Multi-criteria p-cycle network design

Transcription

1 Multi-ritri p-yl ntwork sin Hmz Dri, Brnr Cousin, Smr Lou, Miklos Molnr To it tis vrsion: Hmz Dri, Brnr Cousin, Smr Lou, Miklos Molnr. Multi-ritri p-yl ntwork sin. r IEEE Conrn on Lol Computr Ntworks (LCN 8), Ot 8, Montrél, Cn. Proins o r IEEE Conrn on Lol Computr Ntworks, pp.6-66, 8, <.9/LCN >. <l-848> HAL I: l-848 ttps://l.rivs-ouvrts.r/l-848 Sumitt on Au 5 HAL is multi-isiplinry opn ss riv or t posit n issmintion o sintii rsr oumnts, wtr ty r pulis or not. T oumnts my om rom tin n rsr institutions in Frn or ro, or rom puli or privt rsr ntrs. L riv ouvrt pluriisiplinir HAL, st stiné u épôt t à l iusion oumnts sintiiqus nivu rr, puliés ou non, émnnt s étlissmnts nsinmnt t rr rnçis ou étrnrs, s lortoirs pulis ou privés.

2 Multi-Critri p-cyl Ntwork Dsin Hmz Dri, Brnr Cousin, Smr Lou Univrsity o Rnns I -IRISA- Rnns, Frn Emil: {ri, ousin, slou}@iris.r Miklos Molnr INSA o Rnns -IRISA- Rnns, Frn E-mil : molnr@iris.r Astrt T mjor lln o p-yl ntwork sin rsis in inin n optiml st o p-yls prottin t ntwork or ivn workin pity. Existin solutions (xt n uristi ppros), or solvin t prolm, in t st o p-yls prottin t ntwork trou two stps: on stp or nrtin nit p-yls n son stp or sltin t iint ons. In tis ppr, w prsnt novl uristi ppro, wi omputs n iint st o p-yls prottin t ntwork in on stp. Our uristi ppro tks into onsirtion two irnt ritri: t runny n t numr o p-yls involv in t solution. Simultion stuy sows tt our ppro nssitts lowr runny n wr p-yls to prott t ntwork ompr to stt-o-t-rt ppros. Inx Trms WDM prottion, Ntwork survivility, p-yl, Ntwork mnmnt omplxity. I. INTRODUCTION Optil WDM ms ntworks r l to trnsport u mounts o inormtion. T us o su tnoloy, owvr, poss t prolm o prottion inst ilurs su s ir uts or no ilurs. Consquntly, ny ut o su ir my l to u t loss n lot o tri in lok. For tis rson, mtos o prottion soul implmnt to minimiz t t loss wn ilur ours. On o t prinipl prottion mtos propos or optil WDM ntworks is s on p-yls or proniur prottion yls, introu y Grovr n Stmtlkis in []. P-yl ors t vnts o ot rin n ms prottion sms: str rstortion tim s in rin prottion, n i pity iiny s in ms-prottion. Prisly, in p-yl, rstortion tim is str us only t two n nos o t il link n to prorm rstortion. Morovr, pity iiny is u to t t tt p-yl n provi prottion not only or on-yl links ut lso or strlin links. A strlin link is link, wi os not lon to t p-yl ut wos n-nos r ot on t p-yl. In ition, p-yl or two rstortion pts to t il strlin links witout rquirin ny itionl spr pity. Tis proprty rus tivly t rquir prottion pitis. Fiur pits n xmpl tt illustrts p-yl prottion. In iur (), (-----) is p-yl wit on unit o spr pity on on-yl link. Wn t on-yl In t rin prottion, t tri on t il link or no is rrout roun t rin on t prottion irs twn t nos jnt to t ilur. link (-) ils s sow in iur (), t p-yl provis on prottion pt (----). In iur () n xmpl o strlin link ilur is sown. Wn t strlin link (-) rks, t p-yl n prott two workin wvlnts on tis link y proviin rsptivly two ltrnt pts (--) n (---). tri () Fi.. () tri () Prottion usin p-cyl A p-yl wit i iiny is p-yl tt s smll rtio twn its spr n prott pity, i.. it protts mor workin wvlnts usin w spr wvlnts. T mjor lln o tis mto o prottion rsis in inin t optiml st o p-yls in trms o rsour utiliztion tt protts t ntwork or ivn workin pity istriution. T p-yl sin n ormult itr s non-joint or joint optimiztion prolm. In t irst ppro, tr t workin pts r rout (.. usin sortst pts) t optiml st o p-yls is lult usin vill pity [], [], [], [4], [5]. In t son ppro, t routin o t workin pts n t p-yls r omput simultnously optimizin t totl pity [6], [7], []. Solvin t joint optimiztion prolm is mor iiult us o t itionl omputtion omplxity. Svrl solutions v n propos in t litrtur to solv ts optimiztion prolms. Ts solutions n lssii into two lsss: xt n uristi solutions. T irst lss nrlly uss Intr Linr Prormmin (ILP) to in t optiml solution. Howvr, ILP mtos oms unsuitl s t siz o t ntwork inrss, us t

3 numr o p-yls in rp rows xponntilly wit t ntwork siz. T son lss is uristi solutions, ivi into two su-lsss: uristi ppros s on ILP ormultion n pur uristi ppros. In t irst su-lss, limit st o nit p-yls is nrt, n tn t iint su-st o p-yls is slt usin ILP ormultion []. In orr to in oo solution wit tis ppro it is nssry to nrt lr numr o nit p-yls. Tis mns tt t runnin tim or solvin t ILP ormultion inrss rmtilly [8], [9]. T son su-lss (pur uristi) tris to in oo solution witout usin ILP ormultion. T ojtiv o tis kin o solutions is to ru t tim rquir to omput n iint st o p-yls tt protts t ntwork [], [5]. In tis ppr, w ous on loritms propos or t nonjoint prottion prolm. Existin solutions strt y nrtin st o nit p-yls, n tn slt oo su-st o p-yls. Most o t solutions propos in t litrtur onsir t p-yls wi v mor strlin links s t most iint p-yls. Consquntly, t p-yl nrtion is s only on t topoloy o t ntwork n is ompltly inpnnt o t workin pity istriution. Anotr rwk o ts solutions is tt ty nrt lr numr o nit p-yls, wi inrss runnin tim o t loritm. In tis ppr w propos nw uristi to omput st o p-yls tt protts t ntwork witout oin trou t stp o nit p-yl nrtion. Our uristi is inpnnt o ILP n tks into ount t workin pity o ntwork. Ts nl us to in rpily t iint st o p-yl prottin t ntwork. T rst o t ppr is orniz s ollows. In stion, w sri our loritm o p-yl nrtion s on t inrmntl rtion o yls. In stion w vlut our uristi n ompr it wit t xt solution tt uss ILP n wit t min propos uristis. W onlu t ppr in stion 4. II. OUR HEURISTIC A. Ntwork Mol W mol t WDM optil ntwork s n unirt rp G=(V,E)wr no in V rprsnts n optil swit n in E rprsnts ntwork link. E link j s w j workin wvlnt nnls. Tis workin pity is otin y routin t tri mns. Aorin to t p-yl inition p-yl is orm in t spr pity o t ntwork. A p-yl n prott on wvlnt on on o its on-yl link n two wvlnts on o its strlin link usin only on unit o wvlnt on o its on-yl links. In tis s t spr pity o p-yl n in s t numr o on-yl links o t p-yl. In tis ppr, w onsir t most rqunt ilur in t optil ntworks, wi is t sinl link ilur. In tis ppr w onsir t mol tt nsurs ull prottion inst sinl ilurs []. Fiur () sows WDM ntwork wit it optil swits (...), wr link is ssoit wit n intr () 6 Fi.. p-cyl # o opis () P-yl xmpl vlu tt inits t workin pity o t link. Fiur () sows st o p-yls tt prott t ntwork or t ivn workin pity. T tl sown in iur () ontins st o p-yl struturs prottin t ntwork, wr # opis nots t numr o opis o p- yl strutur. For instn, t irst p-yl strutur ( ) in t tl s two opis, on n prott on wvlnt on link lonin to t p-yl n two wvlnts on t strlin links (-) n (-). B. Aloritm Motivtion In tis stion, w sri t motivtion o our loritmi ppro or p-yl sin. W strt y nlyzin t limittions o t uristis propos in t litrtur. Tn w introu t min uilins o our uristi tt nl to ovrom ts limittions. Firstly, t propos uristis, pur or ILP-s, us two-stps ppro tt strts y nrtin st o nit p-yls n tn slts t iint ons or prottion. Hn, t qulity o t solution pns vily on t st o nit p-yls. Prtiulrly, ts uristis n to nrt lrr st o nit p-yls in orr to inrs t sr sp n nn t iiny o t inl solution. Howvr, tis ls to onsirl inrs in omputtion tim n inus srious limittion or ts ppros. In our solution, w propos on-stp loritm tt irtly omputs t p-yls tt r us or prottion. Tis nls to prorm in tunin twn iiny n runnin tim n ovroms t limittion o usin st o nit p-yls. Sonly, t propos uristis nrt t st o nit p-yls s only on t ntwork topoloy. In orr to in n iint solution in trms o rsour utiliztion (runny), ty onsir tt p-yls wit lr numr o strlin links r mor iint, n ty r prrly slt or prottion. Howvr, tis my not lwys vntous: onsir or xmpl t s wr t us o p-yls wit lr numr o strlin links rsults in ovr-prottin som links n tus rus t iiny o t lol solution s sown in iur. T p-yl (-- ()

4 () () limittions tt t t stt-o-t-rt solutions. Tus, our loritm omputs n iint st o p-yls tt protts t ntwork n tks into ount t workin pity n t topoloy o ntwork. Our loritm prorms in sinl stp witout unroin prliminry stp or nit p- yl nrtion. Aitionlly, our loritm xpliitly tks into ount two iiny ritri: t runny n t numr o p-yls. Fi.. Eiint p-yl xmpl C. Aloritm Insits -) sown in iur () is mor iint tn t p-yl (----) sown in iur () in spit tt t lttr on ontins strlin link. In our uristi, w ovrom tis rwk y tkin into ount t workin pity on t ntwork links. Tus t numr o strlin links pr p-yl is pt to t workin pity in orr to nsur ttr iiny o t solution. Tirly, t ojtiv o our uristi is to in oo st o p-yls not only in trm o rsour utiliztion (runny) lik t otr propos uristis. But it tks into onsirtion two irnt ritri: t runny n t numr o p-yls in t inl st o prottion. Ts ritri r in in t ollowin prrp. T runny ritrion nls to msur t prottion iiny: it nots t rtio twn t prottion rsours n t workin rsours in t ntwork. Runny onstituts mjor vlution ritrion or WDM ntwork sin tt ivs n insit on t qulity n t rltiv ost o ntwork prottion. In our ppro, w strntn tis oi y intrtin t runny in t itrtiv prour o t loritm. In t ollowin, t ntwork runny is not y R. Prtiulrly, R() nots t runny o yl n is in s t rtio twn t spr pity (numr o wvlnts us y tis yl) n t workin pity o t yl (numr o wvlnts prott y t yl on its on-yl n strlin links). Similrly, t runny R o t inl solution is omput s t rtio twn t totl ntwork spr pity us or prottion sin n t prott workin pity in t ntwork. T numr o p-yls is notr vlution ritrion or our solution. It is motivt y t t tt solutions wit smll numr o struturs simpliy t ntwork mnmnt tsk []. For xmpl, onsir t s wr t ntwork sinrs mnully implmnt t prottion sin. In tis s, solutions wit smll numr o p-yls llvit t tious oniurtion n mintnn tsks n ru t risk o oniurtion rrors. Tis ppro rmins vli wn usin n utomt ontrol pln su s t GMPLS rittur. In tis s, solution wit ru numr o prottion struturs nls to ru t strss ovr t ontrol pln n ss t mnmnt tsk. T tils o our loritm r prsnt in t ollowin stion. T min rivr o our loritm is to ovrom t ormntion In tis stion w sri our loritm or omputin p- yls. In prliminry stp o our loritm, wit t lp o t loritm introu in [], w strt y omputin t st o sortst yls in t ntwork. Tn, w us tis st o sortst yls to onstrut t st o p-yls (y inrmntl rtions) tt protts t ntwork workin pity s xplin in t ollowin. () Fi. 4. () Fiur 4. St o sortst yls E itrtion o our loritm strts y sltin on sortst yl mon ll t sortst yls in t ntwork. Typilly, w oos t yl ontinin link wit miniml non-zro workin pity (. Fiur 5, stp 4). Lt us not tis urrnt yl y. Tn w sr or yl tt is to rt wit yl. T st o liil yls or rtion wit yl must stisy t ollowin tr onitions: irst, n liil yl srs on n only on link wit yl, son n liil yl srs no nos wit yl xpt t n nos o t sr link (. Fiur 5, stp 5). W oos to rt t liil yl tt ls to t lrst rution in runny, i n only i t runny o t nwly rt yl is lowst tn tt o yl (. Fiur 5, stp 6). W now onsir t nw rt yl s t yl n pro wit notr rtion. T rtion pross vntully stops wn t liil st oms mpty or no rtion ls to rution in t runny. T output rt yl o on itrtion is to t inl prottion p-yls st. T workin pitis tt r prott y t nwly omput p-yl r rmov rom t orrsponin links n t loritm itrts until ll t workin pity is prott or tr is no mor vill wvlnts in t ntwork (. Fiur 5, stp 7 n 8).

5 Aloritm: P-yl nrtion Input : St o sortst yls, workin pity o link Output: St o prottion p-yls ULR = E N Prott Links E () () j E, u j = w j ; () Lt itrtion numr i = ; () j,k E, slt j su tt u j u k ; (4) Slt t sortst yl ontin j. (su tt s t ist numr o unprott links twn ll t sortst yls usin link j); (5) Slt st o sortst yls SSC={ k /k> } su tt sortst yl vriis t ollowin onitions:. Cyls n k sr on n only on link;. Cyls n k o not sr ny no xpt t n nos o t sr link; (6) I ( SSC mpty ) tn. Slt t yl k tt s t smll vlu o R( + k );. I (R( + k ) R( )) tn := + k notostp(5); (7) For link j prott y t P-yl upt t vlu o u j orin to :. u j =u j - or link j on yl ;. u j =u j - or strlin link o yl ; (8) I j E wit u j > n i tr is vill pity in t ntwork. lti=i+inotostp(); w j : Workin pity o link j. u j : Unprott workin pity o link j. (x+y): T rtion o yl x wit t yl y. R(): T runny o yl. Fi. 5. Aloritm sription III. ALGORITHM EXTENSION In tis stion, w introu n xtnsion o our loritm tt moiis t rtion pross. Tis xtnsion nls to improv t ovrll prormn s onirm y t simultion rsults in stion IV. T si i is to rlx t runny onstrint in t inrmntl rtion pross. Prisly, t si vrsion o our loritm introu in stion rquir tt t runny must stritly ru wn two yls r rt. Tus, no yl rtion lin to n inrs in runny is prorm. Howvr, tis onstrint ls to iniiny prtiulrly wn svrl sussiv rtions r n to improv t ovrll runny. Tis is typilly t s wnvr irst rtion is iniint (inrss t runny) ut i prorm, it my ollow y vry iint rtions. On wir sop, tis is wll-known tniqu tt is us in orr to voi lol minim in t sr or lol minimum. In t xtn loritm, w introu nw vril, tt inits i lr p-yls my turn to iint or not. Tis vril, ll ULR, nots t rtio twn t numr o unprott links n t totl numr o links. ULR = N Unprott Links E () Wr E is t numr o links in t ntwork, N Unprott Links is t numr o unprott links in t ntwork, N Prott Links is t numr o prott links in t ntwork. Intuitivly, wn t numr o unprott links is rltivly i, i.. or lrr vlus o URL, lr p-yls my improv t runny n om iint n vi-vrs. Tror, t nw onition or rtin yls tolrts n inrs in t runny wnvr t vlu o ULR is lr. For tis propos, t onition or rtin yl n yl (. iur 5, stp 6) oms: R( + k ) Or x ULR () Wn t rtion os not ru t runny (i.. t irst su-onition o () is not vrii), w nrt rnomly rl numr x twn n (uniorm istriution). I t nrt numr is lss tn ULR, tn w mk t rtion. Tror, lrr vlus o URL (lrr rtio o unprott links) l to mor rtions n nrt ir yls, wrs smllr vlus o URL l to smllr yls. () Fi. 6. () P-yls rtion (Bsi Solution) Lt us s ow tis xtnsion prorms on smll xmpl. Fiur 6() sows t stt o t ntwork t n intrmiry itrtion o t loritm (som links v no workin pity to prott), n t st o sortst yls (----, ----, ----) tt w will us to onstrut p-yls. At tis st, t loritm slts t yl (----) usin t link wit miniml non-zro workin pity s t yl (. iur 5, stp n 4). T SSC st otin tr t sltion o t yl ontins on yl (----) (. iur 5, stp 5). Wit t si loritm t rtion o yl n yl (----) n not prorm us: R (------) =6/4 R (----) =4/4. Howvr, it pprs to usul to prorm tis irst rtion (spit its iniiny) us it ls to n iint rtion trwrs. Wit t nw xtnsion, s sown in iur 7 (), t yl n t yl (----) r prmitt to rt wit t proility o.8 i.. ULR= 8/. I t nrt rnom numr is twn n.8, tn t rtion is prorm n t nw yl

6 oms (------). T nxt rtion o wit its nior (----) ls to lol rution in runny R( )=8/9 s sow in iur 7 (). TABLE I AVERAE TARFFIC COST 9 KL Dmns Workin pity () Fi. 7. () () P-yls rtion (Extn Solution) At t n o itrtion o t xtn solution, t otin yl is not nssrily t yl wit t smll runny, us in tis vrsion o loritm, t rtion is not onition y t rution o t runny. For tis propos, urin t rtion prosss o tis nw vrsion o loritm w kp t yl otin tr rtion, n t t n o t rtion pross w slt t yl vin t smllst vlu o runny. T slt yl is in t inl st o p-yl prottin t ntwork. IV. SIMULATION RESULTS AND ANALYSIS In tis stion, w vlut our loritm in trms o runny n in trms o numr o p-yl struturs otin in t inl prottion st. Simultion xprimnts r rri out usin two wily us ntwork topolois: t pn- Europn COST 9 topoloy n t KL topoloy. T irst is n -no n 6-link ntwork, tkn rom [], t son is 5-no n 8-link tkn rom []. COST9 s i nol r (=4.7) wi l to i numr o istint yls (5 yls). KL ntwork s 6 istint yls, n it nol r is.7. T tst ntworks r sown in iur 8. () COST 9 ntwork Fi. 8. Tst Ntworks () KL ntwork T tri mn is uniormly istriut mon ll sour-stintion pirs. For sour-stintion pirs, n intr numr twn zro n t mximum llowl mn (wi is in our simultion) is rnomly nrt. T workin pity on vry ntwork link is lult tr ll mns v n rout usin t sortst-pt routin. Tn xprimnts r prorm or topoloy ntwork n t vr vlus r prsnt s t inl rsults. In our simultion w ssum tt t pity o t ntwork ws sin to pt ll onntion mns n to nsur % prottion. Trou t simultion w ssum tt no in t ntwork s ull wvlnt onvrsion pility. All simultions r runnin on DELL Quri Dul Cor Xon prossor n 4 GB o RAM. T mins run Winows Srvr. MATLAB is us to solv t ILP ormultions. To vlut our solution in trms o runny n numr o p-yl struturs, w ompr its prormn to t uristis CIDA n SLA introu in [], [8]. W lso ompr our loritm to t xt solution ivn y n ILP ormultion []. CIDA nrts in t irst stp limit st o nit p-yls, tn it slts t st o p-yls tt protts t ntwork usin tul iiny []. SLA nrts lso st o nit p-yls, wi is vry smll, n tn t iint su-st o p-yls prottin t ntwork is slt usin t ILP ormultion. Not tt in t optiml solution, ll yls r us s nit p-yls or t ILP. TABLE II REDUNDANCY Solutions Topolois COST 9 KL Optiml Solution 7.5% 78.% CIDA Solution 89.6% 9.8% SLA Solution 98.4% % Our si Solution 8.9% 88.7% Extnsion Solution 8.% 86.% W n s in Tl II tt t vlu o runny otin y our si solution is ttr tn t vlu otin y t two uristis CIDA n SLA or t two topolois. Wn w rlx t onstrint o t rtion pross y in nw su-onition, our solution oms vn mor iint. T runny o t improv solution is 7.6% n 8.% r rom t ILP solutions, rsptivly or t COST 9 n KL tst ntworks. T rson or tis iiny (ompr to wll know solutions su s CIDA n SLA) is tt in our ppro, t onstrution o p-yl is on y inrmntl rtion yl n tis lst tks into ount t workin pity, tus lt us ontrol t runny o p-yl tt w onstrut. In otr wor w onstrut only yls wit smll runny wi l to ru t totl runny o t ntwork. Wn w rlx t onstrint o t rtion w nlr t sr sp in orr to in

7 n iint p-yl. TABLE III RUNNING TIME Solutions Topolois COST 9 KL Optiml Solution 778(s) 56(s) CIDA Solution,95(s).(s) SLA Solution 59(s) 8.(s) Our si Solution.4(s).8(s) Extnsion Solution.8(s).7(s) W n s lso tt our runnin tim is vry littl wn w ompr it wit t otrs solution. T rson is tt our loritm os not o trou prliminry stp o nit p-yls nrtion. TABLE IV NUMBER OF P-CYCLES Solutions Topolois COST 9 KL Optiml Solution.4 6 CIDA Solution SLA Solution 7 Our si Solution 7,5 Extnsion Solution Sin t numr o istint p-yl struturs in t inl st o p-yls is n importnt prormn or t ntwork mnmnt tsk, w lso vlut our loritm in trms o numr o istint p-yls. Smllr numr o struturs in t inl st o p-yls mns ttr prormn. T simultion sows tt our loritm omputs smll numr o istint struturs. Tt is u to t ritrion tt w v introu rlir. W n not tt SLA nrts smll numr o struturs in t COST9 topoloy wn w ompr it wit our si solution, us its st o nit p-yls is vry smll, ut t vlu o runny otin y SLA is siniintly ir tn t vlu otin y our solution. V. CONCLUSION In tis ppr, w rss t prottion in WDM optil ntworks usin p-yls. W stui t prolm o inin n optiml st o p-yls tt prott t optil ntwork or ivn workin pity. W propos n loritm s on n inrmntl rtion o yls to onstrut t st o p-yls wi prott t ntwork. Our solution onstruts st o p-yls y tkin into ount t workin pity o t ntwork n witout oin trou prliminry stp o nit p-yls nrtion. Tis nls us to nrt n iint st o p-yls witout ny onsirtion or ssumption out t st o nit p-yls. To voi t loritm ttin stuk in lol minim w v propos n xtnsion to our solution y improvin t onitions o t rtion pross. Simultion rsults sow tt our solution s oo tr-o twn rsour utiliztion, runnin tim n s smll numr o p-yls or t two tst ntworks. Rsults lso sow tt t xtn solution improv siniintly t prormns o our loritm. REFERENCES [] W. D. Grovr n D. Stmtlkis, Cyl Orint Distriut Proniurtion: Rin-lik Sp wit Ms-lik Cpity or Sl-plnnin Ntwork Rstortion, In Pro. o IEEE Intrntionl Conrn on Communitions, pp ,998. [] D. A. Supk, C. G. Grur n A. Autnrit, Optiml Coniurtion o p-cyls in WDM Ntworks, In Pro. o IEEE Intrntionl Conrn on Communitions, pp ,. [] J. Doutt, D. H, W. D. Grovr n O. Yn, Aloritmi Appros or Eiint Enumrtion o Cnit p-cyls n Cpitt p-cyl Ntwork Dsin, In Pro. o t Fourt Intrntionl Worksop on t Dsin o Rlil Communition Ntworks DRCN, pp. -,. [4] D. A. Supk, An ILP or Optiml p-cyl Sltion witout Cyl Enumrtion, in Pro. o t Eit Workin Conrn on Optil Ntwork Dsin n Mollin, 4. [5] Z. Zn, W. Zon n B. Mukrj, A Huristi Aloritm or p-cyls Coniurtion in WDM Optil Ntworks, in Pro : Opto- Elrtoinis n Communitions Conrn, pp ,. [6] C. G.Grur, Rsilint Ntworks wit Non-Simpl p-cyls, inpro. o t Intrntionl Conrn on Tlommunitions,. [7] H. N. Nuyn, D. Hii, V. Q. Pun, S. Lowiz, K. Lo n B. Kn, Joint Optimiztion in Cpity Dsin o Ntworks wit p-cyl Usin t Funmntl Cyl St, In Pro. o IEEE GLOBECOM, 6. [8] H. Zn n O. Yn, Finin Prottion Cyls in DWDM Ntworks, In Pro. IEEE Intrntionl Conrn on Communition, pp ,. [9] C. Liu n L. Run, Finin Goo Cnit Cyls or Eiint p-cyl Ntwork Dsin, In Pro. t Intrntionl Conrn on Computr Communition n Ntworks, pp. -6, 4. [] W. D.Grovr n J. Doutt, Avns in Optil Ntwork Dsin wit p-cyls: Joint optimiztion n pr-sltion o nit P-yls, In Pro. o t IEEE-LEOS Summr Topil Mtin on All Optil Ntworkin, pp. WA-49-WA-5,. [] D. P.Onutou n W. D. Grovr, p-cyl Ntwork Dsin: rom Fwst in Numr to Smllst in Siz, In Pro o t 6t Intrntionl Worksop on Dsin n Rlil Communition Ntworks, 7. [] J. L. Mrzo, E. Cll, P. Vil, A. Urr, Prormn vlution o minimum intrrn routin in ntwork snrios wit prottion rquirmnts, Computr Communitions, pp. 6-68, 7. [] J. Vn Luwn, Aloritms n Complxity, Hnook o Tortil Computr Sin,99.