Designing Low-Capacity Backup Networks for Fast Restoration

Transcription

1 Dsigning Low-Capacity Backup tworks for Fast Rstoration Ron Bannr Aril Orda HP Las Dpt of Elctrical Enginring, Tcnion Haifa 3000, Isral Haifa 3000, Isral Astract - Tr ar two asic approacs to allocat protction rsourcs for fast rstoration T first allocats rsourcs upon t arrival of ac connction rqust; yt, it incurs significant st-up tim and is oftn capacity-infficint T scond approac allocats protction rsourcs during t ntwork configuration pas; trfor, it nds to accommodat any possil arrival pattrn of connction rqusts, nc potntially calling for a sustantial ovr-provisioning of rsourcs Howvr, in tis study w stalis t fasiility of tis approac Spcifically, w considr a scm tat, during t ntwork configuration pas, constructs an (additional) low-capacity ackup ntwork Upon a failur, traffic is rroutd troug a ypass in t ackup ntwork W stalis tat, wit propr dsign, ackup ntworks induc fasil capacity ovrad W furtr impos svral dsign rquirmnts (g, op-count limits) on ackup ntworks and tir inducd ypasss, and prov tat, commonly, ty also incur minor ovrad Motivatd y ts findings, w dsign fficint algoritms for t construction of ackup ntworks Kywords Fast Rstoration, twork Dsign, Capacity Assignmnt, Prplannd Span Rstoration ITRODUCTIO Background Transmission capailitis av incrasd to rats of 0 Git/s and yond [7] Wit tis incras, any failur may lad to a vast amount of data loss Accordingly, fast rstoration as com a cntral rquirmnt in t dsign of ig-capacity ntworks g, optical ms ntworks It as n rcognizd tat, for many practical sttings, t spd and capacity of t involvd links do not allow to activat rstoration mcanisms aftr t failur Tus, protction rsourcs must allocatd in advanc i, for a failur occurs [7] Tr ar two asic approacs to allocat protction rsourcs In t first approac, rsourcs ar allocatd on dmand i, upon t arrival of vry andwidt rqust, tus incurring a significant ovrad in trms of connction st-up tim Consquntly, tis approac prsnts a clar tradoff twn t fficincy of t rsulting solution in trms of capacity usag and t tim ndd to comput it; furtrmor, its corrsponding solutions ar usually asd on only partial (or no) information rgarding t ntwork stat and futur connction rqusts A diffrnt approac is to pr-allocat t protction rsourcs during t configuration pas of t ntwork Tis approac, at tims trmd prplannd span rstoration or spar capacity assignmnt [4] [7] [8] [9] [8] [9] acts locally in t vicinity of a failur to dploy a st of dtouring pat sgmnts twn t nd nods of t faild link Tis as crtain advantags in trms of simplicity and spd, caus only t status of ac working cannl on a faild link nds to known, not t ovrall dstination or routing of t affctd srvic pats Yt, it rquirs allocating protction rsourcs for any potntial pattrn of connction rqusts; nc, it may potntially call for a sustantial ovr-provisioning of protction rsourcs Wil som studis providd indications tat tis toll migt aral, ty focusd on particular topologis and mainly on numrical findings Otr complications tat av n osrvd ar t typical proiitiv complxity of t rlatd dsign prolms, as wll as t additional toll, in particular in trms of rsourc provisioning, incurrd y practical considrations suc as t nd to limit t op-count of t ypass pats In tis study w mak t following contriutions, wic significantly mitigat t potntial disadvantags of t pr-allocation approac, nc strngtning its cas as a practical solution for ntwork rstoration First, w sow tat, in typical ntwork scnarios, t provisioning ovrad is fasil and scalal in t ntwork siz; morovr, w sow tat t toll incurrd y svral practical considrations, suc as limiting t op count, is typically small In addition, w provid algoritmic scms for t rlatd dsign prolms, wic offr itr xact solutions or fficint approximations, witin polynomial running tim In t following, w furtr laorat on t invstigatd approac and t findings of our study W invstigat a variant of t span rstoration prolm [8] Our main goal is to xploit t major nfits of t prplannd span rstoration approac wil minimizing t incurrd toll of protction rsourcs Spcifically, givn a primary ntwork tat is usd in normal opration mod to rout dmands (in an unprotctd mannr), w stalis a (low-capacity) ackup ntwork tat can protct against any singl failur xprincd y t primary ntwork; i, upon a failur of any primary link =(u,v), t traffic on is rroutd from u to v troug ypass pats tat xclusivly long to t ackup ntwork W formulat tis notion as a ntwork dsign prolm wit t ojctiv of minimizing t total spar capacity of t ackup ntwork T following xampl illustrats tis ida Exampl : Considring Fig, t solid lins rprsnt t connctivity in a givn (unprotctd) primary ntwork Assum tat t ntwork is undirctd and t capacity of all links (in ac dirction) is xcpt for t (old) links (a,f) and (f,) tat av a

2 4 a c f Fig : A primary ntwork and a corrsponding ackup ntwork capacity of 5 T dasd lins wit indicatd capacitis rprsnt a ackup ntwork It is asy to vrify tat tis ackup ntwork indd provids protction against any singl link failur in t primary ntwork For xampl, upon a failur of t (unit capacity) link (,), it is possil to rrout (xclusivly ovr t ackup ntwork) on flow unit troug t ypass pat (,c,d,) Similarly, wn link (a,f) tat as a capacity 5 fails, it is possil to snd on flow unit ovr t ypass pat (a,,f) and 4 flow units ovr t ypass pat (ackup link) (a,f) ot tat ypass pats tat protct diffrnt links in tis primary ntwork can intrsct ac otr and sar t sam amount of andwidt to aciv fficincy For xampl, t links (,) and (c,) ar protctd y t ypasss (,c,d,) and (c,d,), rspctivly, wit t sard links (c,d) and (d,) Our Contriution As mntiond, t us of ackup ntworks poss svral important advantags, most notaly simplicity (upon a failur, t inducd control ffort is small and localizd) and spd (affctd traffic is immdiatly rroutd to t ackup ntwork, troug prconfigurd ypasss) Ts advantags com at a pric, namly, capacity tat nds to scurd xclusivly for t ackup ntwork Howvr, our analysis clarly indicats tat tis ovrad is typically small Spcifically, w indpndntly prov tis claim for two diffrnt typs of wll-stalisd modls of ral-world ntworks, namly Waxman ntworks [] and Powr-Law ntworks [5] Our rsults indicat tat, in ordr to construct a ackup ntwork for an -nod primary ntwork, it is sufficint to incras t total capacity allocatd to t primary ntwork y a factor of O( ) for Waxman ntworks and y a factor of O( ) ln for Powr-Law ntworks Hnc, t analyss for ot (indpndnt) modls stalis tat, as ntworks grow in siz, t pric of xtra capacity diminiss to zro Motivatd y tis finding, w considr svral (indpndnt) dsign rquirmnts tat ar important for t fficint dploymnt of ackup ntworks First, ac link in t primary ntwork sould oftn protctd y a ypass pat (in t ackup ntwork) wit a oundd op-count Indd, imposing a small op-count limit on ac ypass pat is ssntial for supporting QoS rquirmnts; for xampl, it as n notd [] tat t quuing dlay in congstd ntworks 4 d incrass xponntially wit t numr of ops Anotr xampl of t importanc of suc op-count limits is providd y optical ntworks, wr t signal quality dtriorats as it travls ovr multipl ops Anotr fundamntal dsign rquirmnt for ackup ntworks considrs t numr of ypass pats tat protct ac primary link (i, among ow many ypasss t rroutd traffic is split) It is oftn important to ound tis numr du to svral rasons: first, splitting traffic ovr multipl ypass pats can caus packts to arriv out of ordr, tus incrasing dlivry latncy and uffring rquirmnts [0]; scond, t complxity of a scm tat distriuts traffic among multipl ypasss considraly incrass wit t numr of pats [6]; tird, oftn tr is a limit on t numr of xplicit ypass pats (suc as lal-switcd pats in MPLS) tat can st up twn a pair of nods [6] Finally, w addrss a dsign rquirmnt tat considrs t topology of t rsulting ackup ntwork; spcifically, tis rquirmnt rstricts ac ackup topology to a sugrap of t primary topology Wit tis rquirmnt, t construction of t ackup ntwork is muc asir for practical purposs For xampl, wn t ackup ntwork is a sugrap of t primary ntwork, t duct systms tat contain t communication cals of t primary ntwork can usd to trad all communication cals of t ackup ntwork, tus avoiding an xtnsiv and xpnsiv digging procss In fact, wit tis rquirmnt, it is possil to avoid ardwar installation altogtr Eac of ts dsign rquirmnts lvis a toll in trms of t rquird ackup capacity Accordingly, w turn to considr t xtra capacity tat must allocatd to t ackup ntwork du to t imposition of ac comination of t aov dsign constraints Spcifically, w quantify t incras in xtra capacity as follows Givn a st of (on or mor) dsign constraints, wat is t worst-cas ratio twn t minimum capacity tat nds to allocatd to a ackup ntwork tat satisfis tis st of constraints and t minimum capacity tat nds to allocatd to a ackup ntwork tat as no constraints to satisfy? For all possil cominations of constraints w provid (rigorous) uppr and lowr ounds on tis worst-cas ratio Tal summarizs t corrsponding Hop-Count Limit Unsplittal Routing Sugrap Constraint Constraints Pric o o o (y dfinition) o o +Ys Ys At last ( ) at most - ; o +Ys Ys o Exactly ( - ) Ys o o At most ( - ) Ys o +Ys Ys W( ) Tal : T incras in xtra capacity du to ac comination of dsign constraints (ratio wrt t unconstraind cas) T op count limit is st to (t tigt valu) i, all ypass pats in t ackup ntwork must consist of at most links T numr of ypass pats is rstrictd to (t tigt valu), i, upon a failur on any primary link =(u,v), t traffic is rroutd from u to v ovr a singl ypass pat

3 rsults for -nod ntworks T rsults summarizd in Tal provid insigts as wll as important dsign ruls for t fficint construction of ackup ntworks First, all cominations of dsign constraints sav two (namly, t cominations tat includ ot t op-count and t sugrap constraints) incras t xtra capacity y a factor of at most ; tus, tir nforcmnt provids important prformanc nfits, and, at t sam tim, it incurs only a small cost in trms of xtra capacity; in particular, altoug w av sown in our analysis tat ackup ntworks induc minor ovrad only for t unconstraind cas, tis ovrad rmains small also wn t corrsponding cominations of dsign constraints ar imposd Yt, wn t op-count limits and t sugrap constraints ar concurrntly imposd, t xtra capacity is (dramatically) incrasd y a factor of W() Our rsults dmonstrat tat y using only on of t two constraints (i, itr op count limit or sugrap constraint ut not ot), it is possil to rduc t xtra capacity y a factor of Finally, it is intrsting to not tat t cost incurrd y imposing t rquirmnt to support unsplittal routing at t ackup ntwork is qual to tat incurrd y satisfying tat rquirmnt and t rquirmnt for small op-counts; nc, wn t rquirmnt to support unsplittal routing is imposd, op-count limits can also imposd at no cost For t aov dsign constraints, w dsign svral polynomial running tim algoritms tat aim at minimizing t capacity allocatd for t ackup ntworks wil satisfying a givn st of constraints Spcifically, w dsign two typs of algoritms T first imposs t rquirmnt to support unsplittal routing at t ackup ntwork wil t otr allows traffic to split among svral ypasss For t splittal cas, w prsnt a polynomial running tim algoritm tat optimally solvs t prolm wil considring itr on or ot of t otr dsign constraints (namly, t op-count limits and t sugrap constraint) For t unsplittal cas, w prsnt two algoritms tat approximat t optimal solution y a factor of at most T first approximation is dsignd to mt t sugrap constraint wil running in a tim complxity of O( M) for M- link -nod ntworks, and t otr is dsignd to mt t opcount limit wil running in a (linar) tim complxity of O() Finally, w sow ow to modify som of t proposd scms in ordr to construct ackup ntworks tat protct against corrlatd failurs MODEL W ar givn a primary ntwork G(V,E) wit a capacity c for ac link ÎE Lt = V and M= E As commonly don in studis on survivaility and on optical ntworks, w assum tat t ntwork is undirctd 3 T goal is to construct a ackup ntwork G (V,E ) tat protcts t traffic carrid y ac link in G(V,E) To tat nd, w av to find a Indd, ac comination involv only O() incras in xtra capacity It is notworty tat rlatd mpirical indications av n otaind for long-aul ntworks [7] 3 In optical ntworks, adjacnt nods ar usually connctd y a pair of (idntical) firs carrying information in opposit dirctions [] Trfor, undirctd graps fficintly modl ral optical ntworks and tus av n t focus of many studis on survivaility, g, [3],[],[3],[] st of links E ÍV V and ackup capacitis { c } for ts links so tat G (V,E ) can usd to rrout t traffic of any primary link ÎE onc it fails As xplaind, w considr tr typs of rstrictions on t ackup ntworks T first is a op count rstriction imposd on ac ypass pat in t ackup ntwork T scond rstriction is on t topology of t ackup ntwork; spcifically, tis rstriction limits t ackup ntwork to a sugrap of t primary ntwork T tird rstriction is imposd on t numr of ypass pats tat protcts ac primary link Rfrring to t tird constraint, w formulat t following typs of ackup ntworks Unsplittal ackup ntworks av a ypass pat p twn t nd-nods of ac (primary) link ÎE suc tat p can carry all t traffic of onc it fails i, t capacity of p (dnotd y c(p)) is at last c Splittal ackup ntworks av a collction of ypass pats P() twn t nd-nods of ac (primary) link ÎE suc tat t total capacity of all protction pats in P() is at last t capacity of i, c pîp() c(p) Finally, among all fasil ackup ntworks tat satisfy t aov rstrictions, w wis to construct on wit minimum total capacity i, c is minimizd Î E In t asic vrsion of t prolm, w focus on t singl link failur modl i, at any givn tim tr xists at most on faild link ot tat tis assumption nals to construct ackup ntworks tat ar significantly mor fficint in trms of capacity, sinc ypass pats in t ackup ntwork for two diffrnt (primary) links can intrsct ac otr and sar t sam amount of capacity In Sction 4, w xtnd t singl link failur assumption and considr ackup ntworks tat protct against corrlatd failurs 3 DESIGRULES FOR BACKUP ETWORKS In tis sction w invstigat t incras in t total capacity of ackup ntworks wn t dsign constraints mntiond in t Introduction ar imposd Tis invstigation nals to undrstand som fundamntal tradoffs in ackup ntworks, and it provids important dsign ruls for tir fficint construction To tat nd, w quantify t incras in t total capacity allocatd for t ackup ntwork du to ac comination of t following dsign constraints: (i) imposing a op-count limit of on ac ypass pat; (ii) supporting unsplittal routing at t ackup ntwork (i, upon any failur of a primary link, tr is at last on ypass pat tat can rrout all affctd traffic); (iii) rstricting t topology of t ackup ntwork to a sugrap of t topology of t primary ntwork For convninc, w rprsnt any comination of constraints as a triplt of inary indicators (H,U,S) wit rols and possil valus as follows Indicator H taks t valu H + if t -op count limit is imposd, and t valu H otrwis

4 Indicator U taks t valu U + if it is rquird to support unsplittal routing, and t valu U otrwis Indicator S taks t valu S + if t sugrap constraint is imposd, and t valu S otrwis Tn, for any comination of constraints (H,U,S), w dnot y r(h,u,s) t worst-cas ratio twn t minimum capacity allocatd for a ackup ntwork tat satisfis t comination (H,U,S) and t minimum capacity allocatd for a ackup ntwork tat as no imposd rstrictions (i, (H,U,S )) W outlin t organization and t main rsults otaind in tis sction In Susction 3, w stalis a lowr ound on t minimum capacity of any ackup ntwork; tis lowr ound is ssntial for t valuation of r(h,u,s) for ac comination (H,U,S) In Susction 3, w prov tat protction capacity is incrasd (in t worst cas) y a factor of at most ( - ) wn op count limits ar imposd and y a factor of xactly ( ) - wn unsplittal routing must supportd (i, + - r( H, U, S ) ( - ) and - + r( H, U, S ) = ( - ) ) Somwat surprisingly, w stalis tat wn ot constraints ar concurrntly imposd (i, ot H and U ar st) t protction capacity is still incrasd y a factor of xactly ( ) - ; nc, in unsplittal ackup ntworks, t opcount limits can imposd at no pric In Susction 33, w sow tat rstricting t ackup ntwork to a sugrap of t primary ntwork incrass t protction capacity y a factor of at most and at last ( ) -, ot for t splittal and unsplittal cass Finally, in Susction 34, w considr t incras in t protction capacity wn t ackup ntwork must concurrntly satisfy ot t op count limit and t sugrap constraint; altoug ac of tm incrass t protction capacity only y a (small) constant factor, w sow tat wn ot constraints must satisfid (concurrntly), t protction capacity dramatically incrass y a factor of Ω() 3 A Tigt Lowr Bound on Minimum Capacity In tis susction, w stalis a lowr ound on t minimum capacity of unrstrictd ackup ntworks i, t cas wr (H,U,S ) Tis lowr ound is usd in Susctions 3 and 33 to ound from aov t ratio r(h,u,s) In addition, w provid a simpl xampl tat dmonstrats tat tis lowr ound is tigt Lmma : Givn a primary ntwork G(V,E), dnot for ac t maximum capacity of a link tat is incidnt on v y C v = max c v, u wr c(v,u) is t capacity C(v), i, ( ) ( vu, ) ÎE { ( )} of t link (v,u) Tn, C ( v ) is a lowr ound on t minimum capacity of any unrstrictd ackup ntwork Proof: Considr any unrstrictd ackup ntwork G (V,E ) for t primary ntwork G(V,E) Lt E (v) t collction of all links tat ar incidnt on v in G (V,E ) ot tat, for ac, t total protction capacity of all links incidnt on v in G (V,E ) must at last C(v) i, c ³ C v ÎE ( v) ( ) for ac Indd, otrwis t link wit maximum capacity in G(V,E) tat is incidnt on a nod is not protctd Tus, w conclud c ³ C( v) () ÎE ( v) Sinc t ntwork G (V,E ) is undirctd, ac link =(v,u) in E longs to ot E (v) and E (u); nc, c = c Tis, togtr wit (), provs tat ÎE ÎE ( v) c ³ C( v) i, t minimum capacity of vry ÎE unrstrictd ackup ntwork is at last ( ) v V C v As t following simpl xampl sows, t aov lowr ound is tigt Considr a ntwork tat consists of a pair of nods u,v tat ar connctd y a singl link Oviously, for tis cas, t optimal ackup ntwork constituts of a singl paralll link to wit a capacity of c ; sinc y dfinition C(u)=C(v)=c, it follows tat C ( v ) = c ; nc, t total capacity of tis (optimal) ackup ntwork is xactly C ( v ) v V Î 3 T Pric of Hop-Count Rstrictions and Unsplittal Routing is Small In tis susction w invstigat t incras in t protction capacity wn t ackup ntwork must support unsplittal routing and/or satisfy op count rstrictions on ac ypass pat Oviously, imposing ac of t two dsign rquirmnts sustantially rducs t st of fasil solutions Trfor, on could xpct tat t nforcmnt of any of ts constraints rsults in a svr incras in t capacity ndd for protction Yt, in t following, w prov tat dsigning ackup ntworks tat ar rstrictd to rrout traffic unsplittaly ovr ypass pats wit a op limit = nvr incrass t protction capacity y a factor of mor tan ( - ) Spcifically, w sow tat ac of t ratios r(h,u +,S ) and r(h +,U +,S ) is qual to ( ) - For t cas wr only t op count limit is imposd, w + - sow tat r( H, U, S - ) ( - ) T analysis for all ts ratios is asd on t following ntwork construction Î I, t cas (H,U,S )

5 QoS-Backup twork <Primary twork G(V,E)> Initializ t st of links in t ackup ntwork to mpty i, E f Lt s a nod in V incidnt to a link wit maximum capacity i, C(s)=Max {C(v)} 3 For ac nod uîv/{s}, add to t st E a link twn nods s and u wit a capacity C(u) 4 Rturn t ntwork G (V,E ) Fig : Procdur QoS-Backup twork Torm : T minimum protction capacity of an unsplittal ackup ntwork wit a op limit = is largr y a factor of at most ( - ) tan t minimum protction capacity of a splittal ackup ntwork tat as no op count limits i, r( H U S ) ( ) + +,, - - Proof: Givn a primary ntwork G(V,E), in ordr to prov t torm it is sufficint to sow tat Procdur QoS- Backup twork constructs an unsplittal ackup ntwork tat consists of solly -op protction pats and as a total - C v ; tis, togtr wit v V capacity of at most ( ) ( ) Î Lmma (tat staliss a lowr ound of C ( v ) v Î V on t minimum capacity of any splittal ackup ntwork), provs t torm Lt G (V,E ) t ackup ntwork rturnd y Procdur QoS-Backup twork wn applid on t primary ntwork G(V,E) First, it is asy to s tat, y construction, ac link (u,v)îe as a -op ypass pat (u,s,v) in G (V,E ); nc, ac primary link in G(V,E) is protctd y a -op pat in G (V,E ) xt, not tat t total capacity of t links in G (V,E ) C u = C v - C s ; is ( ) ( ) ( ) uîv\{ s} sinc C(s)= = max { C( v) } ³ C( v), it olds tat t total capacity of G (V,E ) is C ( u ) = uîv\{ s} = Cv ( )- Cs ( ) Cv ( )- Cv ( ) = ( - ) Cv ( ) ; vv Î vv Î vv Î vv Î tus, according to Lmma, t capacity of G (V,E ) is largr y a factor of at most ( - ) tan t minimum possil capacity for splittal ackup ntworks It is lft to sown tat ac link ÎE as a protction pat p in G (V,E ) tat can carry all t traffic of i, c c(p), tus stalising tat G (V,E ) is an unsplittal ackup ntwork for G(V,E) To tat nd, not tat, for ac link (u,v)îe, t ypass pat (u,s,v) in G (V,E ) as a capacity of min{c(u),c(v)}; indd, y construction, t capacity of (u,s)îe and (s,v)îe is C(u) and C(v), rspctivly Trfor, sinc y dfinition of C( ), t valu of ot C(u) and C(v) is at last t capacity valu of t link (u,v)îe, it follows tat t capacity min{c(u),c(v)} (of t ypass pat (u,s,v)) is at last t capacity of t link (u,v) As mntiond in t Introduction, splitting t rroutd traffic ovr a small numr of ypass pats, ac wit a oundd op-count, is ssntial for supporting QoS-snsitiv applications ot tat, in t worst cas, t rroutd traffic can split among Ω(M) ypass pats, ac wit Ω() links For suc cass, Torm suggsts a usful dsign rul tat nals to trad t amount of capacity ndd for protction wit t quality of t rroutd traffic ot tat tis tradoff is vry ffctiv, as it involvs O(M) improvmnt in t split ratio and O() improvmnt in t op-count of t ypass pats, at a pric of just O() incras in t (minimum) protction capacity W procd to prsnt otr important insigts and dsign ruls for t fficint construction of unsplittal ackup ntworks wit op count limits To tat nd, w first stalis t following corollary tat stms dirctly from t uppr ound on r(h +,U +,S ) (Torm ) and t ovious rlations r(h,u +,S ) r(h +,U +,S ), r(h +,U,S ) r(h +,U +,S ) Corollary : T ratios r(h,u +,S ), r(h +,U,S ) and - r(h +,U +,S ) ar all oundd from aov y ( ) L (a) L Fig 3: T pric of unsplittal routing is xactly ( - ) W now sow tat t uppr ound otaind in Corollary for t ratios r(h,u +,S ) and r(h +,U +,S ) is tigt To tat nd, it is sufficint to stalis a lowr ound of ( - ) for ts ratios W first focus on stalising a lowr ound for t cost incurrd y t rquirmnt to rrout traffic unsplittaly i, a lowr ound for r(h,u +,S ) Mor spcifically, w prsnt an xampl of optimal unsplittal and splittal ackup ntworks tat sar t sam primary ntwork and diffr from ac otr in t allocatd capacity y a factor of ( - ) T xampl is illustratd troug Fig 3 T solid lins rprsnt a primary ntwork (wit a topology of an -nod ring), and t dasd lins rprsnt an unsplittal ackup ntwork in (a) and a splittal ackup ntwork in () W assum tat t capacitis of all links in t primary ntwork ar qual to Morovr, it is assumd tat ac link as a capacity of in t ackup ntwork of (a) and a capacity of ½ in t ackup ntwork of () () (a) Optimal unsplittal ackup ntwork wit a capacity of - () Optimal splittal ackup ntwork wit a capacity of

6 First not tat any unsplittal ackup ntwork must consist, for ac nod, of at last on link wit a unit capacity tat conncts v to som otr nod in V\{v} Indd, if a nod v is connctd to all its nigors in t ackup ntwork only y links wit a capacity smallr tan, it cannot rrout t traffic unsplittaly upon any failur on t links tat ar incidnt to v in t primary ntwork Tis, togtr wit t fact tat vry ackup ntwork for t givn primary ntwork must consist of at last - links, staliss tat t total capacity of any unsplittal ackup ntwork must of at last - In particular, t unsplittal ackup ntwork prsntd in (a) is optimal Considr now t splittal ackup ntwork prsntd in () ot tat tis ackup ntwork is a ring Trfor, upon a failur of any primary link, it is possil to split t traffic vnly and rrout ac alf in an opposit dirction Tus, sinc t capacity valus of all links ar qual to ½ in t ackup ntwork and qual to in t primary ntwork, t ackup ntwork is fasil as it protcts against any failur on t primary ntwork ot tat t total capacity allocatd for tis ackup ntwork is ; nc, t total capacity of any optimal splittal ackup ntwork (for t givn primary ntwork) is at most Tus,,, ³ = - it olds tat r ( H U S ) ( ) Finally, not tat, sinc r(h +,U +,S ) r(h,u +,S ), t lowr ound otaind for r(h,u +,S ) also applis for r(h +,U +,S ) + + r H U S - ³ - ot tat for ot ratios, tis lowr i, ( ) ( ),, ound coincids wit t uppr ound stalisd in Corollary Hnc, t pric of unsplittal routing is xactly ( - ) and is not affctd y imposing (additional) op-count limits on t ypass pats i, r H U S = r H U S = - W summariz tis (,, ) (,, ) ( ) discussion wit t following corollary Corollary : Eac of t ratios r(h,u +,S ) and r(h +,U +,S ) is xactly ( - ) 33 T Sugrap Constraint as a Small Pric As mntiond in t Introduction, from a practical viwpoint, it may muc asir to construct a ackup ntwork wit a topology tat is a sugrap of t primary ntwork Accordingly, in tis susction w invstigat t incras in protction capacity wn t topology of t ackup ntwork is rquird to a sugrap of t topology of t primary ntwork Spcifically, w sow tat, du to tis "sugrap constraint", t total capacity allocatd for t ackup ntwork incrass (in t worst cas) y a factor tat is twn ( - ) and for ot splittal and unsplittal ackup ntworks i, r( H U S ) r( H U S ) Î é ( - ),,,,, ë,ùû T proof of t uppr ound (Torm ) is asd on t ntwork construction spcifid in Fig 4 Mor spcifically, Indd, otrwis t ackup ntwork is not connctd; and, clarly, if t primary ntwork as a connctd topology tn t ackup ntwork must also connctd givn a primary ntwork G(V,E), w prov tat t following procdur constructs a ackup ntwork tat is a sugrap of G(V,E) and as a capacity of at most twic tat of an optimal ackup ntwork (wit aritrary topology and no op-counts) Sugrap Backup twork <Primary twork G(V,E)> Initialization Lt,,, M dnot a non-dcrasing ordring of t links of E according to tir capacitis Initializ t st of links in t ackup ntwork to mpty i, E f Dfin an indx i and initializ it to zro i, i Wil i M Considr t link i =(u i, v i ) in t primary ntwork G(V,E) If i dos not crat a cycl wit t (currnt) links of E, add to E a link twn nods u i and v i wit capacity c (i, t sam capacity as in t primary ntwork) i i+ 3 Rturn t ntwork G (V,E ) Fig 4: Procdur Sugrap Backup twork Torm : Rstricting t ackup ntwork to a sugrap of t primary ntwork incrass t minimum protction capacity y a factor of at most ot for t splittal and for t unsplittal cas i, r(h,u,s + ) and r(h,u +,S + ) Proof: Givn a primary ntwork G(V,E), in ordr to prov t torm it is sufficint to sow t xistnc of an unsplittal ackup ntwork G (V,E ) tat is a sugrap of G(V,E) and as a total capacity of at most C ( v ) Indd, y Lmma, vry splittal ackup ntwork as a lowr ound of C ( v ) on t minimum protction capacity; nc, sinc t minimum protction capacity in t splittal cas is not largr tan in t unsplittal cas, t xistnc of suc a ackup ntwork (i, an unsplittal ackup ntwork tat satisfis t sugrap rquirmnt and as a total capacity of at most C ( v ) ) provs t torm ot for t splittal and unsplittal cass Accordingly, in t following w sow tat t ntwork G (V,E ), rturnd y Procdur Sugrap Backup twork (Fig 4), wn applid on t primary ntwork G(V,E), satisfis all t following: (i) G (V,E ) is an unsplittal ackup ntwork for G(V,E); (ii) G (V,E ) is a sugrap of G(V,E); (iii) G (V,E ) as a total capacity of at most C ( v ) Hnc, proving (i), (ii) and (iii) staliss t torm

7 W first sow tat G (V,E ) is an unsplittal ackup ntwork for t primary ntwork G(V,E) To tat nd, w av to sow (y dfinition) tat ac link ÎE as a protction pat p in G (V,E ) tat can carry all t traffic of, i, c c(p) ot tat wn a link ÎE twn nods u and v is considrd in t construction of G (V,E ) (as pr Fig 4), itr a paralll link to wit a capacity c is addd to t ackup ntwork G (V,E ) or tr alrady is a pat in G (V,E ) twn u and v Oviously, in t first cas, t paralll link tat was addd to G (V,E ) as a capacity c tat can support t traffic of t link In t scond cas, link was considrd aftr a pat p from u to v was formd in G (V,E ); nc, sinc t links ar considrd y dcrasing capacity valus, t capacity of ac link along p in G (V,E ) is not smallr tan c ; trfor, t pat p can support t traffic carrid y W turn to sow tat t total capacity of G (V,E ) is at most C ( v ) To tat nd, w sow t xistnc of a on-toon mapping f:e V in t ntwork G (V,E ) Spcifically, w map ac link (u,v)îe into on of its nd-nods (i, itr u or v) suc tat no two links in E ar mappd into t sam nod in V Sinc links ar addd to t ackup ntwork wit capacity valus idntical to tos in t primary ntwork, and sinc C(v) is t maximum capacity of a link incidnt on nod v, suc a mapping nals to allocat for ac link ÎE an xclusiv nod suc tat c C(v); oviously, tis provs t torm, sinc it implis tat c C( v) ÎE W turn to spcify t mapping W partition t st of nods V into two disjoint sts S and T suc tat SÇT=V Initially, w pick an aritrary nod sîv and assign S={s} and T=V/{s} At ac stp w coos a link =(u,v) in E, wic crosss t cut (S,T) (i, uîs, vît and ÎE ) and map (u,v) into t nod vît; tn, w mov t nod v from t st T to t st S T procss is rpatd until tr is no link in t cut (S,T) Oviously, it follows y construction tat at t nd of t procss ac link (u,v) tat was considrd y t procss is mappd into on of its nd nods (itr u or v) Morovr, onc a link (u,v) is mappd to v, t procss rmovs v from t st T; nc, sinc t procss maps links only to nods in T, ac nod is always associatd wit at most on link i, tr is no pair of links tat ar mappd to t sam nod Trfor, it is lft to sown tat at t nd of t procss all t links in E ar mappd to som nod in V To tat nd, it is sufficint to sow tat ac link is considrd y t procss at last onc To tat nd, w first sow tat G (V,E ) is connctd Rcall tat w assum tat G(V,E) is connctd Hnc, for ac pair of nods u,, tr xists a pat in G(V,E) tat conncts u and v; nc, sinc w av sown tat G (V,E ) as a pat tat conncts t nd nods of ac link in E, tr is also a pat twn vry pair of nods u, in G (V,E ); nc, G (V,E ) is connctd ow assum y way of contradiction tat tr xists a link in E tat as not n considrd y t procss and dnot y AÍE t st of all suc links Sinc G (V,E ) is connctd, it follows tat, for ac link (u,v)îa, itr u or v (or ot) must connctd to at last on nod in V\{u,v} y a link from E Among t links in A, coos a link =(u,v) tat is closst in trms of op count to a link tat as alrady n considrd y t procss; y t slction of (u,v) tr must xist a nod wîv\{u,v} suc tat itr t link (v,w) or t link (u,w) longs to E and it as n considrd y t procss Witout loss of gnrality, assum tat link (u,w) as n considrd y t procss Trfor, y construction, wn t procss nds, nod u is in t sust S On t otr and, sinc y construction G (V,E ) is acyclic, it is impossil tat a link incidnt on t nod v as n considrd y t procss, sinc tis implis tat tr xists in G (V,E ) a pat twn s and v and also a pat twn s and u tat, togtr wit t link (u,v), form a cycl in G (V,E ) Hnc, nod v is in T and t link (u,v) crosss t cut (S,T); tus, y construction, it sould av n considrd y t procss Oviously, tis contradicts t slction of t link (u,v) W tus conclud tat all t links in E ar considrd y t procss at last onc It rmains to sown tat t topology of G (V,E ) is a sugrap of G(V,E) By construction, Procdur Sugrap Backup twork dfins a link twn a pair of nods in G (V,E ) only if t pair of nods is connctd y a link in t primary ntwork G(V,E); nc, G (V,E ) is a sugrap of G(V,E), tus complting t proof T proprty stalisd in torm sould considrd in cass wr uilding a compltly indpndnt infrastructur for t ackup ntwork is too costly (or impossil) Spcifically, in suc cass, Torm suggsts an important dsign rul tat maks t construction of t ackup ntwork sustantially asir at t pric of incrasing t total protction capacity y a factor of at most W now turn to sow tat tis rsult is almost tigt for ot t splittal and t unsplittal cass Spcifically, w sow tat ac of t ratios r(h,u,s + ) and r(h,u +,S + ) is largr tan ( - ) To tat nd, w prsnt an xampl of a primary ntwork wit two corrsponding ackup ntworks tat du to t sugrap constraint diffr from ac otr in Spcifically, w av sown at t ginning of t proof tat G (V,E ) is an unsplittal ackup ntwork for G(V,E) Hnc, it must av at last on pat tat conncts t nd nods of ac link in E By construction, t procss considrs only links from E ; morovr, it is asy to s tat all considrd links dfin a connctd componnt Trfor, tr must xist in G (V,E ) a pat twn s and v and also a pat twn s and u if t procss as considrd a link incidnt on t nod v and a link incidnt on t nod u

8 t allocatd capacity y a factor of ( ) - T xampl is illustratd in Fig 5 T solid lins (in (a) and ()) rprsnt a primary ntwork wit a topology of an -nod pat grap, t dasd lins in (a) rprsnt a ackup ntwork tat satisfis t sugrap constraint, and t dasd lins in () rprsnt a ackup ntwork wit an aritrary topology W assum tat t capacitis of all links in t primary ntwork ar qual to Morovr it is assumd tat ac link as a capacity of in t ackup ntwork of (a) and a capacity of ½ in t ackup ntwork of () (a) W first ound from low t ratio r(h,u,s + ) To tat nd, not tat wn t sugrap constraint is imposd, t ackup and t primary ntworks must av idntical topologis (i, t ackup ntwork must consist of a paralll link to ac primary link) Trfor, t topology of t ackup ntwork is a pat grap and tr is only on pat twn ac pair of nods Tus, upon a failur of any primary link, t rroutd traffic cannot split and is carrid unsplittaly ovr t link tat is paralll to in t ackup ntwork Trfor, sinc t capacity of ac primary link is, t capacity of t ackup links must of at last ; nc, t total capacity of any ackup ntwork must of at last -, and in particular t ackup ntwork prsntd in (a) is optimal On t otr and, t total capacity of any (unrstrictd) optimal ackup ntwork for t givn primary ntwork is at most Indd, t ackup ntwork prsntd in () is ot fasil and allocatd a ackup capacity of units Tus, it olds tat ( H, U, S ) ³ = ( - ); also, sinc r(h,u +,S + ) r(h,u,s + ) r L must old, it also olds tat r( H U S ) ( ),, L Fig 5: T sugrap constraint as a pric of at last ( - ) ³ - Ts lowr ounds, togtr wit t uppr ounds of Torm, stalis t following corollary Indd, assum y way of contradiction t xistnc of a ackup ntwork wit a topology tat is not a duplicat of t givn primary ntwork ut satisfis t sugrap constraint; clarly, tis ackup ntwork is disconnctd as it contains lss tan - links Yt, as mntiond, any ackup ntwork of a connctd primary ntwork must also connctd Indd, t ackup ntwork is a ring wit link capacitis qual to ; nc, upon a failur of any primary link, it is possil to split t traffic vnly and rrout ac alf in t opposit dirction () (a) Optimal ackup ntwork satisfying t sugrap constraint () A ackup ntwork wit aritrary topology Corollary 3: T ratios r(h,u,s + ) and r(h,u +,S + ) ar oundd from low y ( - ) and from aov y 34 A Proiitiv Cost for Hop-Count Limits Comind wit Sugrap Constraints In t prvious susctions w av sown tat t minimum protction capacity incrass y a factor of at most wn itr t ackup ntwork must satisfy op-count limits or t ackup ntwork is rquird to a sugrap of t primary ntwork; similar fficint guarants ar otaind wn ac of ts constraints is comind wit a rquirmnt to support unsplittal routing ovr t ackup ntwork Yt, in contrast to ts positiv rsults, in tis susction w sow tat wn t op-count limit and t sugrap constraint ar concurrntly imposd, t protction capacity can incras y a factor as larg as Ω() i, ot r(h +,U,S + )=Ω() and r(h +,U +,S + )=Ω() old Oviously, tis dramatic incras in protction capacity is proiitiv for practical purposs and implis tat only on (i, t mor significant) of t two constraints sould imposd W gin wit t ratio r(h +,U,S + ) Considr t primary ntwork G(V,E) prsntd in Fig 6 Assum tat all link capacitis ar qual to Dnot y V t uppr st of nods and y V t lowr st of nods T ntwork is a complt ipartit grap i, tr is a link twn ac pair of nods uîv and ut not twn any pair of nods tat ar ot in V or in V Lt G (V,E ) a splittal ackup ntwork for t primary ntwork G(V,E) tat satisfis ot a op-count limit = and t sugrap constraint W first V V L L Fig 6: A primary ntwork wit a topology of a complt ipartit grap sow tat G (V,E ) and G(V,E) must sar t sam topology, i, G (V,E ) must consist of a paralll link for ac primary link ÎE Assum, y way of contradiction, tat G(V,E) and G (V,E ) do not sar idntical topologis Hnc, tr must xist a link in t primary ntwork wit no corrsponding paralll link in t ackup ntwork i, a link (u,v)îe suc tat (u,v)ïe Upon a failur of t link (u,v)îe, t ackup ntwork must provid an altrnativ -op ypass pat twn t nods u and v Yt, it is asy to s tat suc a ypass cannot xist in t ackup ntwork Indd, tr is no common nigor v k to any two nods v i, v j in G(V,E) tat ar connctd y a link; nc, any pat twn u and v tat dos not includ t dirct link (u, v) must av a op count largr tan in t primary ntwork G(V,E) Tus, sinc G (V,E ) is a sugrap of G(V,E), t ypass for t link (u,v) in G (V,E ) must largr tan Oviously, tis

9 contradicts t assumption tat G (V,E ) is a ackup ntwork tat consists of ypasss wit at most ops Tus, w conclud tat G (V,E ) and G(V,E) must sar idntical topologis W now mploy t fact tat G (V,E ) and G(V,E) sar t sam topology to sow tat t total capacity of G (V,E ) is at last To tat nd, not tat upon a failur of any 4 primary link, only t link tat is paralll to in t ackup ntwork satisfis t op-count rstriction =; indd, w av alrady sown tat all otr ypass pats violat tis op-count limit Hnc, upon a failur of t link only t link paralll to (in t ackup ntwork) is mployd Trfor, sinc t capacity of ac primary link is, t capacity valu of ac paralll link (i, a link in t ackup ntwork) must of at last ; nc, c ³ E ÎE Howvr, sinc G (V,E ) and G(V,E) sar t sam topology it olds tat E = E ; nc, c ³ E = E = 4 ÎE Litrally, any splittal ackup ntwork for G(V,E) tat concurrntly satisfis t op-count limit and t sugrap constraint must allocatd wit a capacity of at last On t otr and, it is asy to s tat wn no rstrictions on t ackup ntwork ar imposd, any spanning tr (dfind ovr V) wit link capacitis qual to on, is a fasil ackup ntwork for G(V,E); sinc t total capacity in suc a cas is -, t total capacity of any (unrstrictd) optimal ackup ntwork for G(V,E) is at most - Tus, w conclud tat r( H, U, S + ) = =W( ) i, wn nitr t - op-count limits nor t sugrap rquirmnts ar considrd, t capacity of t ackup ntwork can dcras y a factor of Ω() Tis lowr ound, togtr wit t fact tat r(h +,U +,S + ) r(h +,U,S + ), stalis t following corollary Corollary 4: T ratios r(h +,U,S + ) and r(h +,U +,S + ) ar oundd from low y Ω() 4 COSTRUCTIG BACKUP ETWORKS In tis sction w sow ow to construct ackup ntworks for any givn primary ntwork W first prsnt optimal construction algoritms for ackup ntworks tat satisfy t dsign constraints mntiond in t Introduction Tn, w considr t computational complxity of ac algoritm For t cass wr t computational complxity is proiitiv, w prsnt constant approximation algoritms tat construct t ackup ntworks in polynomial tim; w also prsnt som numrical rsults tat sow tat t running tim of all t proposd optimal algoritms is fasil for practical purposs Finally, w sow ow to modify t proposd algoritms in ordr to construct ackup ntworks tat protct against corrlatd link failurs 4 Optimal Algoritms In tis sction w formulat linar and intgr programs, t solution of ac idntifis an optimal ackup ntwork for any givn primary ntwork For as of prsntation, w 4 transform ac undirctd link in t primary ntwork G(V,E) into two dirctd links wit opposit dirctions tat av ac t sam capacity as t original (undirctd) link Dnot y f ( ) t total flow rroutd ovr t ackup link =(w,w )äe upon a failur on t link ; lt c dnot t capacity of t ackup link äe Upon a failur on a primary link =(u,v)äe, t flow f ( ) carrid ovr t ackup link =(w,w )äe is a composition of flows tat ar rroutd from u to w troug ypass pats of diffrnt op counts Lt f ( ) t total flow ovr =(w,w )äe tat is rroutd from u to w troug ypasss (from u to w ) wit a op-count of xactly upon a failur on t primary link =(u,v)äe If t topology of t ackup ntwork must a sugrap of t primary ntwork G(V,E), w st E E (yt, it is possil to assign zro capacitis to ackup links, i, c 0 = for som äe ) On t otr and, wn tr is no rstriction on t rsulting topology (i, t sugrap constraint is not imposd and aritrary topologis ar allowd) w st E V V For ac väv, w dnot y O(v) t st of all links in E tat manat from v, and y I(v) t st of all links in E tat ntr tat nod, namly O(v)={v l v läe } and I(v)={w v w väe } Finally, for vry link =u v, lt s =u and t =v Tn, a corrsponding linar program wos solution is an optimal splittal ackup ntwork tat satisfis t op-count and sugrap constraints can formulatd using t varials f ( ), c, as spcifid in Fig 7 {{ } { } } Rcall tat an undirctd link wit a capacity c rprsnts two dirctd links suc tat ac of t links can transfr at most c flow units Trfor, t total capacity of all links in t undirctd rprsntation and in t dirctd rprsntation diffrs y a factor of In particular, minimizing t total capacity of all links in t dirctd rprsntation also minimizs t total capacity in t undirctd rprsntation

10 Program Backup twork (G(V,E), E, H) Paramtrs: G(V,E) t primary ntwork; E t connctivity of t ackup ntwork; H t op-count limit Minimiz Sujct to: c () ÎE - f ( ) f ( ) - = 0 ÎO( v) ÎI( v) 0 f ( ) ³ c E ÎO( s ) H f ( ) ³ c E ÎI( t ) = 0 H f ( ) c, = 0 ( ) 0 T ojctiv function () minimizs t total capacity allocatd to t ackup ntwork G (V,E ) Constraint () is t nodal flow consrvation constraint of t ackup flow (i, flow on t ackup ntwork) Mor spcifically, quation () stats tat upon a failur of any primary link =s t, t total ackup flow tat ntrs into any nod väv\{s, t } and as travrsd pats (from s to v) of opcount - must qual t total ackup flow manating out of tat nod troug pats of op count Equations (3) and (4) nsur tat, for ac primary link äe, t ackup ntwork rsrvs at last a total capacity of c along t pats tat connct t nd-nods of ; spcifically, constraint (3) mits from t nd-nod s of ac primary link =s t a ackup flow of at last c flow units; similarly, constraint (4) asors at t nd nod t of ac primary link =s t, a total ackup flow of at last c flow units Equation (5) nsurs tat, upon a failur of any primary link äe, t total flow rroutd ovr ac ackup link äe "Î E, " Î[0, H], { s t} "Î v V \, () "Î (3) "Î (4) "Î E " Î E (5) f = " Î E, "Î E, Ï[0, H- ] (6) ( ) 0 f ³, "Î E " ÎE, Î [0, H] (7) c ³ 0 " Î E (8) ( ) = ( ) f f "Î E, " Î [0, H], ", ÎE st s = t and t = s Fig 7: Constructing ackup ntworks wit minimum capacity (9) is at most c Exprssion (6) ruls out non-fasil flows tat violat t op rstriction, and Exprssions (7) and (8) rstrict all varials to non-ngativ Finally, Exprssion (9) rstricts t solution to symmtrical; nc, it ruls out all nonfasil solutions tat ar not fasil for t (original) undirctd ntwork { } ot tat, sinc Equation (7) allows t varials f ( ) to tak any non-ngativ valu, t rroutd flow upon a failur of any link ÎE can split among svral pats; nc, t solution (tat consists of t varials { c }) constituts an optimal splittal ackup ntwork for t givn op count rstriction H All tat is ndd in ordr to transfr t ackup flow unsplittaly is to modify Equation (7) so tat ac varial f ( ) would tak itr t valu 0 or t valu c for ac ÎE, ÎE and Î[0,H] ot tat, y doing so, w otain an intgr program tat constructs an optimal unsplittal ackup ntwork wil satisfying t givn dsign constraints Bot t linar program (tat corrsponds to t splittal cas) and t intgr program (tat corrsponds to t unsplittal cas) can solvd y commrcial softwar tools suc as CPLEX or MOSEK [3] W now considr t running tim of ac of ts programs To tat nd, it is important to not tat t numr of varials and constraints in Program Backup twork is polynomial in t ntwork siz Indd, t op count rstriction H is at most -; trfor, t numr of varials f ( ), c and {{ } { } } t numr of constraints is in t ordr of 3 M Tus, sinc t complxity incurrd y solving a linar program is polynomial in t numr of constraints and t numr of varials [0], Program Backup twork as a polynomial running tim for t splittal cas On t otr and, for t unsplittal cas Program Backup twork is an intgr program tat as no polynomial solution in t gnral cas 4 Fast Approximation Algoritms In t prvious susction, w av sown tat tat t running tim of t proposd optimal construction scms is polynomial in t input for t splittal cas ut may intractal for t unsplittal cas Fortunatly, in tis susction, w osrv tat Procdur QoS-Backup twork (Fig ) and Procdur Sugrap Backup twork (Fig 4) can usd as altrnativ approximations for t construction of unsplittal ackup ntworks Spcifically, ts approximations stalis unsplittal ackup ntworks tat satisfy itr t sugrap constraint or t op-count constraint; morovr, w sow tat ot of tm ar opratd in low polynomial tim and produc ackup ntworks wit a total capacity of at most twic t optimum Procdur QoS-Backup twork and Procdur Sugrap Backup twork ar sown to fficintly approximat t (corrsponding) optimal solutions in t proof of Torms and Spcifically, wil ot ar sown to rturn an unsplittal ackup ntwork wit a capacity of at most twic

11 t optimum, Procdur QoS-Backup twork is guarantd to rturn a ackup ntwork tat consists of solly -op protction pats and Procdur Sugrap Backup twork is guarantd to rturn a ackup ntwork tat satisfis t sugrap constraint Morovr, it is asy to s tat t xcution tims of Procdur QoS-Backup twork and Procdur Sugrap Backup twork ar O() and O(M ), rspctivly Tus, ts procdurs can considr itr t op-count constraint or t sugrap constraint (ut not ot ), wil providing attractiv prformanc guarants on ot running tim and allocatd protction capacity 43 Construction Algoritms for Backup tworks: Slow & Optimal or Fast & Suoptimal? Limiting t running tim of ntwork algoritms tat ar frquntly xcutd (g, routing protocols, scduling and switcing algoritms, w srvics, tc) is usually of major practical importanc Accordingly, in suc contxts uristics and approximations improv tir running tim at t pric of dtriorating t quality of t rturnd solution Yt, in our contxt suc a compromis is usually not rquird First, t singl computation of an optimal ntwork is oftn followd y an installation and configuration procss tat can last for days At t sam tim, t consquncs of a poorly dsignd ntwork may impos a proiitiv toll (as opposd, g, to an occasional packt tat is not snt along t st pat or scduld in t st tim slot) Trfor, unlss t computation tim of t construction algoritm is infasil for practical purposs (g, days), t quality of t solution may wll favord ovr t computation tim; nc, t optimal construction algoritms (spcifid in susction 4), would favord ovr t constant approximation scms (spcifid in susction 4) In t following, w indicat tat Program Backup twork (spcifid in suction 4) can applid to stalis optimal ackup ntworks (for typical primary ntworks) in t ordr of minuts Following t lins of [5], w gnratd 3 random topologis wit nods and invokd Program Backup twork ovr ac topology Tn, w masurd t tim ndd to construct an unsplittal ackup ntwork wit a op-count Bot procdurs av sown to rturn an unsplittal ackup ntwork wit a capacity of at most Cv ( ) ; Tis, togtr wit vv Î on t Lmma tat staliss a lowr ound of ( ) Cv vv Î minimum capacity of any ackup ntwork, prov tat t procdurs rturn a solution tat is witin a factor of away from t optimum W av sown in Sction 33 tat t incras in t rquird protction capacity wn ot constraints ar concurrntly satisfid is proiitiv for practical purposs Tus, altoug t procdurs can satisfy only on of t two constraints, a unifying scm for ot constraints is of limitd intrst from a practical viwpoint 3 T construction is spcifid in Sction 5 undr t Powr-Law topology modl limit H=; t program was implmntd in Matla 60 and run in a Gz Intl 4 macin Finally, w avragd t masurmnts ovr 50 runs for ac, Î{0, 5, 0, 5, 30, 35, 40, 45, 50, 55, 60, 65, 70} In Fig 8 w dpict t avrag running tim vrsus ntwork siz (numr of nods) ot tat t running tim is incrasd linarly wit t numr of nods; nc, from practical point of viw, it scals vry fficintly wit Morovr, for ntworks wit lss tan 70 nods t ackup ntwork is constructd in a tim smallr tan a minut Tim (Sc) umr of nods Fig 8: Running tim incrass linarly wit ntwork siz 44 Coping wit Corrlatd Failurs It as n rportd [5] tat narly 30% of all link failurs ar corrlatd Accordingly, in tis susction w xtnd our work to a framwork wr a failur of on primary link affcts otr primary links To tat nd, w assign to ac primary link ÎE a failur corrlatd st F()ÍE suc tat, upon a failur of t link, all links in F() fail Tn, givn a primary ntwork G(V,E) and a corrlatd st F() for ac ÎE, our goal is to dsign an optimal ackup ntwork tat considrs t dsign constraints suc tat, upon a failur of a link ÎE, t ackup ntwork provids protction against t failurs of all links in F()È{} Our solution for t aov prolm is to xtnd Program Backup twork (spcifid in Fig 7) to dal wit corrlatd failurs To tat nd, not tat wn no corrlation among t failurs xists (and only singl failurs can tak plac), t ypass pats can intrsct ac otr and sar t sam amount of capacity, providd ty ar usd for t failurs of diffrnt primary links in t ntwork On t otr and, for corrlatd failurs ac ackup link ÎE must al to carry t ackup flows inducd y all link failurs in t corrlatd st F() Hnc, if a st of primary links (say,,,, k ) concurrntly fail and tir associatd ackup flows cross t sam link äe tn, it is rquird tat = ( ) + ( ) + ( ) Tus, w only av to f f f f k c modify Equation (5) in Program Backup twork tat rstricts t ackup capacitis { c } to larg noug to carry t rroutd traffic causd y singl failurs to larg noug to carry t rroutd traffic causd y corrlatd failurs Mor spcifically, y rplacing Equation (5) wit

12 H f f c for ÎF( ) È{ } ÎF( ) È{ } = 0 t rstriction ( ) = ( ) ac primary link ÎE and ackup link ÎE, vry ackup link as sufficint capacity c to carry all ackup flows inducd y t failur of t corrlatd st F() and t link ; nc, t rsulting ackup ntwork can protct against corrlatd failurs wil minimizing t allocatd protction capacity 5 HOW EFFECTIVE IS THE PROPOSED APPROACH? In tis sction, w sow tat t total capacity tat must allocatd to protct a primary ntwork is minor compard to t capacity of t primary ntwork To tat nd, w valuat t ratio twn t total capacity of a primary ntwork and t total capacity of a corrsponding optimal ackup ntwork Clarly, a larg valu for tis ratio indicats tat, y propr dsign, it is possil to allocat a singl unit of protction capacity against a larg amount of capacity in t primary ntwork, tus implying tat t proposd approac inducs only a small ovrad in trms of capacity Givn a primary ntwork G(V,E) and t corrsponding optimal unsplittal ackup ntwork G (V,E ), lt r(g) t ratio twn t total capacity of G (V,E ) and t total capacity of G(V,E) i, r ( G) ÎE c In t following w analytically sow tat, for ot Waxman topologis [] and Powr-Law topologis [5], t xpctd valu of tis ratio is larg ot tat, sinc t total protction capacity of splittal ackup ntworks is nvr largr (and is usually smallr) tan tat of unsplittal ackup ntworks, t ound r(g) olds also for t splittal cas W first prsnt our analysis for Waxman topologis [] In tis modl, a pair of nods is locatd at t diagonally opposit cornrs of a squar ara of unit dimnsion Tn, - nods ar uniformly sprad ovr t squar Finally, a link twn ac two nods u and v, is introducd wit t following proaility, wic dpnds on t distanc d(u,v) twn tm: é-d ( uv, ) ù puv (, ) = a xp ê ú, (*) ë û wr a and ar paramtrs in t rang (0,] W assum (for ot t Waxman and t Powr-Law modls) tat t capacitis of all links in t constructd ntwork av t sam ordr of magnitud, wic is t typical cas Indd, in optical ntworks, tr ar a fw standard sizs of andwidt for optical links, ac wit a c transmission capaility twn svral undrds of Mit/sc to a fw Git/sc Torm 3: Assuming a Waxman topology, t xpctd ratio twn t total capacity of a primary ntwork and tat of t corrsponding optimal ackup ntwork is W() i, E[r(G)]=W() Proof: In t proof of Torm, w av sown tat Procdur QoS-Backup twork constructs, for any givn primary ntwork G(V,E), an unsplittal ackup ntwork wit a total capacity of at most C ( v ) In particular, for any givn primary ntwork G(V,E), it olds tat ( ) c ÎE r G ³ C( v) ; tus, sinc c ³ M c Î E min and Cv ( ) c vv Î max for M= E, c ÎE {c } and c ÎE {c }, ac primary ntwork G(V,E) must satisfy r ( G) ÎE ³ ³ c ( ) C v M c c Assum now tat G(V,E) is a primary ntwork of a Waxman topology In t following, w mploy () to sow tat t xpctd valu of r(g) is W() To tat nd, dfin for ac pair of nods v i,v j ÎV, an indicator I i,j, wic is if link (v i,v j ) xists in G(V,E), and is 0 otrwis; not tat, y dfinition, P{I i,j =}=p(v i,v j ), wr t lattr is givn y (*) Tn, t total numr of links M can writtn as a sum of indicators i, M = I Trfor, EM [ ] = E[ Ii, j] = ij, ( vi, vj ) Î V V min max ( vi, vj) ÎV V = EI [ ij, ] = [ 0 PI { ij, = 0} + PI { ij, = } ] = PI { ij, = } ( vi, vj ) Î V V ( vi, vj) Î V V ( vi, vj ) Î V V Finally, not tat, sinc t distanc twn any two nods in t squar ara is at most, it olds tat é ( v, v) i j P{ Iij, } pvv ( i, j) a xp ù = = = a ê ú³ ë û for ac v i,v j ÎV Hnc, a - () EM [ ] = P{ Ii, j= } ³ ( - ) (3) ( vi, vj) Î V V W ar finally rady to driv t xpctd valu of r(g) Sinc is a prdfind paramtr in t construction of t ntwork, it olds according to () tat ém cmin ù min ( ) é c ù Eéër G ù û ³ Eê E M c ú = max ê c ú (4) ë û ë max û Morovr, sinc w assum tat all link capacitis av t sam ordr of magnitud, tr xists som positiv constant Eg, most links twn ackon routrs ar itr OC48 or OC9 (wit lin rats of 48 Git/s and 995 Git/s, rspctivly) wras t links twn accss routrs and ackon routrs ar itr OC3 or OC (wit lin rats of 555 Mit/s and 6 Mit/s, rspctivly) As suc, all link capacitis in a typical optical ntwork av t sam ordr of magnitud Rcall tat C(v) dnots t maximum capacity of a link incidnt on v