed [9 parbace too epesal co be practival, Por exp, pocidaa Zor a alternate outing ed for 2 oeeal dzcet eonncetsne freon To la ep ih high mut ftathe, Pierce nas shosn Faw each af tee oifcuies eon to rome trtent, be alleviated, Theme resvan, Irseve, the fartaer probleme of ‘he configuration of ue vs hehe tn give ging, aad of the fonfigurnticr of rion! nee therertves. I ie quit likey hat the Toca! oopmataches tom given gional loop heve any taut avitehing points umn bel ven a lle within ame gion see not aormall Expeesed fo use the regional Toop. How sould we addoss auch lees Toop so at fo sake rating esay? Mab of the yest of tis paper wil be devote 1 thin publena We eval, int and he next oo eoetons, apenle of "oops generally, but ran :aystem ef leo oops athe most ely reaization, We note in pacing thes « completely gencelsasionel ‘onfiguratian of loopeon ‘whiea wo hierarchical structure as been Imposed will have the sme sddresing problem but probably « mush larger tambr of lope. Westar tothe earshieal situation in Section W. In soace vesy simpl rangement of laops ii way a ao how seddresring night sacersfully be acomplised, Consider, for example, ig. shoving four loop which tush aif they were czele of radius at oe) the vertices ofthe unit squeze Ifthe adress of eas lop wee st the (rosie numeral 4, j= O or 1, representing the coordinates ofits center, then routing could be dost in the folowing extremely simple tnonrer: at cach jonotion, go into the mew loon if thie deeoacen thc “larg istaen® brteon where you anc and your destination. Hit Adon deorete he Hani distane, dou? gu, Th, you wih 0 9 from loa 10 has 1 then te ming sen fx 1s Vou wll not tele the eit fron 0 20 GD see rea feat, foe tines rater than deereaes the Warning ditenes. You will, however, exit ilo TT ‘whe yon roa holiness ogo Fre 1 400 either ext, eo 00 oe to 1, improsea the Haram dstaner and ther rennet equal good ‘Acnple potential rating adem ew Hus be desebed as fll Bach lap bes binary wry it lng, You ree un ext Goo on Taop to anata fend ony if derense the Hmoming detanos betwne ‘where yo aye and weet sou ware to go. If several asta do the sume ub then each ne mat eed to en aqualy eho ptm pall fy sen lnop to ressving loop. Yurkermere, the umber of loons travel sould f pout, be est the Hacningdietee betwvon sender and wowive, with each trenton dermasig iene Tor Ube rewiver by racy (Can seh on adessing sohrmae be devil for every eileen lamp swith whatavee adjeneney struct Hite teleton shows that there nil ceretiniy bo dieu Lets thile of the eoletion of lope ab Seracty ase gush, with cach Joop a vertex, and to vertices connected if and only i the two loops have w mutual transfer point. Thus, the trap of the previa explo sys om Wig 2, We have mumbened such vertex with a prof binary digits vo that adjacent vertices difer in exactly one postion, the nuzzber of exges reused to pass from one point to another is exautly he Hemming distance between the cor responding numbering, and all shore: paths beewoen two pits are ‘chicved by fellowing routes of deereming Llamming dirance to te Astnntion. Another explo (Fig. 8) if we wanted eolletion of x owe swarcHNe 2a Joupsarengedexeleally we would ae the nuenberng 190, 100, 301,122, (11, wn O10, Wr ape that we age Tooking for seleeed pach on the 3 Aenengional eube with Tie wilted properey that twa points ace fy far apace ia Humming, einer the namie of eds to be tavenl betwen them there, the renting fogie would be ruined, ‘Ths we oa ues the nelization fr 1 ale uf sx loops chown in Hig 4a ‘The maining shown in Fig, tb, however, woul a validation, Tih later pete, 2 an HU have Hatening zemce Xan cheee- fovr shold be divert once, Te ph elene em, Rowever, bas Tenth % the Best lik out inerew ralher thin decreases Hemsaing Aistnnory wid therfore would nat representa uefuladaezsing sebeme ‘We hnn so a diflealty ese by pois coming olese together ox the eu for sh addsesng sozeme vo worl, tl hone ste eve deer fieatine, Suppose we with to eonetsuce an addzesting eckems for & fyeten: consisting of (hn plswiswedaceat loops (ae Dig. 3). Tis cxn fever be drawn on ene of any dmension. Por any closed path of frdge on a eube has oven length, anc 3 sod Ie the seem thevfire xp? ‘Nak nite, Wo ea etl imagine ths S-ycle embed on x eube ina peropsiate dimension (in five we wnquare I we are wil to geD- fate what we menu, We sal ate to 4 the le OD, the ede 20, fad to € both IL and Ol, We shall denote the puis 11 and O1 by the fyntbol dl, where dene ion can.” Flaming distanee between twa ‘tape of U's, a coreg hy ering fr ry pst: ab oy as CAS ioe) ig Cycle arms be 250072 HRUE oveOW cHeaKIee, sOCRSAL, oemMa=e Which one nanple bas a 0 end the other a 1, and 0 for every other Denil, Ths, the Hamning dstanen between OU and 1010 is 2 ‘th the contributions eoming from the fst and four postions, With {his sonveatian, the Flamming dtanee heteeen az two of te de aise 00,10, an lie eartcinky I, ad gorect routing rome fessctly of docraing by 1 the amming divtase ot eteh junction a= ‘Which a sraneer iy made ‘We nowr bave a aumber of fundementel questions to anewer. Can «227 clleton ofloops be nuntboro by essigning a eu oop an adress consisting of a saquenco of O', 1's, aul le? Wo require that every het snatahetreen to loop esa be Zon mutematically by wos from a lop to an adjacent one if and only i his dacresss the Hamming distance to the final destinetion by 1 Low maay bits long would auch sn adress have to be? Lats state right sway the fundamental thearsm ft this papers very wolecion of w Joey with maximum distance + Tetaoon aay te Hep em Fedde rei ying eel oop 2 addres of 20 more than ein ~ 1) U's Ve or ds Ta fet, we naw of no example were more thu (w ~ 1) "bits are rowed, und we sail give ‘construction fit has found addzessee ao more than n — 1 bis ong in every easton which it hae been fred, The eonetration, kowere, is ‘ot quien algeaihm and we do ct have a prosttht iv ean acweye be doce wich ae for aan — 1 bi Fig 5h pina op eo swisease oan How i this routing algun yng to wore in peti? Hoe we have nly the very esnlleet ead simploet sugeretions, The bse iden of the “leven chinin tha greater: pole simplicity of rong etrategy at the expense ofthe legt of a2 lop cadres. “hus, for example, you ‘owl plysinlyrestite adresses exnssting of 6, I's, nd. le by freeding OO, a6 01, and d 2 thre 10 ge 21. The lage then sey: Th tie 24 — Tet digit of bor lows iv 0 than vente bre Haman isan Boren tow 2th dpe, I the 2% — Int dig of either ndzeoe fe 1 ore it Ada up over all ky und woe if going into the new loop Alereesse Hemming diene to ths delim, This ould be vere ey Iirmceianize; te aobitaey bit fllow'ugo 1 a odd pution eould be tial for pity cheeks or other parpess. Tis aot inmediotey clea who songs the loop adres too i Aisidonl messge. The “pone book” may conteia a shorar code chat ie foul in the hers junction vou come to, of the eending comer itvelf mn’ use the destination’ eorrct loop une, hie problems i= toned! wth that of system growt, How many numbers do you have vu charge is Ina ia elded to the system? Tho eoxsequers dere fr & iturin! oop address struct at Lage extent falls and wi be dims in Seaton TV. Before we prom with she general eony, In aes how a ptienlr ‘aud ot go emp exammple wks st "This rose te tren of lange ia Fig, . The aster Betwoer yi of seiions i ver Dy the following (symmetric) table ca 0 to pad aot Baro. 20 2 Fieo2 ot ao We shall asgn asequanes of fire's 1's, snd sto cach verte in sel a say Eb the Heamasing dite hetrsen the Stplescormspondng to ‘bio vortices is ext the dnt in he tale One matin, the reader should verity, i the followin 2602 Tee nim aver Thos, sMERAE, SoM tt au Bonde eters pon 1040 enone Ja the sveuel, e all ave Tose we soon ean in Tat be Touna Fa ‘very posable eetem of loops A relly surprciag amouat of interesting rmuthematies stems, at poser, o be involve in the probets, Tomer uses ow 2 seb af ations coop ances ea eoay be ‘oustruot, lol ue ansfove the previous example in more dota‘ fst feus of te lon po ro Wesee that A, C, an hve he vale 1 this oontinate while, Dy snd Five ue yale 0, Tha Un gourd wl eoutibuse 1 tothe Hlnaning disinnoe fv ony 0° ACE ta any of BDF. We may denote 2508 this ACE X BDF. Therefore, the fv column makes the fullowing ontebution bv tae gvenal. distor mane F “The eso wlan my following tothe dst ° 1 ° 1 ° way be waktea ae ACP x ADE sad eoneebutes the Abe D oo on ro Lo wore d er) oor ok er) re at ane op fo 1 on er) vor ed roo 1 0 er) oor od ° “The Fist Ivo etme Ge, coors) then coneibate she sur nf the previous miatrice fa the distaneo mais 4 B cD 2804 rn ARLE seem MnHHTe, ZONA, econ ‘The third column i: ca Do we ro 1 ouwaon A or H acl Do 8 Sine C al anv th Tl cunt lied eourdinate vale df canoe eonteiute vo the Hamming danoe ftom C or B to ony other poin, We ean write AU X DF and obtain the follwing oretet on e the dives mst Roo DR P eo ton oo 1 oF oo 0 oo roo no oo Oo ao boo 9 0 “The fn es clues Coordinte) thu coutsibuts the following tothe Aisne mate 4 Ac 3 an or Ds ot r| tae ‘The It two euumen ane dC ome > reapestvely IF