Multipath Routing Algorithms for Congestion Minimization

Transcription

1 Multipath Routing Algorithms for Congstion Minimization Ron Bannr and Aril Orda Dpartmnt of Elctrical Enginring, Tchnion Isral Institut of Tchnology, Haifa 32000, Isral Abstract. Unlik traditional routing schms that rout all traffic along a singl path, multipath routing stratgis split th traffic among svral paths in ordr to as congstion. It has bn widly rcognizd that multipath routing can b fundamntally mor fficint than th traditional approach of routing along singl paths. Yt, in contrast to th singl-path routing approach, most studis in th contxt of multipath routing focusd on huristic mthods. W dmonstrat th significant advantag of optimal solutions. Hnc, w invstigat multipath routing adopting a rigorous (thortical) approach. W formaliz problms that incorporat two major rquirmnts of multipath routing. Thn, w stablish th intractability of ths problms in trms of computational complxity. Accordingly, w stablish fficint solutions with provn prformanc guarants. Kywords: Routing, Congstion, Algorithms, Optimization, Combinatorics. 1 Introduction Currnt routing schms typically focus on discovring a singl "optimal" path for routing, according to som dsird mtric. Accordingly, traffic is always routd ovr a singl path, which oftn rsults in substantial wast of ntwork rsourcs. Multipath Routing is an altrnativ approach that distributs th traffic among svral "good" paths instad of routing all traffic along a singl "bst" path. Multipath routing can b fundamntally mor fficint than th currntly usd singl-path routing protocols. It can significantly rduc congstion in "hot spots", by dviating traffic to unusd ntwork rsourcs, thus improving ntwork utilization and providing load balancing [13]. Morovr, congstd links usually rsult in poor prformanc and high varianc. For such circumstancs, multipath routing can offr stady and smooth data strams [6]. Prvious studis and proposals on multipath routing hav focusd on huristic mthods. In [16], a multipath routing schm, trmd Equal Cost MultiPath (ECMP), has bn proposd for balancing th load along multipl shortst paths using a simpl round-robin distribution. By limiting itslf to shortst paths, ECMP considrably R. Boutaba t al. (Eds.): NETWORKING 2005, LNCS 3462, pp , IFIP Intrnational Fdration for Information Procssing 2005

2 Multipath Routing Algorithms for Congstion Minimization 537 rducs th load-balancing capabilitis of multipath routing; morovr th qual partition of flows along th (shortst) paths (rsulting from th round robin distribution) furthr limits th ability to dcras congstion through load balancing. OSPF-OMP [21] allows splitting traffic among paths unvnly; howvr, th traffic distribution mchanism is basd on a huristic schm that oftn rsults in an infficint flow distribution. Both [22] and [24] considrd multipath routing as an optimization problm with an objctiv function that minimizs th congstion of th most utilizd link in th ntwork; howvr, thy focusd on huristics and did not considr th quality of th slctd paths. In [17], a schm is prsntd to proportionally split traffic among svral widst paths that ar disjoint with rspct to th bottlnck links. Howvr, hr too, th schm is huristic and valuatd by way of simulations. Simulation rsults clarly indicat that multipath solutions obtaind by optimal congstion rduction schms ar fundamntally mor fficint than th solutions obtaind by huristics. For xampl, in Sction 5, w show that if th traffic distribution mchanism in th ECMP schm had bn optimal, th ntwork congstion would hav dcrasd by mor than thr tims; morovr, if paths othr than shortst had bn allowd, th optimal partition would hav dcrasd th ntwork congstion by mor than tn tims. Hnc, th full potntial of multipath routing is far from having bn xploitd. Accordingly, in this study w invstigat multipath routing adopting a rigorous approach, and formulat it as an optimization problm of minimizing ntwork congstion. Undr this framwork, w considr two fundamntal rquirmnts. First, ach of th chosn paths should usually b of satisfactory "quality". Indd, whil bttr load balancing is achivd by allowing th mploymnt of paths othr than shortst, paths that ar substantially infrior (i.., "longr") may b prohibitd. Thrfor, w considr th problm of congstion minimization through multipath routing subjct to a rstriction on th "quality" (i.., lngth) of th chosn paths. Anothr practical rstriction is on th numbr of routing paths pr dstination, which is du to svral rasons [17]: first, stablishing, maintaining and taring down paths pos considrabl ovrhad; scond, th complxity of a schm that distributs traffic among multipl paths considrably incrass with th numbr of paths; third, oftn thr is a limit on th numbr of xplicitly routing paths (such as labl-switchd paths in MPLS [19]) that can b st up btwn a pair of nods. Thrfor, in practic, it is dsirabl to us as fw paths as possibl whil at th sam tim minimiz th ntwork congstion. Our Rsults: Considr first th problm of minimizing th congstion undr th rquirmnt to rout traffic along paths of "satisfactory" quality. W first show that th considrd problm is NP-hard, yt admits a psudo-polynomial solution. Accordingly, w dsign two algorithms. Th first is an optimal algorithm with a psudo-polynomial running tim, and th scond approximats th optimal solution to any dsird dgr of prcision at th (proportional) cost of incrasing its running tim (i.., an ε-optimal approximation schm). In addition, w show that ths algorithms can b xtndd to offr solutions to rliability-rlatd problms.

3 538 R. Bannr and A. Orda Considr now th rquirmnt of limiting th numbr of paths pr dstination. W show that minimizing th congstion undr this rstriction is NP-hard as wll. Accordingly, w stablish a computationally fficint 2-approximation schm 1. Thn, w gnraliz th 2-approximation schm into a bicritria rsult and stablish a (1+1/r)-approximation schm that, for any givn r 1, violats th constraint on th numbr of routing paths by a factor of at most r. Finally, w broadn th scop of this problm and stablish an fficint approximation schm for th dual problm, which rstricts th lvl of congstion whil minimizing th numbr of paths pr dstination. Du to spac limits, svral proofs and tchnical dtails ar omittd from this vrsion and can b found (onlin) in [4]. 2 Modl and Problm Formulation A ntwork is rprsntd by a dirctd graph G(V,E), whr V is th st of nods and E is th st of links. Lt N= V and M= E. A path is a finit squnc of nods p=(v 0,v 1, v h ), such that, for 0 n h-1, (v n,v n+1 ) E. A path is simpl if all its nods ar distinct. A cycl is a path p=(v 0,v 1,,v h ) togthr with th link (v h,v 0 ) E i.., (v 0,v 1,,v h,v 0 ). 2 Givn a sourc nod s V and a targt nod t V, P (s,t) is th st of (all) dirctd paths in G(V,E) from s to t. For ach path p P (s,t) and link E, lt (p) count th numbr of occurrncs of in p. For xampl, givn a non-simpl path p=(v 0,v 1,v 2,v 3,v 1,v 2,v 4 ) and a link =(v 1,v 2 ), w hav (p)=2. Each link E is assignd a lngth l Z + and a capacity c Z +. W considr a link stat routing nvironmnt, whr ach sourc nod has an imag of th ntir ntwork. Dfinition 1: Givn a (non-mpty) path p, th lngth L(p) of p is dfind as th sum of lngths of its links, namly, L(p) p l. Dfinition 2: Givn a (non-mpty) path p, th capacity C(p) of p is dfind as th capacity of its bottlnck link, namly, C( p) Min{ c }. Dfinition 3: Givn ar a ntwork G(V,E), two nods s,t V and a dmand γ. A path flow is a ral-valud function f:p (s,t) R + {0} that satisfis th flow dmand rquirmnts, i.., ( st, ) f p = γ. p P p 1 i.., an algorithm that provids a solution that, in trms of congstion, is within a factor of at most 2 away from th optimum. 2 As shall b shown, all our solutions consist of simpl paths xclusivly. Cycls and nonsimpl paths ar includd in our trminology to simplify th prsntation of th solution approach.

4 Multipath Routing Algorithms for Congstion Minimization 539 Dfinition 4: Givn is a path flow f:p (s,t) R + {0} ovr a ntwork G(V,E). A link flow is a ral-valud function f:e R + {0} that satisfis, for ach link E, f ( st, ) ( p) f. p P p Dfinition 5: Givn a ntwork G(V,E) and a link flow {f }, th valu f congstion factor and th valu max is th ntwork congstion factor. E c f c is th link As notd in [3], [13], [22] th ntwork congstion factor provids a good indication of congstion. In [4], w show that th problm of minimizing th ntwork congstion factor is quivalnt to th wll-known Maximum Flow Problm [1]. Hnc, whn thr ar no rstrictions on th paths (in trms of th numbr of paths or th lngth of ach path), on can find a path flow that minimizs th ntwork congstion factor in polynomial tim through a standard max-flow algorithm. W ar rady to formulat th two problms considrd in this study. Th first problm aims at minimizing th ntwork congstion factor subjct to a rstriction on th "quality" (i.., lngth) of ach of th chosn paths. Problm RMP (Rstrictd Multipath) Givn ar a ntwork G(V,E), two nods s,t V, a lngth l >0 and a capacity c >0 for ach link E, a dmand γ>0 and a lngth rstriction L for ach routing path. Find a path flow that minimizs th ntwork congstion factor such that, if PŒP (s,t) is th st of paths in P (s,t) that ar assignd a positiv flow, thn, for ach p P, it holds that L(p) L. Rmark 1: For convninc, and without loss of gnrality, w assum that th lngth l of ach link E is not largr than th lngth rstriction L. Clarly, links that ar longr than L can b rasd. Th nxt problm considrs th rquirmnt to limit th numbr of diffrnt paths ovr which a givn dmand is shippd whil at th sam tim minimizing th ntwork congstion factor. Problm KPR (K-Path Routing) Givn ar a ntwork G(V,E), two nods s,t V, a capacity c >0 for ach link E, a dmand γ>0 and a rstriction on th numbr of routing paths K. Find a path flow that minimizs th ntwork congstion factor, such that, if PŒP (s,t) is th st of paths in P (s,t) that ar assignd a positiv flow, thn P K. Rmark 2: In both problms, th sourc-dstination pair (s,t) is assumd to b ( st, ) connctd i.., P 1. 3 Minimizing Congstion Undr Path Quality Constraints In this sction w invstigat Problm RMP, i.., th problm of minimizing congstion undr path quality constraints. W bgin by stablishing its intractability.

5 540 R. Bannr and A. Orda Thorm 1: Problm RMP is NP-hard. Th proof [4] is basd on a rduction to th Partition Problm [11]. 3.1 Psudo-Polynomial Algorithm for Problm RMP Th first stp towards obtaining a solution to Problm RMP is to dfin it as a linar program. To that nd, w nd som additional notation. Rcall that w ar givn a ntwork G(V,E), two nods s,t V, a lngth l >0 and a capacity c >0 for ach link E, a dmand γ>0 and a lngth rstriction L for ach routing path. Lt α b th ntwork congstion factor. Dnot by f λ th total flow along =(u,v) E that has bn routd from s to u through paths with a total lngth of λ. Finally, for ach v V, dnot by O(v) th st of links that manat from v, and by I(v) th st of links that ntr that nod, namly O(v)={(v,l) (v,l) E} and I(v)={(w,v) (w,v) E}. Thn, Problm RMP can b formulatd as a linar program f λ, α, as spcifid in Fig 1. ovr th variabls { } } Program RMP ( GV (, E),{ st, },{ l},{ c}, γ, L) Minimiz α (1) Subjct to: = 0 v V \{ s, t }, λ [ 0, L ] (2) l f λ f λ O( v) I( v) = 0 λ [ 1, L] (3) l f f λ O( s) I( s) 0 f = γ (4) O( s) L λ f c α E λ= 0 (5) f λ = 0 E, λ [ 0, L l ] (6) f λ 0 E, λ [ 0, L] (7) α 0 (8) Fig. 1. Program RMP Th objctiv function (1) minimizs th ntwork congstion factor. Constraints (2), (3) and (4) ar nodal flow consrvation constraints. Equation (2) stats that th traffic flowing out of nod v, which has travrsd through paths p P (s,v) of lngth L( p) = λ, has to b qual to th traffic flowing into nod v, through paths p' P (s,u) and

6 Multipath Routing Algorithms for Congstion Minimization 541 links =(u,v) E, such that L( p') l λ + = ; sinc λ [ 0,L], th lngth rstriction is obyd; finally, quation (2) must b satisfid for ach nod othr than th sourc s and th targt t. Equation (3) xtnds th validity of quation (2) to hold for traffic that ncountrs sourc s aftr it has alrady passd through paths with non-zro lngth. Informally, quation (3) stats that "old" traffic that manats from s not for th first tim (through a dirctd cycl that contains th sourc s) must satisfy th nodal flow V \ s, t. consrvation constraint of quation (2), which solly focuss on nods from { } Equation (4) stats that th total traffic flowing out of sourc s, which has travrsd paths of lngth L = 0, must b qual to th dmand γ. Informally, quation (4) stats that th total "nw" traffic that manats from th sourc s for th first tim must satisfy th flow dmand γ. Equation (5) is th link capacity utilization constraint. It stats that th maximum link utilization is not largr than th valu of th variabl α i.., th ntwork congstion factor is at most α. Exprssion (6) ruls out non-fasibl flows and Exprssions (7) and (8) rstrict all variabls to b non-ngativ. W can solv Program RMP (Fig. 1) using any polynomial tim algorithm for linar programming [15]. Th solution to th problm is thn achivd by dcomposing th output of Program RMP (i.., link flow { f λ } ) into a path flow that satisfis th lngth rstriction L. This is don by modifying th flow dcomposition algorithm [1] (that transforms link flows { f } into path flows { f p } ) in ordr to considr lngth rstrictions i.., transform link flows with "lngths" { f λ } into path flows that oby th lngth rstrictions. Du to spac limits, th dscription of this algorithm is omittd and can b found in [4]. In th rmaindr of this subsction w considr th complxity of th ovrall solution (hncforth, Algorithm RMP), which is dominatd by th complxity of Program RMP [4]. It follows from [15] that th complxity incurrd by solving Program RMP is polynomial both in th numbr of variabls { f λ } and in th numbr of constraints ndd to formulat th linar program. Thus, sinc both of ths numbrs ar in th ordr of M L, th complxity of Algorithm RMP is polynomial in O(M L) i.., Algorithm RMP is a psudo-polynomial algorithm [11]. Thus, whnvr th valu of L is polynomial in th siz of th ntwork, Algorithm RMP is a polynomial optimal algorithm for Problm RMP. On such cas is whn th hop count mtric is considrd (i.., l 1), sinc thn L N ε-optimal Approximation Schm for Problm RMP In th prvious subsction w stablishd an optimal polynomial solution to Problm RMP for th cas whr th lngth rstrictions ar sufficintly small. In this subsction w turn to considr th solution to Problm RMP for arbitrary lngth rstrictions. As Thorm 1 stablishs that Problm RMP is NP-hard for this gnral cas, w focus on th dsign of an fficint algorithm that approximats th optimal solution.

7 542 R. Bannr and A. Orda Our main rsult in this stting is th stablishmnt of an ε optimal approximation schm, which is trmd th RMP Approximation Schm. This schm is basd on Algorithm RMP, spcifid in th prvious subsction, which was shown to hav a complxity that is polynomial in M L. Givn an instanc of Problm RMP and an approximation paramtr ε, w rduc th complxity of Algorithm RMP by first L ε scaling down th lngth rstriction L by th factor and thn rounding it into N an intgr valu. Obviously, as a rsult, w must also scal down th lngth of ach link. Howvr, in ordr to nsur that th optimal ntwork congstion factor dos not incras, w rlax th constraints of th nw instanc with rspct to th constraints of th original instanc. Spcifically, aftr w scal down th lngth rstriction and th lngth of ach link by th factor, w round up th lngth rstriction and round down th lngth of ach link. Thn, w invok Algorithm RMP ovr th nw instanc, in ordr to construct a path flow that minimizs congstion whil satisfying th rlaxd lngth rstrictions. Finally, w convrt ach non-simpl path in th output of Algorithm RMP into a simpl path by liminating loops; this is ssntial, sinc th total rror in th valuation of th lngth of ach path dpnds on th hop count. In Thorm 2, w stablish that th rsulting path flow violats th lngth rstriction by a factor of at most (1+ε) and has a ntwork congstion factor that is not largr than th optimal ntwork congstion factor. Th proof can b found in [4]. Thorm 2: Givn an instanc <G,{s,t},{c },{l },γ,l > of problm RMP and an approximation paramtr ε, th RMP Approximation Schm has a complxity that is polynomial in 1 ε and th siz of th ntwork; morovr, th output of th schm is a path flow f that satisfis th following: a. ( st, ) f p P p = γ i.., th flow dmand rquirmnt is satisfid. b. If α is th ntwork congstion factor of th optimal solution, thn, for ach * E, it holds that (, ) ( p) f α c, i.., th ntwork congstion st p P p factor is at most α. c. For ach path p P (s,t), if f p >0 thn p is simpl and L(p) (1+ε) L, i.., th lngth rstriction is violatd by a factor of at most (1+ε). 3.3 Furthr Rsults In th following, w outlin two important xtnsions to Problm RMP. Multi-commodity Extnsions: In [4], w considr a multi-commodity xtnsion of Problm RMP, i.., a problm with svral sourc-dstination pairs. Following basically th sam lins as in Subsctions 3.1 and 3.2, w prsnt a psudopolynomial solution for this problm and stablish an ε-optimal approximation. End-to-End Rliability Constraints: In [4], w also considr th incrasd vulnrability to failurs whn multipath routing is mployd in ordr to balanc th

8 Multipath Routing Algorithms for Congstion Minimization 543 ntwork load. Indd, whn traffic is split among multipl paths, a failur in ach routing path may rsult in th failur of th ntir transmission. In [4] w formulat th problm and show that it is computationally intractabl. Howvr, w show thr that th RMP Approximation Schm can b modifid in ordr to constitut an ε optimal approximation schm for th rliability problm. 4 Minimizing Congstion with K Routing Paths In this sction w solv Problm KPR, which minimizs congstion whil routing traffic along at most K diffrnt paths. In [4], w show that Problm KPR admits a (straightforward) polynomial solution whn th rstriction on th numbr of paths is largr than th numbr of links M (i.., K M). Howvr, w show in [4] that, in th mor intrsting cas whr K<M, th problm is NP-hard. Accordingly, in this sction w prsnt a 2-approximation schm for K<M. Our approximation schm is basd on solving an auxiliary problm that minimizs congstion whil rstricting th flow along ach path to b intgral in γ/κ. In ordr to formulat th corrsponding problm, considr first th following dfinition. Dfinition 7: Givn ar a ntwork G(V,E), a capacity c >0 for ach link E, a dmand γ and an intgr K. A path flow f:p R + {0} is said to b γ/κ-intgral, if for ach path p P (s,t), it holds that f p is a multipl of γ/κ. Problm Intgral Routing: Givn ar a ntwork G(V,E), two nods s,t V, a capacity c >0 for ach link E, a dmand γ>0 and an intgr K. Find a γ/κ-intgral path flow that minimizs th ntwork congstion factor, such that th dmand γ is satisfid. 4.1 Solving th Intgral Routing Problm Th following obsrvation shall b usd in ordr to construct a polynomial solution to th Intgral Routing Problm. Th proof can b found in [4]. Lmma 1: Givn an instanc <G,{s,t},{c },γ,k> of th Intgral Routing Problm, th optimal ntwork congstion factor is includd in th st i γ α { E, i [ 0, K] K c } 1. W now introduc Procdur Tst, which is givn an instanc <G,{s,t},{c },γ,k> of th Intgral Routing Problm and a rstriction α on th ntwork congstion factor. Procdur Tst prforms thr squntial stps. Initially, it multiplis all link capacitis by a factor of α in ordr to impos th rstriction on th ntwork congstion factor; indd, multiplying all capacitis by α assurs that th flow f along ach link 1 Obsrv that th siz of α is polynomial in th ntwork siz, namly: 2 ( 1) ( ) α M K + = O M.

9 544 R. Bannr and A. Orda E is at most α c ; thrfor, for ach E, th link congstion factor f /c, and, in particular, th ntwork congstion factor { f c} max, E ar at most α. Nxt, th procdur rounds down th capacity of ach link to th narst multipl of γ/κ; sinc th flow ovr ach path in vry solution to th Intgral Routing Problm is γ/κintgral, such a rounding has no ffct on th capability to transfr th flow dmand γ. Finally, th procdur applis any standard maximum flow algorithm that rturns an intgral link flow whn all capacitis ar intgral. Sinc all capacitis ar γ/κ-intgral, th maximum flow algorithm dtrmins a γ/κ-intgral link flow that transfrs th maximum amount of flow without violating th rstriction α on th ntwork congstion factor. If this link flow succds in transfrring at last γ flow units from s to t, thn th procdur rturns it. Othrwis, th procdur fails. Thorm 3: Givn is an instanc <G,{s,t},{c },γ,κ > of th Intgral Routing Problm. Dnot by α * th corrsponding optimal ntwork congstion factor. Thn, Procdur Tst succds for th input <G,{s,t},{c },γ,κ,α> iff α α *. Th proof appars in [4]. Thorm 3 has two important implications that nabl to construct an fficint solution to th Intgral Routing Problm. First, th thorm stablishs that th smallst α for which Procdur Tst succds with th input <G,{s,t},{c },γ,κ,α > is qual to α *. Thrfor, if S is a finit st that includs th optimal ntwork congstion factor α * and α is th smallst ntwork congstion factor in S such that Procdur Tst succds for th input <G,{s,t},{c },γ,κ,α >, thn α=α *. This fact, togthr with * th fact that th st α includs α (as pr Lmma 1), imply that, for vry instanc <G,{s,t},{c },γ,κ > of Problm Intgral Routing, th optimal ntwork congstion * factor α is th smallst α α such that Procdur Tst succds for th input <G,{s,t},{c },γ,κ,α >. Morovr, sinc in cas of a succss Procdur Tst rturns th corrsponding link flow, finding th smallst α α such that Procdur * Tst succds idntifis a link flow with a ntwork congstion factor of at most α. Th scond implication of Thorm 3 nabls to mploy a binary sarch whn w sk th smallst α α such that Procdur Tst succds. Indd, it follows from Thorm 3 that, whn Procdur Tst succds for α1 α, it succds for all α α, α α ; and whn it fails for 1 α2 α, it fails for all α α, α α2 ; thus, if Procdur Tst succds for α1 α (altrnativly, fails for α2 α ) it is possibl to liminat from furthr considration all th lmnts of α that ar largr than α 1 (corrspondingly, smallr than α 2 ). Rmark 3: Not that prforming a binary sarch ovr α rquirs sorting all th 2 lmnts of α, which consums O( α log α ) O( M log N) oprations [10]. Thus, w conclud that th mploymnt of a binary sarch so as to find th smallst α α for which Procdur Tst succds, stablishs a link flow that has

10 Multipath Routing Algorithms for Congstion Minimization 545 th minimal ntwork congstion factor. Th optimal solution is thn achivd by dcomposing th rsulting link flow into a path flow via th flow dcomposition algorithm [1]. Du to spac limits, th formal dscription of this algorithm, trmd Algorithm Intgral Routing, is omittd and can b found in [4]. Our discussion is summarizd by th following thorm, which stablishs that Algorithm Intgral Routing solvs Problm Intgral Routing. Its proof appars in [4]. Thorm 4: Givn is an instanc <G,{s,t},{c },γ,κ > of Problm Intgral Routing. If Algorithm Intgral Routing rturns Fail, thn thr is no fasibl solution for th givn instanc; othrwis, th algorithm rturns a γ/κ-intgral path flow that transfrs at last γ flow units from s to t along simpl paths, such that th ntwork congstion factor is minimizd. Rmark 4: It is asy to show [4] that th computational complxity of Algorithm Intgral Routing is O(MÿlogNÿ(M+ NÿlogN)). 4.2 A 2-Approximation Schm for Problm KPR Finally, w ar rady to stablish a solution for Problm KPR. To that nd, w show that th solution of th Intgral Routing Problm can b usd in ordr to stablish a constant approximation schm for Problm KPR. Th approximation schm is basd on th following ky obsrvation, which links btwn th optimal solution of Problm Intgral Routing and th optimal solution of Problm KPR. Thorm 5: Givn ar a ntwork G(V,E) and a dmand of γ flow units that has to b routd from s to t. If f 1 is a γ/κ-intgral path flow that has th minimum ntwork congstion factor and f 2 is a path flow that minimizs its ntwork congstion factor whil routing along at most K paths, thn th ntwork congstion factor of f 1 is at most twic th ntwork congstion factor of f 2. Proof: Suppos that f 1 and f 2 satisfy th assumptions of th Thorm. Lt α 1 and α 2 dnot th ntwork congstion factor of path flows f 1 and f 2, rspctivly. W hav to show that α 1 2ÿα 2. Out of th path flow f 2, w construct a γ/κ-intgral path flow that ships at last γ flow units from s to t and has a ntwork congstion factor of at most 2ÿα 2. Clarly, such a construction implis that th ntwork congstion factor of vry optimal γ/κintgral path flow that ships γ flow units from s to t is at most 2ÿα 2 ; in particular, sinc f 1 is on such optimal γ/κ-intgral path flow, such a construction stablishs that α 1 2ÿα 2. With this goal in mind, dfin th following construction. First, doubl th flow along ach routing path that f 2 mploys; obviously, th rsulting path flow transfrs 2ÿγ flow units from s to t along at most K routing paths whil yilding a ntwork congstion factor of 2ÿα 2. Thn, round down th (doubld) flow along ach routing path to th narst multipl of γ/κ; in this procss, th flow along ach path is rducd by at most γ/κ flow units. Hnc, sinc thr ar no mor than K routing paths, th total flow from s to t is rducd by at most γ units; thrfor, sinc bfor th

11 546 R. Bannr and A. Orda rounding opration xactly 2ÿγ flow units wr shippd from s to t, it follows that aftr rounding is prformd, th rsulting path flow transfrs at last γ flow units from s to t. Thus, w hav idntifid a γ/κ-intgral path flow that transfrs at last γ flow units from s to t. In addition, sinc prior to th rounding opration th ntwork congstion factor is 2ÿα 2 and th rounding can only rduc flow, th ntwork congstion factor of th constructd path flow is at most 2ÿα 2. Not that, givn a ntwork G(V,E) and a dmand γ that nds to b routd ovr at most K paths, vry γ/κ-intgral path flow satisfis th rquirmnt to ship th dmand γ on at most K diffrnt paths. On th othr hand, it has bn stablishd in Thorm 5 that th ntwork congstion factor obtaind by an optimal γ/κ-intgral path flow is at most twic th ntwork congstion factor of an optimal flow that admits at most K routing paths. Thus, computing a γ/κ-intgral path flow that has th minimum ntwork congstion factor satisfis th rstriction on th numbr of routing paths and obtains a ntwork congstion factor that is at most twic largr than th optimum. W summariz th abov discussion in th following corollary, which yilds an approximation schm for Problm KPR. Corollary 1: Givn ar a ntwork G(V,E), a dmand γ and a rstriction on th numbr of routing paths K. Th mploymnt of Algorithm Intgral Routing for th stablishmnt of a γ/κ-intgral path flow that minimizs th ntwork congstion factor provids a 2-approximation schm for Problm KPR with a complxity of O(MÿlogNÿ(M+ NÿlogN)). 4.3 Furthr Rsults In [4], w gnraliz th rsult of this sction into a bicritria rsult. Spcifically, for any givn r 1, w stablish a (1+1/r)-approximation schm that violats th constraint on th numbr of paths by a factor of at most r. Not that, for r = 1, th corrsponding schm obtains th sam prformanc guarants as in Subsction 4.2 abov. In addition, in [4] w considr th dual problm, which rstricts th ntwork congstion factor whil minimizing th numbr of routing paths, and prsnt a corrsponding approximation schm. 5 Simulation Rsults In this sction, w prsnt a comparison btwn an optimal solution to multipath routing and that providd by a huristic schm such as th (popular) Equal Cost MultiPath (ECMP) routing schm. W gnratd 10,000 random topologis, following th lins of [23] 1. For ach topology, w conductd th following masurmnts: (a) w masurd th ntwork congstion factor producd by invoking ECMP; (b) w masurd th ntwork 1 Du to spac limits, w omit th dtails of this construction, which can b found in [4].

12 Multipath Routing Algorithms for Congstion Minimization Congsdtion Ratio Lngth Rstriction ( normalizd to L * ) Fig. 2. Th ratio btwn th ntwork congstion producd by an optimal multipath routing assignmnt (for svral lngth rstrictions) and th ntwork congstion producd by ECMP congstion factor producd by an optimal assignmnt of traffic to shortst paths and to paths with a lngth that is qual to 1.17 L *, 1.33 L *, 1.5 L *, 1.67 L *, 1.83 L *,2 L * and 2.17 L *, whr L * is th lngth of a shortst path. Our rsults ar summarizd in Fig. 2. Not that if th ECMP schm had an optimal traffic distribution mchanism, th ntwork congstion factor could b rducd by a factor of 3. Morovr by rlaxing th rquirmnt to rout along shortst paths by 33%, th ntwork congstion factor is 10 tims smallr than with th standard ECMP. Thus, by mploying Algorithm RMP * or its -optimal approximation with L 1.33 L, congstion can b rducd by a factor of 10 with rspct to that producd by ECMP. 6 Conclusion Prvious multipath routing schms for congstion avoidanc focusd on huristic mthods. Yt, our simulations indicat that optimal congstion rduction schms ar significantly mor fficint. Accordingly, w invstigatd multipath routing as an optimization problm of minimizing ntwork congstion, and considrd two fundamntal problms. Although both hav bn shown to b computationally intractabl, thy hav bn found to admit fficint approximation schms. Indd, for ach problm, w hav dsignd a polynomial tim algorithm that approximats th optimal solution by a (small) constant approximation factor. Whil this study has laid th algorithmic foundations for two fundamntal multipath routing problms, thr ar still many challngs to ovrcom. On major challng is to stablish an fficint unifying schm that combins th two problms. Furthrmor, as in practic thr may b a nd for simplr solutions, anothr rsarch challng is th dvlopmnt of approximations with lowr computational complxity. Finally, as discussd in [4], multipath routing offrs rach ground for rsarch also in othr contxts, such as survivability, rcovry, ntwork scurity and nrgy fficincy. W ar currntly working on ths issus and hav obtaind svral rsults rgarding survivability [5].

13 548 R. Bannr and A. Orda Rfrncs [1] R. K. Ahuja, T. L. Magnanti and J. B. Orlin, "Ntwork Flows: Thory, Algorithm, and Applications", Prntic Hall, [2] D. Awduch, J. Malcolm, M. O'Dll, and J. McManus, "Rquirmnts for traffic nginring ovr MPLS", Intrnt Draft, April, [3] D. Awduch, J. Malcolm, J. Agogbua, M. O'Dll and J. McManus, "Rquirmnts for Traffic Enginring Ovr MPLS", IETF RFC 2702, Sptmbr [4] R. Bannr and A. Orda, "Multipath Routing Algorithms for Congstion Minimization", CCIT Rport No. 429, Dpartmnt of Elctrical Enginring, Tchnion, Haifa, Isral, Availabl from: [5] R. Bannr and A. Orda, "Th Powr of Tuning: a Novl Approach for th Efficint Dsign of Survivabl Ntworks", In Proc. IEEE ICNP 2004 (Bst Papr Award). [6] D. Brtskas and R. Gallagr, Data ntworks, Prntic-Hall, [7] Allan Borodin and Ran El-Yaniv, "Onlin Computation and Comptativ Analysis", Cambridg Univrsity Prss, [8] I. Cidon, R. Rom and Y. Shavitt, "Analysis of Multi-Path Routing", IEEE Transactions on Ntworking, v. 7, No. 6, pp , [9] S. Chn and K. Nahrstdt, An Ovrviw of Quality-of-Srvic Routing for th Nxt Gnration High-Spd Ntworks: Problms and Solutions, IEEE Ntwork, v. 12, No. 6, pp , Dcmbr [10] T. Cormn, C. Lisrson, and R. Rivst, Introduction to Algorithms. Cambridg, MA: Th MIT Prss, [11] M. R. Gary and D. S. Johnson, "Computrs and Intractability", W.H. Frman and Co., [12] A. V. Goldbrg and R. E. Tarjan, "A Nw Approach to Th Maximum Flow Problm", Journal of ACM, v.35, No. 4, pp , [13] S. Iyr, S. Bhattacharyya, N. Taft, N. McKon, C. Diot, A masurmnt Basd Study of Load Balancing in an IP Backbon, Sprint ATL Tchnical Rport, TR02-ATL , May [14] Y. Jia, Ioanis Nikolaidis and P. Gburzynski, "Multipl Path QoS Routing", Procdings of ICC'01 Finland, pp , Jun [15] N. Karmarkar, "A Nw Polynomial-Tim Algorithm For Linar Programming", Combinatorica, vol. 4, pp , [16] J.Moy, OSPF Vrsion 2, IETF RFC 2328, April [17] S. Nlakuditi and Zhi-Li Zhang, On Slction of Paths for Multipath Routing, In Proc. IWQoS, Karlsruh, Grmany, [18] V. Paxson, "End-to-End Routing Bhavior in th Intrnt", in proc. ACM SIGCOMM, [19] E. Rosn, A. Viswanathan, and R. Callon. "Multiprotocol Labl Switching Architctur". IETF RFC 3031, [20] D. Thalr and C. Hopps, Multipath Issus in Unicast and Multicast Nxt-Hop Slction, IETF RFC 2991, [21] C. Villamizar, OSPF Optimisd Multipath (OSPF-OMP), Intrnt Draft, [22] Y. Wang and Z. Wang, Explicit Routing Algorithms For Intrnt Traffic Enginring, In Proc. ICCN'99, Boston, Octobr [23] B. M. Waxman, Routing of Multipoint Connctions, IEEE Journal on Slctd Aras in Communications, 6: ,1988. [24] A. E. I. Widjaja, "Mat: MPLS Adaptiv Traffic Enginring", Intrnt Draft, August, 1998.