Link-State Routing with Hop-by-Hop Forwarding Can Achieve Optimal Traffic Engineering

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1 Lnk-Sae Roung wh Hop-by-Hop Forwardng Can Acheve Opmal Traffc Engneerng Daha Xu AT&T Labs - Research dahaxu@research.a.com Mung Chang Dep. of EE, Prnceon Unversy changm@prnceon.edu Jennfer Rexford Dep. of CS, Prnceon Unversy jrex@cs.prnceon.edu Absrac Lnk-sae roung wh hop-by-hop forwardng s wdely used n he Inerne oday. The curren versons of hese proocols, lke, spl raffc evenly over shores pahs based on lnk weghs. However, opmzng he lnk weghs for o he offered raffc s an NP-hard problem, and even he bes seng of he weghs can devae sgnfcanly from an opmal dsrbuon of he raffc. In hs paper, we propose a new lnk-sae roung proocol,, ha spls raffc over mulple pahs wh an exponenal penaly on longer pahs. Unlke s predecessor, DEFT [], our new proocol provably acheves opmal raffc engneerng whle reanng he smplcy of hop-by-hop forwardng. A gan of 5% n capacy ulzaon over s demonsraed usng he Ablene opology and raffc races. The new proocol also leads o sgnfcan reducon n he me needed o compue he bes lnk weghs. Boh he proocol and he compuaonal mehods are developed n a new concepual framework, called Nework Enropy Maxmzaon, whch s used o denfy he raffc dsrbuons ha are no only opmal bu also realzable by lnk-sae roung. Keywords: Ineror gaeway proocol, raffc engneerng, roung,, opmzaon, nework enropy maxmzaon. I. INTRODUCTION A lnk-sae roung proocol has hree componens. Frs s wegh compuaon: he nework-managemen sysem compues a se of lnk weghs hrough a perodc and cenralzed opmzaon. Second s raffc splng: each rouer uses he lnk weghs o decde raffc splng raos for every desnaon among s ougong lnks. Thrd s packe forwardng: each rouer ndependenly decdes whch ougong lnk o forward a packe based only on s desnaon prefx, n order o realze he desred raffc splng. The populary of lnksae proocols can be arbued o her ease of managemen; n parcular, each rouer s decson on raffc splng s conduced auonomously whou furher asssance from he nework-managemen sysem, and each packe s forwardng decson s made n a hop-by-hop fashon whou memory or end-o-end unnelng. Such smplcy seems o carry a cos on opmaly. In a procedure known as Traffc Engneerng (TE), nework operaors mnmze a convex cos funcon of he lnk loads, by unng he lnk weghs o be used by he rouers. Wh Open Shores Pah Frs (), he major varan of lnksae proocol n use oday, compung he rgh lnk weghs s NP-hard and even he bes seng of he weghs can devae The work was done when Xu was n Dep. of EE, Prnceon Unversy. TABLE I Comparson of varous TE schemes (new conrbuons n alcs). Commody Lnk-Sae Roung Roung Traffc Splng Arbrary Even among shores pahs Exponenal Scalably Low Hgh Hgh Opmal TE Yes No Yes Complexy Convex Convex Class Opmzaon NP Hard Opmzaon sgnfcanly from opmal TE [2]. Conrary o some popular belef, he opmal TE mehod n [3] s no dsrbued lnk-sae roung, and he followng queson remans open: can a lnksae proocol wh hop-by-hop forwardng acheve opmal TE? Ths paper shows ha he answer s n fac posve, by developng a new lnk-sae proocol, Penalzng Exponenal Flow-splTng (), provng ha acheves opmal TE, and demonsrang ha lnk wegh compuaon for s hghly effcen n heory and n pracce. In, packe forwardng s jus he same as : desnaon-based and hop-by-hop. The key dfference s n raffc splng. spls raffc evenly among he shores pahs, and spls raffc along all pahs bu penalzes longer pahs (.e., pahs wh hgher sums of lnk weghs) exponenally. Whle hs s a dfference n how lnk weghs are used n he rouers, also enables a change n how lnk weghs are compued by he operaor. I urns ou ha usng lnk weghs n he way acheves opmal raffc engneerng. Usng he Ablene opology and raffc races, we observe a 5% ncrease n he effcency of capacy ulzaon by over. Furhermore, exponenal penaly n raffc splng s he only penaly ha can lead o hs opmaly resul. The correspondng bes lnk weghs for can be effcenly compued: as effcenly as solvng a lnearly consraned concave maxmzaon and much faser han he exsng wegh compuaon heurscs for. Clearly, f he complexy of managng a roung proocol were no a concern, oher approaches could be used o acheve opmal TE. One possbly s mul-commody-flow ype of roung, where an opmal raffc dsrbuon s realzed by dvdng an arbrary fracon of raffc over many pahs. Ths can be suppored by he forwardng mechansm n Mul- Proocol Label Swchng (MPLS) [4]. However, opmaly hen comes wh a cos for esablshng many end-o-end unnels o forward packes. Second, oher sudes explored

2 more flexble ways o spl raffc over shores pahs [3], [5], bu hese soluons do no enable rouers o ndependenly compue he flow-splng raos from lnk weghs. Insead, a cenral managemen sysem mus compue and confgure he raffc-splng raos, and updae hem when he opology changes, sacrfcng he man benef of runnng a dsrbued lnk-sae roung proocol. Clearly, here s a enson beween opmal bu complex roung or forwardng mehods and he smple bu o-dae subopmal lnk-sae roung wh hop-byhop forwardng. Recen works [], [6] aemped o aan opmaly and smplcy smulaneously bu neher proved opmaly for TE nor developed suffcenly fas mehods for compung lnk weghs. A summary s provded n Table I. There are several new deas n hs paper ha enable a proof of opmaly and a much faser compuaonal beyond, for example, he heory and algorhm n DEFT []. One of hese deas s o develop boh he raffc splng and he wegh compuaon mehods from he concepual framework of Nework Enropy Maxmzaon (NEM). As a proof echnque, we wll consruc an opmzaon called NEM ha s solved neher by he operaor nor by he rouers, bu by us, he proocol developers. The opmaly condon of NEM reveals he srucure of hop-by-hop forwardng and s laer used o gude boh he rouer s raffc splng and he operaor s wegh compuaon. In shor, urns ou ha a ceran noon of enropy can denfy hose opmal raffc dsrbuons ha can be realzed by lnk-sae proocols. The res of he paper s organzed as follows. Background on opmal raffc engneerng s nroduced n Sec. II. The heory of Nework Enropy Maxmzaon n Sec. III leads o he roung proocol n Sec. IV and he assocaed lnk wegh compuaon algorhm n Sec. V. Exensve numercal expermens are hen summarzed n Sec. VI. Dfferences from our prevous resuls [] are summarzed n Sec. VII, before we conclude wh furher observaons and exensons n Sec. VIII. The key noaon used n hs paper s shown n Table II. II. BACKGROUND ON OPTIMAL TE A. Defnons of Opmaly Consder a wrelne nework as a dreced graph G =(V, E), where V s he se of nodes (where N = V ), E s he se of lnks (where E = E ), and lnk (u, v) has capacy c u,v.the offered raffc s represened by a raffc marx D(s, ) for source-desnaon pars ndexed by (s, ). The load f u,v on each lnk (u, v) depends on how he nework decdes o roue he raffc. An objecve funcon enables quanave comparsons beween dfferen roung soluons n erms of he load on he lnks. Traffc engneerng usually consders a lnk-cos funcon Φ(f u,v,c u,v ) ha s a ncreasng funcon of f u,v. For example, Φ(f u,v,c u,v ) can be he lnk ulzaon f u,v /c u,v, and he objecve of raffc engneerng can be o mnmze max (u,v) E Φ(f u,v,c u,v ). As anoher example, le Φ(f u,v,c u,v ) be a pecewse-lnear approxmaon of he M/M/ delay formula [7], e.g., TABLE II SUMMARY OF KEY NOTATION Noaon Meanng D(s, ) Traffc demand from source s o desnaon c u,v Capacy of lnk (u, v) f u,v Flow on lnk (u, v) c u,v Necessary capacy of lnk (u, v) fu,v Flow on lnk (u, v) desned o node fu Toal ncomng flow (desned o ) au w u,v Wegh assgned o lnk (u, v) w mn Lower bound of all lnk weghs d u The shores dsance from node u o node. d = h u,v Gap of shores dsance, h u,v d v + w u,v d u Γ(h u,v) Traffc splng funcon Φ(f u,v,c u,v) = f u,v f u,v/c u,v /3 3f u,v 2/3 c u,v /3 f u,v/c u,v 2/3 f u,v 6/3 c u,v 2/3 f u,v/c u,v 9/ 7f u,v 78/3 c u,v 9/ f u,v/c u,v 5f u,v 468/3 c u,v f u,v/c u,v / 5f u,v 638/3 c u,v / f u,v/c u,v, () and he objecve s o mnmze (u,v) Φ(f u,v,c u,v ). More generally, we use Φ({f u,v,c u,v }) o represen any ncreasng and convex objecve funcon. The opmaly of raffc engneerng s wh respec o hs objecve funcon. A hs pon we can already observe ha here s a gap beween he objecve of TE and he mechansm of lnksae roung. Opmaly s defned drecly n erms of he raffc flows, whereas lnk-sae proocols represen he pahs ndrecly n erms of lnk weghs. Brdgng hs gap s one of he challenges ha have prevened researchers from achevng opmal raffc engneerng usng lnk-sae roung hus far. B. Opmal TE Va Mul-Commody Flow Consder he followng convex opmzaon problem: mnmzng he TE cos funcon over flow conservaon and lnk capacy consrans: COMMODITY: mn Φ({f u,v,c u,v}) (2a) s.. fs,v fu,s = D(s, ), s (2b) v:(s,v) E u:(u,s) E f u,v V f u,v c u,v, (u, v) (2c) vars. f u,v,f u,v. (2d) The above mul-commody problem can be readly solved n polynomal-me, where he flow desned o a sngle desnaon s reaed as a commody, and f u,v s amoun of flow on lnk (u, v) desned o node. The resulng soluon, however, may no be realzable hrough lnk-sae roung and hop-by-hop forwardng. Indeed, To preven bandwdh wase, we can elmnae flow loop n he opmal roung wh a O(E log N)-me algorhm for each commody [8]. The loopfree propery s mporan n desgnng lnk-sae roung [3] as demonsraed laer n Sec. V.

3 for a nework wh N nodes and E lnks, he mul-commodyflow soluon may requre up o O(N 2 E) unnels (.e., explc roung) [9], makng dffcul o scale. In conras, lnk-sae roung s much smpler, requrng only O(E) parameers (.e., one per lnk). Furhermore, whle s rue ha, from he soluon of he COMMODITY problem, a se of lnk weghs can be compued such ha all he commody flow wll be forwarded along he shores pahs [3], [5], he flow-splng raos among hese shores pahs are no relaed o he lnk weghs, forcng he operaor o specfy up o O(NE) addonal parameers (one parameer on each lnk for each desnaon) as he flowsplng raos for all he rouers. The res of hs paper shows ha opmal raffc engneerng can, n fac, be acheved usng only E lnk weghs. III. THEORETICAL FOUNDATIONS: NEM In hs secon, we presen he heory of realzng opmal TE wh lnk-sae proocols. We frs compue he mnmal load ha each lnk mus carry o acheve opmal raffc dsrbuon, hen examne all he raffc splng choces subjec o necessary (mnmal) lnk capaces. The raffc splng confguraons ha s realzable wh hop-by-hop forwardng can be pcked ou by explong a propery has: maxmzng a weghed sum of he enropes of raffc splng vecors. In addon, he correspondng lnk weghs can be found by solvng he new opmzaon problem usng graden projecon. I s mporan o realze ha he proposed NEM framework developed n hs secon s used o desgn he proocol he NEM problem self s no solved by he operaor or rouers. I s consruced as a proof echnque and an nermedae sep owards he resuls n he nex wo secons. A. Necessary Capacy Gven he raffc marx and he objecve funcon, he soluon o he COMMODITY problem (2) provdes he opmal dsrbuon of raffc. We represen he resulng flow on each lnk (u, v) as he necessary capacy c u,v f u,v (or c as a vecor). The necessary capacy s a mnmal 2 se of lnk capaces o realze opmal raffc engneerng. There could be numerous ways of raffc splng ha realze opmal TE. If we replace lnk capacy c u,v n COMMOD- ITY (2) wh he necessary capacy c u,v, we are free o mpose anoher objecve funcon o pck ou a parcular opmal soluon o he orgnal problem. A key challenge here s o desgn a new objecve funcon, purely for he purpose of proocol developmen, such ha he resulng roung of flow can be realzed dsrbuvely wh lnk-sae roung proocols. B. Nework Enropy Maxmzaon Denoe P as he se of pahs from s o, and x as he probably (fracon) of forwardng a packe of demand D(s, ) o he -h pah (P). Obvously, x =. To be realzed wh hop-by-hop forwardng, he values of x should sasfy 2 Bu may no be he mnmum capacy. c s mnmal f c : c c c c whereas c s he mnmum f c : c c. (3) below where w u,v s he wegh assgned o lnk (u, v), and g( ) s a known funcon for all he rouers. ( ) g (u,v) P w u,v x x j = g ( (u,v) P j w u,v ). (3) We fnd ha he se of values of x sasfyng (3) maxmzes a nework enropy defned as follows. Consder he enropy funcon z(x ) = x log x for source-desnaon par (s, ). The weghed sum, ( D(s, ) P z(x ) ), s defned as he nework enropy. 3 Now we defne he Nework Enropy Maxmzaon (NEM) problem under he necessary capacy consrans as follows: NEM: max s.. D(s, ) P,:(u,v) P x =, s, z(x ) D(s, )x c u,v, (u, v) (4a) (4b) (4c) vars. x. (4d) From he opmal soluon of he COMMODITY problem, we know he feasbly se of NEM s non-empy. For a concave maxmzaon over a non-empy, compac consran se, here exs globally opmal soluons o NEM. C. Solve NEM by Dual Decomposon We wll connec he characerzaon of opmal soluons o NEM wh hop-by-hop forwardng and exponenal penaly. Towards ha end, and o provde a foundaon for lnk wegh compuaon n Sec. V, we frs nvesgae he Lagrange dual problem of NEM and a dual-graden-based soluon. Denoe dual varables for consrans (4b) as λ u,v for lnk (u, v) (or λ as a vecor). The maxmzaon of he Lagrangan over x can be solved as a TRAFFIC-DISTRIBUTION problem (5): TRAFFIC-DISTRIBUTION: max λ u,v c u,v + D(s, ) z(x ) (5a) (u,v) E P λ u,v D(s, )x s.. (u,v) E x =.,:(u,v) P (5b) Then, he dual problem can be solved by usng he graden projecon algorhm as follows for eraons ndexed by q, 3 The physcal nerpreaon of enropy for IP roung and he unqueness of choosng he enropy funcon o pck ou he rgh flow dsrbuons are presened n [].

4 λ u,v(q +) [ = λ u,v(q) α(q) ( c u,v )] +,:(u,v) P D(s, )x (q) =[λ u,v(q) α(q)( c u,v f u,v(q))] +, (u, v) E. where α(q) > s he sep sze, x (q) are soluons of he TRAFFIC-DISTRIBUTION problem (5) for a gven λ(q), and f u,v (q) s he oal flow on lnk (u, v). Afer he above dual decomposon, he followng resul can be proved wh sandard convergence analyss for graden algorhms []: Lemma : By solvng he TRAFFIC-DISTRIBUTION problem for he NEM problem and he dual varable updae (6), λ(q) converge o he opmal dual soluons λ and he correspondng prmal varables x are he globally opmal prmal soluons of (4). D. Solve TRAFFIC-DISTRIBUTION Problem Noe ha, he TRAFFIC-DISTRIBUTION problem s also separable,.e., he raffc splng for each demand across s pahs s ndependen of he ohers snce hey are no coupled ogeher wh lnk capacy consran (4b). So we can solve a subproblem (7) below for each demand D(s, ) separaely: DEMAND-DISTRIBUTION for D(s, ): max s.. D(s, ) P (u,v) E z(x ) λ u,v x =. :(u,v) P D(s, )x (6) (7a) (7b) We wre he Lagrangan assocaed wh he DEMAND- DISTRIBUTION subproblem n (8). = ( L r (x,µ ) D(s, ) ) P z(x ) µ ( x ) (u,v) E λu,v( :(u,v) P D(s, )x ) where µ s he Lagrangan varable assocaed wh (7b). Accordng o Karush-Kuhn-Tucker (KKT) condons [2], a he opmal soluon of he DEMAND-DISTRIBUTION subproblem, we have z (x ) (u,v) P λ u,v µ D() =. (9) For he enropy funcon, z(x) = x log x, z (x) = log x, wehave (8) x = e ( (u,v) P λu,v+ µ D() +). () where x,µ are he values of he x,µ respecvely a he opmal soluon. Then for wo pahs, j from s o, wehave x x j = e ( (u,v) P λu,v) e ( (u,v) P j λ u,v). () If we use λ u,v as he wegh w u,v for lnk (u, v), he probably of usng pah P s nversely proporonal o he exponenal value of s pah lengh. I s mporan o observe a hs pon ha, snce () has no facor of µ, an nermedae rouer can gnore he source of he packe n forwardng. Equally mporanly, from (6), n eraon q, he procedure of lnk prce (wegh) updang does no need he values of x (q). Insead, jus needs f u,v (q), he aggregaed bandwdh usage. We wll show how o calculae effcenly n Sec. V-B. Now, combnng he opmaly resuls n Sec. II-B and Lemma wh he dsrbued naure of (), we have Theorem : Opmal raffc engneerng for a gven raffc marx can be realzed wh lnk weghs usng exponenal flow splng (). IV. A NEW LINK-STATE ROUTING PROTOCOL: In hs secon, we ranslae he heorecal resuls n Sec. III no a new lnk-sae roung proocol run by rouers. Each rouer makes an ndependen decson on how o forward raffc o a desnaon (.e., flow-splng raos) among s ougong lnks usng only he lnk weghs. We frs presen from (), and summarze he noaon of raffc-splng funcon [] for calculang flow-splng raos. Then for a flow, we show an effcen way o calculae s raffc-splng funcon, whch can be approxmaed o furher smplfy he compuaon of raffc splng n pracce. A. Based on (), we propose a new lnk-sae roung proocol, called Penalzng Exponenal Flow-splTng (). The fracon of he raffc (from u o ) dsrbued across he -h pah (or probably of forwardng a packe), x u,, snversely proporonal o he exponenal value of s pah lengh p u, (sum of w u,v of he lnks along he pah), as shown n (2). : x u, = e p u,. (2) j e pj u, Theorem n Sec. III shows can acheve opmal TE. A flow can be realzed wh hop-by-hop forwardng. For he sample nework n Fg., for he wo pahs from s o, s u a and s u b, and wo pahs from u o, he flows on hem for (2) sasfy (3). f s u a : f s u b = f u a : f u b (3) Therefore, rouer u can rea he packes from dfferen sources (e.g. s or u) equally by forwardng hem among he ougong lnks wh precalculaed splng raos. Furher dscussons can be found n []. As a lnk-sae roung proocol, we need o defne he raffc splng funcon for as follows.

5 a s Fg.. u Realze a flow usng hop-by-hop forwardng B. Revew: Traffc Splng Funcon The noaon of raffc-splng (allocaon) funcon was nroduced n [] o succncly descrbe lnk-sae roung proocols. In a dreced graph, each undreconal lnk (u, v) has a sngle, confgurable wegh w u,v. Based on a complee vew of he opology and lnk weghs, a rouer can compue he shores dsance d u from any node u o node ; d v + w u,v represens he dsance from u o when roued hrough neghborng node v. Shores dsance gap, h u,v, s defned as d v + w u,v d u, whch s always greaer han or equal o. Then, (u, v) les on a shores pah o f and only f h u,v =. Traffc-splng funcon (Γ(h u,v)) ndcaes he relave amoun of raffc desned o ha node u wll forward va ougong lnk (u, v) 4.Lef u denoe he oal ncomng flow (desned o ) a node u (ncludng he bypassng flow and self-orgnaed flow). The oal ougong flow of raffc (desned o ) raversng lnk (u, v), f u,v, can be compued as follows: f u,v = f u b Γ(h u,v) (4) (u,j) E Γ(h u,j ). Conssen wh hop-by-hop forwardng, u spls he raffc over he ougong lnks whou regard o he source node or he ncomng lnk where he raffc arrved. Implemenaon of on boh daa-plane and conrol-plane can be readly accomplshed usng oday s echnology, as dscussed n []. C. Exac Traffc Splng Funcon for The raffc splng funcon for can be calculaed n polynomal me. From he defnon of (2), more raffc should be sen along an ougong lnk used by more pahs and he pahs should be reaed dfferenly based on her pah lenghs. To compue he raffc splng on each ougong lnk, we frs defne a posve real number Υ u as he equvalen number of shores pahs from node u o desnaon, and le Υ. Inaflow,wehave 4 For example, he raffc-splng funcon for even-splng across shores pahs (e.g., ) s { Γ O (h f h u,v) = u,v =, f h u,v >. Υ u e (p u, d u ) = e (pj u, wu,v d v +d v +wu,v d u ) (u,v) E j:(u,v) P j u, = e (d v +wu,v d u ) e (pj u, wu,v d v ) (u,v) E j:(u,v) P j u, = ( ) e h u,v Υ v (u,v) E (5) The recursve relaonshp represened n (5) can be used n he followng way: e h u,v Υ v s an equvalen number of shores pahs from u o for hose pahs bypassng lnk (u, v) and he rouer should dsrbue he raffc from u on lnk (u, v) n proporon o e h u,v Υ v. Then we have an exac raffc splng funcon 5 for a lnk (u, v): Γ PX (h u,v) =Υ ve h u,v (6) To enable hop-by-hop forwardng, each rouer needs o ndependenly calculae Γ PX (h u,v) for all node pars. Then each rouer frs compues he all-pars shores pahs, usng, e.g., he Floyd-Warshall algorhm wh me complexy O(N 3 ) [3], and calculaes he values of e h u,v. Then for each desnaon, o compue he values of Υ u, each rouer needs o solve N lnear equaons (5), whch requres O(N 3 ) me [3]. Thus he oal complexy s O(N 4 ). D. Traffc Splng Funcon for Downward To preven loops n lnk-sae roung, packes are usually forwarded along a downward pah where he nex hop s closer o desnaon. Ths nspres he followng Downward, whose raffc splng funcon s Γ PD (h u,v) 6 : { Γ PD (h Υ u,v) = v e h u,v f d u >d v, (7) oherwse. Γ PD (h u,v) can approxmae Γ PX (h u,v) and furher smplfy he compuaon of Υ u and raffc splng as dscussed below and ulzed n Sec. V-C. We consder each desnaon ndependenly. Afer emporarly removng lnk (u, v) where d u d v snce here s no flow on, we ge an acyclc nework and do opologcal sorng on he remanng nework. Proceedng hrough he nodes u n ncreasng opologcal order (sarng wh desnaon ), we compue he value of Υ u usng (5). For each desnaon, opology sorng requres O(N + E) me, and summarzng he Υ u across he ougong lnks requres O(N + E) me. Thus, he oal me complexy o calculae Υ u s O(N 3 + N(N + E)) = O(N 3 ). In general, downward does no provably acheve opmal TE, alhough comes exremely close o opmal TE n pracce, wh he assocaed lnk wegh compuaon even faser han ha for exac. In he case where he 5 P n he subscrp emphaszes ha he calculaon of raffc splng consders he pahs owards desnaon, and X means he exacness. 6 D n he subscrp emphaszes downward.

6 lower bound of all lnk weghs, w mn, s large enough, he downward s same as exac 7. V. LINK WEIGHT COMPUTATION FOR The las secon descrbed raffc splng under. A new way o use lnk weghs also means he nework operaor needs a new way o compue, cenrally and off-lne, he opmal lnk weghs. I urns ou ha he NP-hard problem of lnk wegh compuaon n can be urned no a convex opmzaon when lnk weghs are used by. To do ha, we wll conver he erave mehod of solvng he NEM problem n Sec. III no a smple and effcen algorhm. We frs presen an algorhm ha eravely chooses a enave se of lnk weghs and evaluaes he correspondng raffc dsrbuon by smulang he exac raffc splng run by he rouers. From Theorem, he algorhm s guaraneed o converge o a se of lnk weghs, whch realzes opmal TE wh. To furher speed up he calculaon, he raffc dsrbuon wh exac for each eraon can be approxmaed wh downward. The smulaon n Sec. VI show ha such an approxmaon s very close o opmal and provdes subsanal speedup n pracce. A. Algorhm Framework for Opmzng Lnk Weghs The erave algorhm consss of wo man pars: ) Compung he opmal raffc dsrbuon (necessary capaces) for a gven raffc marx by solvng he COMMODITY problem (2). 2) Compung he lnk weghs ha would acheve he opmal raffc dsrbuon. Sarng wh an nal seng of lnk weghs, he algorhm (see Algorhm ) repeaedly updaes he lnk weghs unl he load on each lnk s he same as he necessary capacy. Each seng of he lnk weghs corresponds o a parcular way of splng he raffc over a se of pahs. The Traffc Dsrbuon procedure compues he resulng lnk loads f u,v, based on he raffc marx. Then, he Lnk Wegh Updae procedure (see Algorhm 2) ncreases he wegh of each lnk (u, v) lnearly f he raffc exceeds he necessary capacy, or decreases oherwse. The parameer α s a posve sep sze, whch can be consan or dynamcally adjused; we fnd ha seng α o he recprocal of he maxmum necessary lnk capacy ( max c u,v ) performs well n pracce. Algorhm s guaraneed o converge o he global opmal soluon as saed n Lemma. In erms of compuaonal complexy, we know ha COMMODITY can be solved effcenly. The complexy of Algorhm 2 s O(E). The remanng queson s how o solve he subproblem Traffc Dsrbuon(w) effcenly. 7 For lnk (u, v), f he shores dsance o of u, d u d v, hen h u,v = d v + wu,v d u wu,v and Γ PX(h u,v ) Υ v e wu,v,and he flow desned o on (u, v) s close o f w u,v s large enough, e.g., e.5%. Therefore, mos flow n always makes forward progress owards he desnaon,.e., from rouer u wh larger d u o rouer v wh smaller d v. : Compue necessary capaces c by solvng (2) 2: w Any se of lnk weghs 3: f Traffc Dsrbuon(w) 4: whle f c do 5: w Lnk Wegh Updae(f) 6: f Traffc Dsrbuon(w) 7: end whle 8: Reurn w /*fnal lnk weghs*/ Algorhm : Opmze Over Lnk Weghs : for each lnk (u, v) do 2: w u,v w u,v α ( c u,v f u,v ) 3: end for 4: Reurn new lnk weghs w Algorhm 2: Lnk-Wegh Updae(f) B. Compue Traffc Dsrbuon wh Exac To compue he raffc dsrbuon for, we should frs compue he shores pahs beween each par of nodes and all he values Γ PX (h u,v) as n Sec. IV-C. Compung he resulng dsrbuon of raffc s complcaed by he fac ha Γ PX ( ) may drec raffc backwards o a node ha s furher away from he desnaon. To capure hese effecs, recall ha fu s he oal ncomng flow a node u (ncludng raffc orgnang a u as well as any raffc arrvng from oher nodes) ha s desned o node. In parcular, he raffc D(s, ) ha eners he nework a node s and leaves a node sasfes he followng lnear equaon: fs ( ) fx Γ PX (h x,s) x:(x,s) E (x,j) E Γ PX(h x,j ) = D(s, ). (8) Tha s, he raffc D(s, ) enerng he nework a node s maches he oal ncomng flow fs a node s (desned o node ), excludng he raffc enerng s from oher nodes. The rans flow s capured as a sum over all ncomng lnks from neghborng nodes x, whch spl her ncomng raffc fx over her lnks based on he raffc-splng funcon. The N lnear equaons (8) for each ypcally requre O(N 3 ) me [3] o solve. Thus he oal complexy s O(N 4 ). C. Approxmae Traffc Dsrbuon wh Downward To furher reduce he compuaonal overhead, we realze ha he opmal raffc dsrbuon should be loop free. Thus, n he las eraon n Algorhm, he flow loop should be neglgble. In addon, he accurae soluon for each nermedae eraon s no necessary n pracce, we can approxmae Exac (Γ PX ( )) wh Downward (Γ PD ( )) o forward raffc only on downward pahs, he raffc dsrbuon for each nermedae eraon can be compued usng a combnaoral algorhm, whch s sgnfcanly faser han solvng lnear equaons (8). As n Sec. V-B, we frs compue he shores pahs beween all pars of nodes, as well as he values of Γ PD (h u,v),asshown n he frs sep of Algorhm 3. The followng procedure s

7 : For lnk weghs w, consruc all-pars shores pahs and compue Γ PD (h u,v) 2: for each desnaon do 3: Temporarly remove lnk (u, v) where d u d v 4: Do opologcal sorng on he remanng nework 5: for each source s n he decreasng opologcal order do 6: fs D(s, )+ x:(x,s) E f x,s 7: fs,v fs 8: end for 9: end for Γ PD(h s,v ) (s,j) E ΓPD(h s,j ) : f u,v V f u,v : Reurn f /*se of f u,v */ Algorhm 3: Traffc Dsrbuon(w) wh Γ PD ( ) very smlar o bu subly dfferen from ha for calculang Γ PD (h u,v). We consder each desnaon ndependenly, snce he flow o each desnaon can be compued whou regard o he oher desnaons. Afer emporarly removng lnk (u, v) where d u d v snce here s no flow on, we ge an acyclc nework and do opologcal sorng on he remanng nework. The compuaon sars a he node whou ncomng lnk n he acyclc nework, snce hs node would never carry any raffc o ha orgnaes a oher nodes. Proceedng hrough he nodes s n decreasng opologcal order, we compue he oal ncomng flow a node s (desned o ) as he sum of he flow orgnang a s (.e., D(s, )) and he flow arrvng from neghborng nodes x (f x,s). Then, we use he oal ncomng flow a s o compue he flow of raffc oward on each of s ougong lnks (s, v), usng he raffcsplng funcon Γ PD ( ). In Algorhm 3, compung he all-pars shores pahs wh he Floyd-Warshall algorhm has me complexy O(N 3 ) [3]. For each desnaon, opology sorng requres O(N + E) me, and summarzng he ncomng flow and splng across he ougong lnks requres O(N + E) me. Thus, he oal me complexy o run Algorhm 3 n each eraon of Algorhm s O(N 3 + N(N + E)) = O(N 3 ). Fnally, he oal runnng me for Algorhm depends on he me requred o solve (2) and he oal number of eraons requred for Algorhms 2 and 3. Alhough he orgnal NEM problem nvolves an exponenal number of varables, he complexy of Algorhm s sll comparable o solvng a convex opmzaon wh polynomal number of varables (lke he COMMODITY problem (2)) usng graden projecon, snce we do no need o solve NEM drecly. VI. PERFORMANCE EVALUATION How well can he new roung proocol perform and how fas can he new lnk wegh compuaon be? has been proven o acheve opmal TE n Sec. III, wh a complexy of lnk wegh compuaon smlar o ha of solvng convex opmzaon (wh a polynomal number of varables). In hs secon, we numercally demonsrae ha s approxmae verson, Downward, can make convergence very fas n pracce whle comng exremely close o TE opmaly. A. Smulaon Envronmen We consder wo nework objecve funcons (Φ({f u,v,c u,v })): maxmum lnk ulzaon, and oal lnk cos () (as used n operaor s TE formulaon). For benchmarkng, he opmal values of boh objecves are compued by solvng lnear program (2) wh CPLEX 9. va AMPL, and serve as he performance benchmarks. To compare wh, we use he sae-of-he-ar localsearch mehod n [2]. We adop TOTEM. [4], whch follows he same approach as [2], and has smlar qualy of he resuls 8. We use he same parameer seng for local search as n [2], [7] where he lnk weghs are resrced as negers from o 2 snce a larger wegh range would slow down he searchng [7], nal lnk weghs are chosen randomly, and he bes resul s colleced afer 5 eraons. To deermne lnk weghs under, we run Algorhm wh up o 5 eraons of compung raffc dsrbuon and updang lnk weghs. Abusng ermnology a lle, n hs secon we use he erm o denoe he raffc engneerng wh Algorhm (ncludng wo sub-algorhms 2 and 3). We run he smulaon on a real backbone nework and several synhec neworks. Frs s he Ablene nework, whch has nodes and 28 dreconal lnks wh Gbps capacy. The raffc demands are exraced from he sampled Neflow daa on Nov. 5h, 25. To smulae neworks wh dfferen congeson levels, we creae dfferen es cases by unformly decreasng he lnk capacy unl he maxmal lnk ulzaon reaches % wh opmal TE. We also es he algorhms on he same opologes and raffc marces as hose n [2]. The 2-level herarchcal neworks were generaed usng GT-ITM, whch consss of wo knds of lnks: local access lnks wh 2-un capacy and long dsance lnk wh -un capacy. In he random opologes, he probably of havng a lnk beween wo nodes s a consan parameer and all lnk capaces are uns. In hese es cases, for each nework, raffc demands are unformly ncreased o smulae dfferen congeson levels. B. Mnmze Maxmum Lnk Ulzaon Snce we creae dfferen levels of congeson for he same nework by unformly decreasng lnk capaces or unformly ncreasng raffc demands, we jus need o compue he Maxmum Lnk Ulzaon (MLU) for one es case n each nework because MLU s proporonal o he rao of oal demand over oal capacy. In addon o MLU, we are parcularly neresed n he merc effcency of capacy ulzaon, η, whch s defned as he followng rao: he percenage of he raffc demand sasfed when he MLU reaches % under a raffc engneerng scheme over ha n opmal raffc engneerng. The mprovemen n η s referred o as he Inerne capacy ncrease n [2]. 8 Propreary enhancemens can brng n facors of mprovemen, bu as we wll see, s advanage of compuaonal speed s orders-of-magnude.

8 For any es case of a nework, f MLU of opmal TE,, and are ξ, ξ O and ξ D respecvely, hen η O = ξ ξ O, and η D = ξ ξ D. Thus can ncrease Inerne capacy over by η D η O. Fg. 2 llusraes he effcency of capacy ulzaon of he hree schemes. They show ha s very close o opmal raffc engneerng n mnmzng MLU, and ncreases Inerne capacy over by 5% for Ablene nework and 24% for her5b nework, respecvely. Opmaly Gap (%) Opmaly Gap (%) Opmaly Gap (%) Effcency of Capacy Ulzaon Opmal TE ablene her5a her5b rand5 rand5a rand Nework Opmaly Gap (%) Nework Loadng (a) Ablene Nework Nework Loadng (b) Rand Nework Opmaly Gap (%) Opmaly Gap (%) Nework Loadng (c) Her5b Nework Fg. 2. Effcency of capacy ulzaon of opmal raffc engneerng, and local Search C. Mnmze Toal Lnk Cos We also employ he cos funcon () as n [2]. Comparson s on he opmaly gap, n erms of he oal lnk cos, compared agans he value acheved by opmal raffc engneerng. Typcal resuls for dfferen opologes wh varous raffc marces are shown n Fg. 3, where he nework loadng s he rao of oal demand over oal capacy. From he resuls, we observe ha he gap beween and opmal raffc engneerng can be very sgnfcan (up o 82%) for he mos congesed case of Ablene nework. In conras, can acheve almos he same performance as he opmal raffc engneerng n erms of oal lnk cos. Noe ha, whn hose fgures, he maxmum opmaly gap of s only up o 8.8% n Fg. 3(b), whch can be furher reduced o.5% wh a larger sep-sze and more eraons (whch s feasble as he algorhm runs very fas o be shown n Sec. VI-D). D. Runnng Tme Requremen The ess for and local search were performed under he me-sharng servers of Redha Enerprse Lnux 4 wh Inel Penum IV processors a Ghz. Noe ha he runnng me for local search s sensve o he raffc marx snce a near-opmal soluon can be reached very fas for lgh raffc marces. Therefore, we show he range of her average runnng mes per eraon for qualave reference. Fg. 4 shows he opmaly gap (n a log scale) acheved by local search and, whn he frs 5 eraons for a ypcal scenaro (Fg. 3(c)). I demonsraes ha Algorhm for converges much faser han local search for. Table III shows he average runnng me per eraon for dfferen neworks. We observe ha our algorhm s very fas, Nework Loadng (d) Her5a Nework Nework Loadng (e) Rand5 Nework Nework Loadng (f) Rand5a Nework Fg. 3. Comparson of and Local Search n erms of opmaly gap on mnmzng oal lnk cos Opmaly Gap (log scale) 3 2 GAP GAP Ieraon Fg. 4. Comparson of he drop n opmaly gap beween Local Search and n a 2-level opology wh 5 nodes and 22 lnks requrng a mos 2 mnues even for he larges nework (wh nodes) esed, whle he local search needs ens of hours. On average, he algorhm developed n hs paper o fnd lnk weghs for roung s 2 mes faser han local search algorhms for roung. VII. DIFFERENCE BETWEEN AND DEFT Lnk-sae roung proocols can be caegorzed as lnk-based and pah-based n erms of flow splng. Ther dfference s llusraed n Fg. 5, wh a nework ha only has raffc demand

9 TABLE III Average runnng me per eraon requred by and local search o aan he performance n Fg. 3 Ne. ID Topology Node # Lnk # Tme per Ieraon (second) ablene Backbone her5a 2-level her5b 2-level rand5 Random rand5a Random rand Random from s o. Assume he weghs of he lnks are shown n Fg. 5(a). Obvously, he shores dsance from s o s 2 uns and boh nodes and u are on he shores pahs from s o. In a lnk-based splng scheme (e.g., Fong [6] and DEFT []), node s evenly spls raffc across s wo ougong lnks (s, ) and (s, u) as shown n Fg. 5(b). Whereas n a pah-based splng scheme, e.g., here are hree equallengh pahs from (s, ) and s evenly spls raffc across hem as shown n Fg. 5(c). Noe ha, he pah-based model does no mply explc roung o se up unnels for all he possble pahs. Insead, each node jus needs o compue and sores he aggregaed flow-splng rao across s ougong lnks, lke 66% on lnk (s, u) for he sample nework n Fg 5(c). Therefore, pah-based splng schemes can sll be realzed wh hop-by-hop forwardng. 2 s u (a) Lnk Weghs 5% s 25% 5% u 25% (b) Lnk-based Splng 33% s 33% 66% u 33% (c) Pah-based Splng Fg. 5. Dfference n raffc splngs for lnk-based and pah-based lnk-sae roung proocol Ths paper s subsanally dfferen from our prevous work on [], wh he followng key dfferences: ) DEFT s a lnk-based flow splng whle s a pah-based flow splng. 2) The core algorhms for seng lnk weghs are compleely dfferen. [] nroduces a non-convex nonsmooh opmzaon for DEFT and a wo-sage erave soluon mehod, whle he heory for s Nework Enropy Maxmzaon. The wo-sage mehod for DEFT s much slower han he algorhms developed for n hs paper. 3) [] numercally shows DEFT can realze near opmal TE n erms of a parcular objecve (oal lnk cos), whle hs paper proves ha can realze opmal TE wh any convex objecve funcon. VIII. CONCLUDING REMARKS Commody-flow-based roung proocols are opmal for any convex objecve n Inerne TE bu nroduce much confguraon complexy. Lnk-sae roung s smple bu does no seem o acheve opmal TE based on pror works. Ths paper proves ha opmal raffc engneerng, n fac, can be acheved by lnk-sae roung wh hop-by-hop forwardng, and he rgh lnk weghs can be compued effcenly, as long as flow splng on non-shores pahs s allowed bu properly penalzed. In [], we also show unqueness of he exponenal penaly n achevng opmal TE, and dscuss nerpreaons of NEM from he vewpons of sascal physcs and combnaorcs. We also hghlgh ha opmzaon s used n hree dfferen ways n hs paper []. Frs, s used when developng algorhms o solve he lnk wegh compuaon problem for. In a more neresng way, he level of dffculy of opmzng lnk weghs for s used as a hn ha perhaps we need o revs he sandard assumpon on how lnk weghs should be used, n he approach of Desgn For Opmzably. In ye anoher way, opmzaon n he form of NEM s nroduced as a concepual framework o develop lnk-sae roung proocols wh hop-by-hop forwardng. ACKNOWLEDGMENT Ths research s n par suppored by DARPA W9NF , ONR YIP N , AFOSR FA , NSF CNS-5988 and CNS We apprecae he helpful dscussons wh D. Applegae, B. Forz, J. He, J. Huang, D. Johnson, H. Karloff, Y. L, J. Lu, M. Pryz, A. Tang, M. Thorup, J. Yu, and J. Zhang. REFERENCES [] D. Xu, M. Chang, and J. Rexford, DEFT: Dsrbued exponenallyweghed flow splng, n INFOCOM 7, Anchorage, AK, May 27. [2] B. Forz and M. Thorup, Increasng Inerne capacy usng local search, Compuaonal Opmzaon and Applcaons, vol. 29, no., pp. 3 48, 24. [3] Z. Wang, Y. Wang, and L. Zhang, Inerne raffc engneerng whou full mesh overlayng, n INFOCOM, Anchorage, AK, 2. [4] D. Awduche, MPLS and raffc engneerng n IP neworks, IEEE Communcaon Magazne, vol. 37, no. 2, pp , Dec [5] A. Srdharan, R. Guérn, and C. Do, Achevng near-opmal raffc engneerng soluons for curren /IS-IS neworks, IEEE/ACM Transacons on Neworkng, vol. 3, no. 2, pp , 25. [6] J. H. Fong, A. C. Glber, S. Kannan, and M. J. Srauss, Beer alernaves o roung, Algorhmca, vol. 43, no. -2, pp. 3 3, 25. [7] B. Forz and M. Thorup, Inerne raffc engneerng by opmzng weghs, n INFOCOM, Tel Avv, Israel, 2, pp [8] D. D. Sleaor and R. E. Tarjan, A daa srucure for dynamc rees, Journal of Compuer and Sysem Scences, vol. 26, no. 3, pp , 983. [9] D. Mra and K. G. Ramakrshnan, A case sudy of mulservce mulprory raffc engneerng desgn for daa neworks, n GLOBE- COM 99, Ro de Janero, Brazl, Dec. 999, pp [] D. Xu, M. Chang, and J. Rexford, Lnk-sae roung wh hop-byhop forwardng can acheve opmal raffc engneerng, n echrepor, hp:// dahaxu/pub/nem/pef.pdf, Jul. 27. [] D. P. Bersekas, Nonlnear Programmng, 2nd ed. Ahena Scenfc, 999. [2] S. Boyd and L. Vandenberghe, Convex Opmzaon. Cambrdge Unversy Press, 24. [3] T. Cormen, C. Leserson, and R. Rves, Inroducon o Algorhms. The MIT Press, Cambrdge, 99. [4] TOTEM, hp://oem.nfo.ucl.ac.be.