A New Efficient Distributed Load Balancing Algorithm for OTIS-Star Networks

Transcription

1 Int'l Con. Pr. n Dst. Pro. T. n Appl. PDPTA' A Nw Ent Dstrut Lo Blnn Alortm or OTIS-Str Ntwors A. Aww 1, J. Al-S 1 Dprtmnt o CS, Unvrsty o Ptr, Ammn, Jorn Dprtmnt o ITC, Ar Opn Unvrsty, Ammn, Jorn Astrt - T OTIS-Str ntronnton ntwor s on o t promsn ntronnton ntwors or utur sp strut omputn, T OTIS-Str topoloy onssts o ot n vrtx symmtry, t s mny ttrtv topolol proprts s on t s own rrl strutur ntur. Mny rsr orts v n vot on ts ttrtv ntwor n ltrtur, ut ts orts not rss t prolm o lo lnn w s n mportnt ssu tt must rss n ny nw topoloy. In ts ppr, w r nvsttn n proposn nw nt lortm or lo lnn prolm or t OTIS-Str ntwor. T nvstt lortm s ll OTIS-Str Eltron-Optl-Eltron Exn Mto; OSEOEM or sort; to mplmnt on t OTIS-Str ntwor. T propos lortm s s on t FOFEM mto prsnt or OTIS-Cu ntwors. T OSEOEM lortm s sown to n nt n rstrutn t lo lnn or OTIS-Str ntwors, t outom rsult n qul struton o los mon ll nos wtn t ntwor. Kywors: Eltron Intronnton ntwors, Optl ntwors, Lo lnn, Prlll Alortms, OTIS-Str ntwor. 1 Introuton T Str rp ws propos y Ars n t l s on o t ttrtv topolos, t ws propos s n ltrntv to t u ntwor [1]. T Str rp s xllnt topolol proprts wn w ompr t wt ntwor smlr szs []. T str rp sow to v mny ttrtv proprts ovr mny ntwors nlun t wll-nown u ntwor nlun: smllr mtr, smllr r, n smllr vr mtr. T str rp prov to v rrl strutur w wll nl t uln lr ntwor sz o smllr ons, t str rp s ot n vrtx symmtry [1,]. Wt t nw vns o tnoloy, nw r o Optl ntwors s n ppr. Mny prvous rsrs rss t mr o t Optl tnoloy wt t trtonl ltron ntronnton topoloy. OTIS-Str ws on o t propos ntwors n ts r u to ts ttrtv proprts n turs[3]. Altou som lortms propos or t OTIS-Str rp su s routn lortms n strut ult-tolrnt routn lortm [3, 4], stll tr s sort o orts to solv t prolm o Lo Blnn usn OTIS-Str ntwors. To our nowl tr s no nou rsults propos n ltrtur out mplmntn n proposn nt lortms or lo lnn on OTIS-Str ntwor. In ts ppr w r lln ts p y proposn n mn t OSEOEM lortm on t OTIS-Str rp w s s on t FOFEM lortm w ws sown to nt on OTIS-Cu ntwors [5]. T mn mnsm o ts lortm s to rstrut t lo qully mon t prossors o t ntwor [6]. Ent mplmntton o t OSEOEM lortm on t OTIS-Str ntwor wll m ts ntwor mor sutl ntwor or rl l pplton n onnton to lo lnn prolm. T rst o t ppr s ornz s ollows: In ston w prsnt t nssry s nottons n ntons, n ston 3 w ntrou som o t rlt wor on lo lnn, n ston 4 w prsnt n suss t mplmntton o t OSEOEM lortm on t OTIS-Str rp, lso w prsnt n xmpl o OSEOEM on OTIS-3-str ntwor, nlly ston 5 onlus ts ppr. Dntons n Bs Proprts Durn t lst numrous numr o ntronnton ntwors or Prlll Computrs v n propos n ltrtur [7, 9]. Som o ts ntwors r t str n t ypru ntronnton ntwor. T OTISstr rp [1,3] s notr xmpl, w s n propos s n ttrtv ltrntv to ts tor ntwor [1]. Sn ts pprn t OTIS-str ntwor s ttrt lot o rsr orts, mny topolol proprts o t OTIS-str ntwor v n nvstt n t ltrtur nlun ts s topolol proprts [1], prlll pt [8], lo lnn lortm [6] n mn [10]. S n Aww [3] v mploy t oo proprts str rp n ts l ovr t proprts o smlr ntwor to prsnt n nt ult-tolrnt routn lortm n lso lo lnn lortm. Ts proprts r nlun lowr r, smllr mtr, smllr vr mtr, n Symnt rttur.

2 Int'l Con. Pr. n Dst. Pro. T. n Appl. PDPTA' 3 T strutur o t OTIS-str ntwor plys n tv stp or proposn ny lortms or t. T utors n [8] v sown tt t OTIS-n-Str rp my sn s n! n!, wr t symol (!) mns t torl o n numr, n no ll wt unqu prmutton on n = {1,,n} o t tor str ntwor. Howvr, tr s n rltvly lmt rsr orts v n t to sn nt lortms or t OTIS-str rp nlun routn n Brostn lortms [11, 1], rostn [13] n lo lnn [14], [5]. In n ttmpt to ovrom ts prolm w prsnt n nt lortm or rstrut t lo lnn prolm on t OTIS-Str ntwor to rstrut t lo lnn mon ll prossors o t ntwor qully s possl. T topolol proprts o t OTIS-Str ntwor lon wt tos o t str ntwor r suss n rv low. Ts topolol proprts v n rv usn t tortl rmwor or nlyzn topolol proprts o t OTIS-Ntwors n nrl w ws propos y Al- Ayyou n Dy [], s otr rlt rsr wor [11]. In ts ppr w wll rr to s t roup rss n p s t prossor rss. An ntrroup o t orm (, p, p, ) rprsnts n optl ln n wll rrr to s OTIS or optl mov. An OTIS s ntwor ontns N nos prtton nto N roups wt N nos. A no s nx y pr x, y, 0 x, y N wr x s t roup nx n y s t prossor nx. Nos wtn t sm roup r onnt y rtn ntronntn topoloy; wl ntr-roup lns r v y trnsposn roup n no nxs. T OTIS-ntwor s onstrut y "multplyn" nown topoloy y tsl. T vrtx st s qul to t Crtsn prout on t vrtx st n t tor ntwor. T st onssts o s rom t tor ntwor n nw s ll t trnspos s. T orml nton o t OTISn-Str rp s vn low. Dnton 1: T OTIS-n-Str Grp, w s not y OTIS-n-Str s n xn nos ll wt sol prmutton n = {1 n, 1 n}. T rst prt o t ll rrs to t roup rss n t son prt rrs to t tor rss o t no wtn t roup. Any two nos o OTIS-n-Str r onnt, n only, tr orrsponn prmuttons r xtly n t rst poston n ny otr poston. Dnton : Lt n-str = (V0, E0) n unrt str rp rprsntn tor ntwor. T OTIS-n-Str = (V, E) ntwor s rprsnt y n unrt rp otn rom n-str s ollows V = { x, y x, y V0 su tt x y} n E = {( x, y, x, z ) (y, z) E0} {( x, y, y, x ) x, y V0 su tt x y }. From t ov ntons t st o s E onssts o two susts, on s rom n-str, ll n-str typ s, n t otr sust ontns t trnspos OTIS s. T OTIS ppro susts mplmntn n-str s y ltron lns sn ty nvolv ntr-p sort lns n mplmntn trnspos s y r sp opts. Trouout ts ppr, t trms ltron mov n t OTIS mov (or optl mov ) wll us to rr to t trnsmsson s on ltron n optl tnolos, rsptvly. F. 1 sows t OTIS-3-Str rp wt 6 roups; 3!=6; ontnn 6 vrts (.. sx ops o 3-str rps). F. 1: T OTIS-3-Str Grp 3 Broun n Rlt Wor T ttrtv proprts o OTIS-n-Str prov y rsrrs n ltrtur o t rp m t on o t stronst ompttor s topoloy or H Sp Prlll Computrs (HSPC) n stron nt ntwor or rl l ppltons [1, ]. Ts t s motvt us to nvstt t lo lnn prolm on t OTIS-n-Str ntwor sn t OTIS-n-Str rp surs rom lmt numr o nt lortms propos or t n nrl n or t Lo Blnn prolm n sp. T lo lnn prolm s n nvstt on vrous typs o nrstrutur rnn rom ltron ntwors [5] n OTIS ntwors [6]. Lo lnn prolm s on o t wll-nown n mportnt typs o prolms w wr stu rom rnt pont vws n rnt ppros. Ts prolm ws stu n nvstt y Rn, Won, n Sn [], ty propos n ntrou t Dmnson Exn Mto (DEM) on t ypru topoloy. T DEM lortm ws s on t o nn t vr lo o nours nos, su tt t mnson o t ntwor s n, t nours w r onnt on t n t mnson ty wll xn tr los to rstrut t lo n v vnly lo lnn s possl, t prossor w v

3 4 Int'l Con. Pr. n Dst. Pro. T. n Appl. PDPTA' mor lo wll rost t xtr mount o t lo to ts rt nour no. T mn vnt o t Dmnson Exn Mto s tt vry no wll l to rstrut tss to ts rt nours to r vn lo lnn mon ll nos. Rn n t l v v tt n t worst s n t DEM mto to rstrut lo lnn ws lon on t u ntwor []. T rsrrs Zo, Xo, n Qn v prsnt yr sm o uson n mnson xn ll DED-X or lo lnn on Optl Trnspos Intronnton Systm (OTIS) [], []. T propos lortm wors y vn t lo lnn ts to tr rnt pss. T rsults v on OTIS ntwors sow tt t lo ln ntly rstrut lmost vnly. On t otr n t v rsults o t smulton rom Zo t l o t propos lortms on lo lnn s sown onsrly mjor vnmnt n nnmnt o ny n stlty [], []. In notr rsr on y Zo n Xo ty v prsnt rnt DED-X sms or lo lnn on omonous OTIS ntwors n ty propos nw lortm strutur ll Gnrlz Duson- Exn- Duson Mto, t propos sm nl lo lnn on Htronous OTIS ntwors []. Anotr lortm prsnt n [6 ] ll SCDEM s s on t Clustr Dmnson Exn Mto CDEM or lo lnn or Optl Trnspos Intronnton systm on Hypru tor ntwor [6]. T worst tm s omplxty o CDEM or lo lnn on OTIS-Hypru ws O(Sqrt(p)*M lo p). Also t numr o ommunton stps w s rqur y CDEM prov to 3lo p [6]. Furtrmor utors o [5] prsnt nw lo lnn lortm or OTIS-Cu ntwors, ty v sown tt t uslty o t nw propos lo lnn mtos to ttr tn t DED-X trtonl lo lnn lortm []. 4T Propos Alortm T mn ojtv o ts ppr s to propos n prsnt nw lo lnn lortm or t OTIS-n-Str ntwors nm OSEOEM s on t lortm o FOFEM prsnt n [5]. T mn vmnt o t nw prsnt OSEOEM s to otn vn lo lnn or t OTIS- n-str ntwor y rstrutn numr o tss twn rnt nos on rnt roups. T numr o xns n twn rnt nos n t OSEOEM s (n!)+1, wr n s t mtr o tor ntwor; n-str. F. prsnts t OSEOEM lortm or lo lnn prolm on OTIS- 3- Str ntwor T OSEOEM lo lnn lortm s s on t ollown 3 pss: ٠Ps 1: T mn m o ts ps s to ln t lo o ll nos wtn tor roup o t ntwor. T 1st st o ts ps s v y rstrutn t lo lnn o ll rt nour nos tt tr orrsponn prmuttons r xtly n t 1 st n n poston. T n st s to rstrut t lo lnn o ny two nour nos tt tr orrsponn prmuttons r xtly n t rst n 3r poston. In nrl w ontnu rstrutn t lo lnn o ny two nour nos tt tr orrsponn prmuttons r xtly n t rst n t poston, wr n. Ts pross o ps 1 soul rpt tms to v n optml qul rstruton o lo mon ll nos wtn roup o t tor ntwor. ٠Ps : Ts ps wll xn t lo o vry two nos onnt v n optl ln. Ts wll yl to rstrut t lo qully mon ll roups ut t lso wll v rnt no lo wtn t roup tsl w ls to t nssty o prormn t tr ps. ٠Ps 3: rpt ps 1 to v n qul struton o lo mon ll nos o t OTIS-n-Str Not tt n-1 s t numr o nors o ny prossor n t tor ntwor Sn: Fur 3 llustrts t tr pss o OSEOEM n tls: 1. or m = ;m n; m++ // strt o ps 1. or ll n-1nour nos; p n p j w ty r n 1 st n m poston o S n o n prlll 3. Gv-n-t p n p j totl lo szs o t two nos 4. TAvrLo p,j = Floor ((Lo p + Lo p )/ ) 5. ( Totllo p >= xss AvrLo p,j ) 6. Sn xss lo p to t nour no p 7. Lo p = Lo p xtr lo 8. Lo p j = Lo p j + xtr lo 9. ls 10. Rv xtr lo rom nour p j 11. Lo p = Lo p + xtr lo 1. Lo p j = Lo p j xtr lo 13. Rpt stps 3 to 1 on mor tm // n o ps or ll jnt nos v n optl ln, xn t los o t nos // ps. Rstrut t wt wtn roup y rptn stps 1 13 // ps 3 F. : T OSEOEM lo lnn Alortm OSEOEM lortm wors on rstrutn t lo lnn mon ll nos o t ntwor, pss: on, two n tr r sr s ollows: ٠Ps 1: T lo lnn twn t nos o S n s on OSEOEM lortm s xn s n stps to 1 n prlll, n rst stp t lo xn wll twn ll t

4 Int'l Con. Pr. n Dst. Pro. T. n Appl. PDPTA' 5 nos n w ty r n 1 st poston n n poston or ll t tor ntwors o S n. Tn t sm pross wll rpt ontnully untl t rs t nours p j tt r n-1 postons r wy rom p. To nn t lo lnn ny twn rnt prossors o n tor ntwors, t lortm susts rptn t stps to 1 s mnton ov ٠Ps : To rstrut t los mon ll t roups o t ntwor, vry two nos tt r onnt v n optl ln wll xn tr los. Ts stp s onut on prlll. ٠Ps 3: As nl ps ll jnt nos w ty r n rst poston n ny otr poston.. p n p j. T lortm wll rstrut t wts mon t nos p n p j. Bslly, ts ps s rptton o ps 1 to rln t wts o ll nos wtn roup on mor tm F. 3: OTIS-3-Str ntwor lo lnn - ntl stt Exmpl: - To xpln t OSEOEM lortm w prsnt n F., t ollown xmpl mplmnts t lo lnn lortm on t OTIS-3-Str wr t tor ntwor s S 3. F. 3 sows t OTIS-3-Str ntwor, tor ntwor S 3 s 6 nos wt sp lo ssn to on. E no onnt to two ltron rt nos wtn t roup, n ton to tr optl onnton to notr roup. Not tt t numr n ol n tl w s ssn nxt to no rprsnts t strtn lo F. 4: OTIS-3-Str ntwor lo lnn ps 1 stp F. 5: OTIS-3-Str ntwor lo lnn ps 1 stp Frst w strt y mplntn ps 1 o t OSEOEM lortm y ollown t stps -1. T urs 4, 5 n 6 rlt ps1: o t OSEOEM lortm, pr o t nos w ty r n 1 st poston, n, 3 r n 4 t poston. At t n o ts ps F. 6 sows t nw lo lnn struton o t ps 1 o t lortm. Fnlly n ps 3 outom s sown n ur 11 wr ll nos wll

5 6 Int'l Con. Pr. n Dst. Pro. T. n Appl. PDPTA' rstrut tr lo lnn y rptn ps 1 sts, ts ur sows tt ll nos wll v lmost t sm lo pproxmtly. T nw v struton o t lo ln sown to nt n optml. T nl struton s v n (n!)+1 ommunton stps wr n s t r o t OTIS-n-str ntwor F. 6: OTIS-3-Str ntwor lo lnn ps 1 stp F. 7:OTIS-3-Str lo lnn ps1 stp 4 F. 8: OTIS-3-Str ntwor lo lnn ps 1 stp 5 5 F. 9: OTIS-3-Str ntwor lo lnn ps 1 stp 6 In ps w rpt t sm stps on mor tm to rstrut t lo lnn mon t nours nos s sust n OSEOEM lortm prsnt n F. y Ps 1 s n! sts (stps -1 n t lortms), Furs 4-9 llustrt ts sts tt v n sro n t lortm ov. Fur 10 llustrts ps wr nos xn tr wts v optl lns.

6 Int'l Con. Pr. n Dst. Pro. T. n Appl. PDPTA' 7 F. 10: OTIS-3-Str ntwor lo lnn ps F. 11: OTIS-3-Str ntwor lo lnn- n o ps 3 5 Conlusons Ts ppr prsnt n nt lortm or strutn t lo lnn o nos n OTIS-n-str ntwor. T propos lortm s ll OSEOEM w s s on t wll-nown lortm FOFEM mto prsnt or OTIS- Cu ntwors. T propos lortm OSEOEM rsult n lmost n vn rstrutons lo lnn mon t ll nos o OTIS-n-str ntwor. T lortm s l to rstrut lo lnn mon ll nos n (n!)+1 ommunton stps w s onsr to nt. As utur xtnson o ts rsr wor w wll o som nlytl stmton n nlyss nlun: xuton tm, lo lnn ury, ommunton stps n sp to prov t OSEOEM s nt mtmtlly. Rrns [1] S. B. Ars, D. Hrl n B. Krsnmurty, T Str Grp: An Attrtv Altrntv to t n-cu Pro. Intl. Con. Prlll Prossn, 87, pp [] K. Dy n A. Trpt, A Comprtv Stuy o Topolol Proprts o Hyprus n Str Grps, IEEE Trns. Prlll & Dstrut Systms, vol. 5. [3] J. AL-S, A. M. Aww, B. F. AlBw, Ent Routn Alortm on OTIS-Str Ntwor, Prons o t IASTED Intrntonl Conrn on Avns n Computr Sn n Tnoloy, Vrn Islns USA, Novmr -, 04, ACTA Prss, pp. 7. [4] Kl Dy n Al-El Al-Ayyou, No-rnn sms or t str ntwors, Journl o prlll n Dstrut Computn, Vol. 63 ssu 3, Mr 03, pp [5] J Al-S, "Implmntn FEFOM lo lnn lortm on t nn OTIS-n-Cu topoloy", T Son Intrntonl Conrn on Avns n Eltron Dvs n Cruts - EDC13, Kul Lumpur, Mlys, My, 13. [6] B.A. Mz n B.A. Jrt, T Lo Blnn prolm n OTIS-Hypru Intronnton Ntwor, J. o Supromputn (08) 46, [7] S. B. Ars, n B. Krsnmurty, A Group Tort Mol or Symmtr Intronnton Ntwors, Pro. Intl. Con. Prlll Pro., 86, pp [8] K. Dy n A. Al-Ayyou, T Cross Prout o Intronnton Ntwors, IEEE Trns. Prlll n Dstrut Systms, vol. 8, no., F. 97, pp [9] A. Al-Ayyou n K. Dy, A Comprtv Stuy o Crtsn Prout Ntwors, Pro. o t Intl. Con. on Prlll n Dstrut Prossn: Tnqus n Appltons, vol. I, Auust 9-11, 96, Sunnyvl, CA, USA, pp [10] I. Jun n J. Cn, Emn Complt Bnry Trs n Str Grps, Journl o t Kor Inormton Sn Soty, vol., no., 94, pp [11] Brtom, P., A. Frrr, n S. Prnns, Optml Inormton Dssmnton n Str n Pn Ntwors, IEEE Trns. Prlll n Dstrut Systms, vol. 7, no. 1, Au. 96, pp [1] P. Fropoulou n S. Al, A Prlll Alortm or Computn Fourr Trnsorms on t Str Grp, IEEE Trns. Prlll & Dstrut Systms, vol. 5, no. 5, 94, pp [13] Mn V. n D. Srr, Optml Brostn on t Str Grp, IEEE Trns. Prlll n Dstrut Systms, Vo;. 3, No. 4, 9, pp [14] N. Imn t l, Prt lo lnn on str ntronnton ntwor, J. o supromputrs, Volum 41 Issu 3, Sptmr 07. pp [] Rn, Y. Won, S. Sn, Prormmn Hypru Multomputr, IEEE Sotwr, 5 (5): 69 77, 98. [] Zo C, Xo W, Qn Y (07), Hyr uson sms or lo lnn on OTIS ntwors, In: ICA3PP, pp 4 43 [] G. Mrsn, P. Mrn, P. Hrvy, n S. Esnr, Optl Trnspos Intronnton Systm Arttur, Opts Lttrs, (13), 93, pp [] Qn Y, Xo W, Zo C (07), GDED-X sms or lo lnn on tronous OTIS ntwors, In: ICA3PP, pp