Precomputation for Multi-constrained QoS Routing in High-speed Networks

Preomputtion for Multi-onstrined QoS Routing in High-speed Networs Yong Cui, Ke Xu, Jinping Wu Deprtment of Computer Siene, Tsinghu University, Beijing, PRChin, 100084 {y, xue}@snet1stsinghuedun; jinping@ernetedun Astrt As one of the most hllenging prolems of the next-genertion high-speed networs, qulity-of- servie routing (QoSR with multiple ( onstrints is n NP-omplete prolem In this pper, we propose multi-onstrined energy funtionsed preomputtion lgorithm, MEFPA It res eh QoS 1 weight to degrees, nd omputes numer (B= C + 2 of oeffiient vetors uniformly distriuted in the -dimensionl QoS metri spe to onstrut B liner energy funtions Using eh LEF, it then onverts QoS onstrints to single energy vlue At lst, it uses Dijstr's lgorithm to rete B lest energy trees, sed on whih the QoS routing tle is reted We first nlyze the performne of energy funtions with onstrints, nd give the method to determine the fesile nd unfesile res for QoS requests in the -dimensionl QoS metri spe We then introdue our MEFPA for -onstrined routing with the omputtion omplexity of O(B(m+n+nlogn Extensive simultions show tht, with few oeffiient vetors, this lgorithm performs well in oth solute performne nd ompetitive performne In onlusion, for its high slility, high performne nd simpliity, MEFPA is promising QoSR lgorithm in the next-genertion high-speed networs Keywords QoS routing, preomputtion, slility, liner energy funtion, performne evlution I INTRODUCTION Providing different qulity-of-servies (QoS support for different pplitions in the Internet is hllenging issue [1], of whih QoS Routing (QoSR is one of the most pivotl prolems [2] The min funtion of QoSR is to find fesile pth tht stisfies multiple onstrints for QoS pplitions Although QoSR ws initilly proposed for the IntServ model, it ould lso e used in the DiffServ model [3], [4], [5] The QoS onstrints n e divided into lin onstrints nd pth onstrints The lin onstrint of pth n e onverted to the onstrint of the ottlene lin in the pth, suh s ndwidth It n e esily delt with in preproessing step y pruning ll lins tht do not stisfy the onstrint nd omputing pth from the rest su-grph The pth onstrint is the restrition of eh lin long the pth, suh s dely Hene, in this pper we will fous on the pth onstrint prolem Mny heuristis hve een proposed for the multionstrined QoSR prolem euse of its NP-ompleteness [6], [7] However, these lgorithms hve some or ll of the following limittions [2]: (1 High omputtion omplexity prevents their prtil pplitions; (2 Low performne Supported y: (1 the Ntionl Nturl Siene Foundtion of Chin (No 90104002; No 69725003; (2 the Ntionl High Tehnology Reserh nd Development Pln of Chin (No 2002AA103067 mens tht these lgorithms sometimes nnot find fesile pth even when it does exist (3 Some lgorithms wor only for speifi networ Furthermore, most of the on-line routing lgorithms, whih lulte the pth when QoS request rrives, will not e le to fford the high omputtionl lod in high-speed networs In most ses, onsiderle redution in the overll omputtionl lod ould e hieved y preomputtion, espeilly when the rte of QoS requests is muh higher thn tht of (signifint hnges in the networ stte [8] This pper proposes novel lgorithm MEFPA (Multi-onstrined Energy Funtion sed Preomputtion Algorithm for multi-onstrined QoSR prolem sed on the nlysis of liner energy funtions (LEF We ssume tht eh node s in the networ mintins onsistent opy of the glol networ stte informtion This lgorithm res eh 1 QoS metri to degrees It then omputes B (B= C + 2 oeffiient vetors tht re uniformly distriuted in the -dimensionl QoS metri spe, nd onstruts one LEF for eh oeffiient vetor Then sed on eh LEF, node s use Dijstr's lgorithm to lulte lest energy tree rooted y s nd prt of QoS routing tle At lst, s omines the B prts of the routing tle to form the omplete QoS routing tle it mintins For distriuted routing, for pth from s to t, in ddition to the destintion t nd the weights, the QoS routing tle only needs to sve the next hop of eh pth For soure routing, the end-to-end pth from s to t long the lest energy tree should e sved in the routing tle Therefore, when QoS onnetion request rrives, it n e routed y looing up fesile pth stisfying the QoS onstrints in the routing tle Both theoretil nlysis nd experimentl results show tht our MEFPA n e esily implemented with high performne nd high slility There re two mjor ontriutions in this pper (1 We give mthemtil model tht deides, in -dimensionl onstrint spe, the re tht nnot e determined y the ontinuous hnge of -dimensionl LEFs (2 We propose the preomputtion lgorithm MEFPA for multi-onstrined QoSR prolem The rest of this pper is orgnized s follows Relted wor is disussed in Setion II We nlyze the reltion of LEFs nd the onstrint spe in Setion III, nd then propose MEFPA in Setion IV In Setion V MEFPA is evluted y extensive simultions Finlly, onlusions pper in Setion VI 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

II RELATED WORK Finding fesile pth tht stisfies multiple onstrints is NP-omplete prolem, for whih mny heuristi lgorithms hve een proposed If some sheduling shemes [9], [10] (eg weighted fir queuing re used, sed on the dependenies mong QoS prmeters, the onstrints of queuing dely, jitter, nd loss n e formulted s funtion of ndwidth Then the originl NP-omplete prolem n e redued to the stndrd shortest pth prolem [11], [12] Bsed on this pproh, Ord did n extensive study [13] However, this is not the se for propgtion dely, whih needs to e ten into ount for QoS routing in high-speed networs [14] Furthermore, suh lgorithms n only e pplied in networs with speifi sheduling shemes In order to inrese the ppliility, pseudo-polynomiltime nd pproximte lgorithms hve een proposed For 2-onstrined prolems, Jffe proposed distriuted lgorithm with the omplexity O ( n 5 log n, where is the upper ound of the weight [15] Beuse its omplexity depends on the vlues of weights (eg, the mximum lin weight in ddition to the size of networ, it is lled pseudopolynomil-time lgorithm Approximte lgorithms hve een proposed for the DCLC prolem [16], [17] For n ritrry ε > 0, these lgorithms n find pth in polynomil time Not only is the dely onstrint stisfied in the pth, ut lso its ost is less thn ( 1+ ε times the optiml ost For exmple, Lorenz proposed the lgorithm with the omputtion omplexity O ( nmlog n log log n + ( nm ε [17] In order to improve the performne, the omplexity of suh lgorithms will inrese hevily In ddition to the ove lgorithms, heuristis sed on the onvergene of multiple weights hve een proposed Jffe proposed the liner funtion g ( p = 1w1 ( p + 2w2 ( p to solve 2-onstrined prolems first [15] As his onlusion, for given onstrint vetor, of QoS request, when 2 1 1 2 ( 1 2 =, the pth found y Dijstr's lgorithm with minimizing g ( p n e fesile with mximum proility λ Bsed on the nonliner funtion g λ ( p = l = 1( w1 ( p 1, Neve proposed the TAMCRA [18] for multi-onstrined prolems for PNNI protool, nd its reformtion SAMCRA [19] for IP networs The lgorithm tries to find the pth with minimum g λ ( p y heuristis Beuse of the nonliner hrteristis, suh pth nnot e found in polynomil time The lgorithm uses the vrint of Dijstr's lgorithm to see to find nd store K undominted pths on eh node with the omplexity of O ( Knlog( Kn + K 3 m Bsed on mring lels reversely, Kormz proposed H_MCOP for multi-onstrined optiml-pth prolems [20] This lgorithm mrs eh node y running Dijstr's lgorithm reversely with g 1( p Then when it runs Dijstr's lgorithm forwrd with g >1 ( p, it onsiders oth the lels mred reversely nd the λ tree prtly onstruted forwrdly However, suh forwrd proess nnot gurntee to find the pth with minimum g >1 ( p λ An extensive survey on QoSR n e found in [2], [21] Among the proposed QoSR lgorithms, most of them use n on-line sheme Tht is to sy, when the QoS request rrives, the lgorithms hve to ompute the pth sed on the networstte informtion for eh request, respetively The nextgenertion networs hve high speed, so tht suh routing shemes re diffiult to e used On the other hnd, QoS requests rrive fster thn networ stte hnges Thus, the preomputtion sheme is fitter for high-speed networs thn the on-line sheme [8] However, urrent preomputtion lgorithms re often sed on distne vetors nd use the extended Bellmn-Ford lgorithm Yun presented limited grnulrity heuristi nd limited pth heuristi [22] The former limits the weight of eh lin nd the ltter limits the size of the routing tle diretly The ltter hs less omputtion omplexity O ( n 3 m log n with the routing tle size O ( n 2 log n Pointing to the multi-onstrined optiml prolem, Ord proposed preomputtion lgorithm y mpping the ost to disrete spe [8] Its omplexity is O( 1 ε HmlogC, where H is the longest hop numer nd C is the upper ound of the ost Some other erly lgorithms hve lrger omplexity [23] Additionlly, distne-vetor lgorithms hve some inherent prolems, eg the ount-to-infinity prolem, inevitle routing loops nd lrge quntity of updting informtion tht my overlod the networ There re three mjor differenes etween MEFPA proposed in this pper nd other similr lgorithms (1 From the viewpoint of ojetives nd funtions, MEFPA is to solve the generl multi-onstrined routing prolem, while some other lgorithms require some speifi sheduling shemes [12], [13] or limited set of QoS weights [15], [18] (2 Viewed from the methods, MEFPA is n off-line lgorithm sed on networ lin sttes while some others re on-line lgorithms sed on lin sttes [15], [18], [19], [20] or off-line ones sed on distne vetors [22] (3 MEFPA uses multiple energy funtions tht re independent of QoS requests, nd the liner hrteristi gurntees to find the lest energy pth esily Some similr lgorithms only use one energy funtion They either see for prtiulr liner energy funtion for speifi QoS request [15], or emphsize how to find the lest energy pth with non-liner energy funtion y heuristis [20] III LINEAR ENERGY FUNCTION ANALYSIS A Prolem Formultion A direted grph G ( V, E presents networ V is the node set nd the element v V is lled node representing router in the networ E is the set of edges representing lins tht onnet the routers The element e ij E represents the 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

edge e = v i v j in G In QoSR, eh lin hs group of independent weights ( ( e, w2 ( e, L, w ( e, whih is lso lled QoS metri w (e The pth onstrints n e divided into dditive onstrints (eg ost, dely nd multiplitive onstrints (eg loss rte Beuse either type n e trnsformed into the other, we only onsider dditive onstrints in the pper Aordingly, for pth p = v0 v1 L v nd 1 l, the weight + n w l ( e R stisfies the dditive hrteristi if ( p = n = wl ( vi vi i 1 1 Definition 1 Multi-onstrined pth For given grph G ( V, E, soure node s, destintion node t, 2 nd onstrint vetor = ( 1, 2, L,, the pth p from s to t is lled multi-onstrined pth (MCP, if w ( p for ny 1 l We write w( p in rief l l Note: w (e nd re oth -dimensionl vetors For given QoS request nd its onstrints, QoSR sees to find fesile pth p stisfying w( p sed on the urrent networ-stte informtion B Liner Energy Funtion Dijstr gve the Shortest Pth Tree (SPT lgorithm, whih hs low omputtion omplexity [24] However, QoSR prolem is relted to multiple weights simultneously Thus the prolem is hnged to the one, in whih the omplexity is NP-omplete, nd the originl Dijstr's lgorithm nnot e used to solve it In this se, one fesile method is to onvert the multiple weights to single vlue, s follows: g 1 Definition 2 Liner Energy Funtion (LEF The LEF of lin e is defined s the liner funtion g ( e = l = l w 1 l, (1 whih represents the "ost" of e Here, the oeffiient l [0,1] is independent of e for l = 1,2, L,, nd it stisfies =1 = 1 l l The vetor = ( 1, 2, L, tht stisfies the ove onditions is lled n energy oeffiient Bsed on LEF, we onvert the originl multi-onstrined pth prolem to lest-energy pth prolem Eh oeffiient of LEF represents the importnt degrees of different elements in the weight vetor when the SPT is omputed Theorem 1: For given grph G nd vetor, Dijstr's 1 We use the term "energy" insted of the trditionl nme "ost", so tht it is esy to distinguish the funtion vlue nd lin ost A trditionl ost onfigured for lin n e n element of the weight vetor LEF here onverts multiple weights to single vlue, whih does not hve prtil mening w l SPT lgorithm with respet to g (e n rete lest-energy tree T rooted y s The pth p long the tree T from s to n ritrry node t stisfies g ( p p( s, t G t = min g ( p( s, (2 Proof: The originl Dijstr's lgorithm n gurntee tht the pth from s to n ritrry node t long the tree hs the lest ost Beuse g (e is liner funtion, whih stisfies g ( e 1 + e2 = = + l l wl ( e e 1 1 2 = + l = l wl ( e 1 1 l = 1l wl e 2 ( = g ( ( e1 + g e2, we n lulte g (e for eh lin e first, nd then run the Dijstr's SPT lgorithm with respet to g (e rther thn the originl ost Thus, lest-energy tree n e reted nd p is the lest-energy pth from s to t, viz g p = min p ( s, t G g ( p( s, t ( If we te the energy oeffiient s the independent vrile, the question is hnged: For given G nd soure-destintion pir ( s, t, when energy oeffiient rehes to ll of the fesile vlues, wht hrteristis does the set { p } hve? For exmple, how mny elements does it hve nd how re they distriuted? For onveniene, we will first define the QoS metri spe nd then nlyze it s followings C QoS metri spe Definition 3 QoS metri spe W = W1 W2 L W is lled the QoS metri spe, if w ( p for ny p G l W l For the ommon ondition s + w l ( e R, we n te + W l = R, so wl ( p Wl for ny pth p Thus, w ( p is point in the spe set of points in W W, viz w ( p W, nd { ( p } w is Theorem 2: For given G, soure-destintion pir ( s, t, n energy oeffiient nd ting g opt = l = l wl ( p 1, w ( p of n ritrry pth p from s to t must e on the upside of the hyperplne { l 1 l l opt P = w( p = w ( p = g } (3 in the spe W Proof: We use the redution to surdity If there is point w ( p on the downside of hyperplne P, we hve > = l wl p = g opt = ( l l l wl ( p, whih is ontrry to 1 1 g( p = min p( s, t G g( p( s, t in theorem 1 Thus, if there is ny other pth p ' from s to t, w ( p must e on the upside of the hyperplne P 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

P w( p w ( p θ θ w( p P( P ( ( w( p θ P ( P ( ( w( p P ( P ( θ 1 P ( θ 2 w( p P ( p is in the upper side of P fixed p with different Figure 1 Vetor nd hyperplne P For exmple, when = 2 s shown in Fig 1, for the given vetor, we use Dijstr's lgorithm with respet to g nd rete the lest-energy pth p from s to t Drwing the perpendiulr P of vetor rossing the point w ( p, we 2 get prtition of spe W All of the weight points w ( p of pths p from s to t must e on the upside of P s shown in Fig 1 We should note tht, euse of the disreteness of networ topology grph, the orresponding p is hnged disretely with the ontinuous hnge of vetor Thus, we n see tht the mpping p is not n injetion When the given vetor hnges from to, the lest-energy pth does not hnge, ie p = p for [, ] in Fig 1 Therefore, the single pth p orresponds to the ontinuous set { P ( [, ]} of hyperplnes D Fesiility of the onstrint in W We will give prtition of the QoS metri spe W, inluding unfesile re, fesile re M nd unnown re M in this setion Aordingly, for given QoS request with onstrint in W, we n judge its fesiility nd find fesile pth for fesile request Definition 4 Unfesile re The point set M ( ={ w w W, w is in the lower side of P (} is lled n unfesile re determined y given vetor is lled the unfesile re U = M ( (4 = 1, 0 l = 1 l Theorem 3: For onstrint of request from s to t, there is NO fesile pth p stisfying w( p Proof: Aording to the definition of, is the union of ll the M ( with the ontinuous hnge of with unique p Trnsition from p to Figure 2 Unfesile re with lest energy pths p vetors For given onstrint, there must e hyperplne P ( orresponding to the vetor, so tht is on the downside of P ( Aording to theorem 2, ny pth p from s to t must e on the upside of P ( Therefore, there is no fesile pth p stisfying w( p Definition 5 Aville re The point set U M AVL = = M ( (5 is lled the ville re in spe = 1, 0 l= 1 l W \ M is the omplement of M W, where M = Fig 2 shows the reltion etween M AVL nd For exmple, if there is only one lest-energy pth, Fig 2 shows the unfesile re nd the ville re, where M = M ( U M ( For the ommon sense where multiple lest-energy pths exist, we should onsider tht (1 multiple vetors my mp to single point w ( p nd (2 single vetor my lso mp to multiple points s shown in Fig 2 When the vetor hnges from to ontinuously, the lest-energy point eeps the sme vlue, ie w ( p It is the sme se of the lest-energy point w ( p when the vetor hnges from to However, the vetor, equl to, hs two lest-energy pths, ie p nd p In this se, disrete hnge of lest-energy pths ours Suh disrete hnge introdues good hrteristi: lthough the lest-energy points re disrete, the unfesile re is ontinuous Theorem 4: The ville re M AVL is onvex set, the weight point of n ritrry lest-energy pth p is on the order of M AVL, nd there must e vetor mpping to vertex of M AVL so tht the point w ( p is the vertex Proof: We prove the ove three prts in turn (1 To prove tht M AVL is onvex set (See [25] for the p 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

definition nd the hrteristis of onvex set: For given vetor, the orresponding hyperplne divides the spe W into two prts nd either of them is hlf-spe Thus, the ville re ( = W \ ( is onvex set for eh give vetor On the other hnd, sine M AVL = M = U = 1, 0 I M ( = M ( nd the intersetion of = 1, 0 onvex sets is still onvex set, M AVL is onvex set (2 To prove tht n ritrry w p is on the order of AVL w( p M AVL = w( p = UM ( M : For n ritrry w p, drwing the hyperplne ( P ( rossing the point w ( p with eing the norml vetor, M is on one side of P ( ording to theorem 2 Therefore, w ( p is on the order of M AVL (3 To prove tht there must e vetor mpping to vertex of M : For ny vertex x of M, euse AVL M AVL is onvex set, there must e hyperplne P ( Here, P ( rosses the point x nd M AVL is on one side of P (, where is norml vetor of the hyperplne Beuse M AVL is n unlimited set, ie (,, L, M AVL, M AVL must e on the upside of P ( Therefore, for given vetor, the point w( p = x must e le to e lulted For exmple, when there re multiple lest-energy pths with the different energy funtions, the ville re M AVL must e onvex set, nd eh vertex of M AVL must e lest-energy pth s shown in Fig 3 We further divide M AVL into two prts: the fesile re M, nd the unnown re M UNKNOW Definition 6 Fesile re M M w( p is lled the fesile re w( p AVL = w( p, w( p w( p } (6 { M Theorem 5: For n ritrry onstrint of QoS request, M w( p = 1, 0 The onvexity of M AVL Fesile onstrints: nd Figure 3 Three prtitions of W : unfesile, unnown nd fesile res if M, there must e fesile pth Proof: From the definition of M, tht stisfies w( p, so p n e fesile pth for the onstrint Definition 7 Unnown re M is lled unnown re M = M M (7 AVL For exmple, in Fig 3 pth p is fesile for onstrint M euse w( p < We don t now the existene of fesile pth for ny onstrint M Thus, we divide the spe W into three res: n unfesile re, n unnown re M nd fesile re M d s (2,3 (4,4 (5,1 (5,1 e (1,5 (1,5 (1,9 (1,9 t Originl networ grph d s (2,3 (4,4 (5,1 (5,1 e (1,5 (1,9 t =(0,1 d (2,3 (5,1 (5,1 s t e (1,5 (1,5 (1,9 =(05,05 d (2,3 (5,1 s t e (1,5 (1,5 (1,9 (1,9 d =(1,0 Figure 4 An exmple of proposed lgorithm IV PROPOSED PRECOMPUTATING MEFPA A The ide of MEFPA Bsed on the ove theory, the set { p } of lest-energy pths n e pre-omputed with different energy oeffiients, nd then the QoS routing tle n e onstruted sed on { p } Fig 4 shows n exmple, where node s is going to onstrut its QoS routing tle First, it uses =(0,1 to ompute the lest energy tree s shown in Fig 4 Then it uses =(05,05 to onstrut nother tree in Fig 4, nd so on in Fig 4d At lst, node s n get three different trees with these different vetors All of these three trees ompose the finl QoS routing tle mintined y s When QoS request rrives, s loos up fesile pth in the routing tle nd 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

forwrds the request Aording to the positions of QoS onstrints in the spe W, there re three possiilities (1 We now fesile pth for M, nd then we n find n element in the { p } s the fesile pth (2 We now tht there is NO fesile pth for Thus we n refuse the QoS request or strt the QoS negotition (3 We don't now whether fesile pth exists for M We expet tht the proility of the third se is smll In the performne evlution (setion V, extensive simultions will show tht the re M is relly smll nd most (95% QoS requests will e in the other two res Furthermore, we will lso demonstrte tht QoS request in M hs smll opportunity to e fesile Therefore, we te this re s the unfesile re without ffeting the performne signifintly, so we refuse this ind of QoS requests A prtil lgorithm nnot use ontinuous hnge of, so we disuss how to onstrut multiple energy oeffiient vetors s followings In order to me the disrete independent of networs, we normlize the weights of eh lin first Tht is to sy, the potentil mximum weight, mx e E w l ( e, is onstnt, whih is independent of l Then, we selet uniform numers in [0,1], ie D={0/(-1, 1/(-1,,1} with totlly elements Thus, we get the uniform vetors A= { D, = 1} l = 1 l (8 s the oeffiients in the suset [ 0,1] of QoS metri spe W At lst, the node, rrying out MEFPA, lultes the QoS routing tle for ll A sed on the networ lin stte it mintins Theorem 6: The numer of the elements in the energy oeffiient set A= {, D = 1} l = 1 l is 1 + 2 A = C (9 Proof: There re elements in set D={0/(-1, 1/(-1,,1}, nd they re uniform in [0,1] We hve D, l = 1l = 1 = ( 1 /( 1 Thus, if we te eh 1/(-1 s ll, the set D n e onsidered s D={0 ll, 1 ll, 2 lls,, (-1 lls} The totl numer of the lls represented y the vetor is (-1 Additionlly, the mening of l1 is different to tht of l 2 for l1 l2 Therefore, we n equte A with the numer of the methods to put (-1 sme lls into different oxes, nd n ritrry numer of lls n e put into eh oxes Aording to the 1 omintoris, there re C + 2 = ( + 2! ( 1!( 1! 1 different methods to put the lls [26], ie A = C + 2 MEFPA (G, s,, 1 IF (==K-1 2 []=-1 // we hve got the oeffiient [K] 3 FOR EACH edge e IN G 4 g ( e = l = l w 1 l 5 dijstr (G,s 6 FOR EACH node t IN G 7 store p ( s, t 8 ELSE 9 FOR(i=0; i<; i++ 10 []=i 11 MEFPA(G, s, +1, -i 1 We denote B s the numer of LEFs, ie B= A = C + 2 For the -onstrined routing prolem, node s first onstruts B energy oeffiients Then s omputes the lest energy tree for eh oeffiient, respetively, nd sves the pth from s to eh destintion node t in the networ long the tree to the QoS routing tle Thus, there re t most B different pths from s to eh destintion t When QoS request rrives t s, s only needs to loo up fesile pth stisfying onstrints in the routing tle B Desription of MEFPA We propose the preomputtion lgorithm, MEFPA, for the -onstrined routing prolem, s shown in Fig 5 G is the networ grph with K weights, s is the node running MEFPA, nd (-1 is the numer of degrees to whih eh weight is red Our MEFPA, running on s, inludes two prts ( A numer B of energy oeffiients = ( 1, 2, L, re onstruted ording to the onfigurtion of (Line 1, 2, 8-11 ( A prt of QoS routing tle is lulted with respet to eh oeffiient (Line 3-7 This inludes the following steps: (i For the given G nd eh vetor, lulte the energy g (e for eh lin (Line 3-4 (ii Use Dijstr's lgorithm to ompute lest-energy tree T rooted y node s with LEF g (Line 6 (iii Sve the lest-energy pth from s to eh node long 6-7 Figure 5 The proposed MEFPA T to QoS routing tle (Line Beuse the liner funtion g (e stisfies the isotoniity, there re no routing loops in the routing tle lulted y node s [27] Therefore, in the soure routing sheme, MEFPA n void routing loops even when different nodes hve the inonsistent opies of networ stte informtion However, in the distriuted routing sheme, we need the onsistene of the networ stte informtion in different nodes If ll nodes use the routing tle entries generted with sme g (e to forwrd speifi QoS request hop y hop, routing loops re voided 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

C Anlysis of the proposed lgorithm We first nlyze the omputtion omplexity of MEFPA to lulte QoS routing tle In networ grph G with QoS metris, the node numer is n = V nd the edge numer is m = E Step (i hs the omplexity of O (m Step (ii is O ( nlog n + m with the improved Dijstr's lgorithm nd step (iii is O (n The numer of oeffiient vetors is 1 B= C + 2 (theorem 6 As result, inluding the reursive prt, the overll omputtion omplexity of MEFPA is 1 O( C 1 + 2 ( m + nlog n + n, whih is B= C + 2 times the originl Dijstr's lgorithm with single weight In generl sitution, there re only few inds of pth onstrints, eg ost, dely, jitter, nd loss rte Therefore, will not e very lrge nd omplexity of MEFPA is eptle We now nlyze the omputtion omplexity to loo up fesile pth in the QoS routing tle As we nlyzed ove, the QoS routing tle is B times the originl routing tle with single weight Beuse the QoS routing tle sves weights, the urrent pet lssifition tehnique in multiple dimensions n hieve onstnt time omplexity [28] Tht is to sy, it only needs to ess the memory for d times to find route, where d is the redution of rnges to prefix looup We n even otin the omputtion omplexity of O(1 if we use some speil hrdwre, eg TCAM [29] 1 Beuse the vlue B= C + 2 is importnt to MEFPA, we will illustrte the reltion etween B nd the performne y extensive simultions s follows, whih show tht MEFPA performs well when B is smll (eg when =2, we let B==7 V PERFORMANCE EVALUATION We first propose method, the unnown-re proportion, to the solute performne evlution of MEFPA Additionlly, in order to ompre it with others diretly, we lso simulte QoS requests to evlute its performne In eh of these experiments with node numer N eing 50, 100, 200 nd 500 respetively, we generte 10 pure rndom networ grphs [30], [31] with weights for eh lin, where w l (e ~ uniform[1,1000] for l = 1,2, L,, nd w l (e hve no orreltion for different e or l In eh grph, we selet soure-destintion node pir (s,t 100 times ( prtiulr node n e seleted more thn one, where we gurntee tht the minimum hop is not less thn two Eh soure node s use 1 our MEFPA to lulte lest energy tree for B= C + 2 times with B different energy oeffiients A The solute performne QoSR lgorithms re often evluted y two methods (1 Competitive rtio, whih indites how well heuristi lgorithm performs, is defined s the rtio of the numer of requests stisfied using heuristi lgorithm nd the numer of requests stisfied using the exhustive lgorithm (2 Suess w( p A rtio (SR is defined s the rtio of the numer of requests stisfied using heuristi lgorithm nd the totl numer of requests generted The differene etween the two methods is the fesiility of the requests eing the denomintor in theory Both hve shortomings euse the evlution depends hevily on the generted onstrints of the requests, eg the distriution of onstrints For lrge-sle networ, it is diffiult to judge the fesiility of request, so SR is used widely in most ses Beuse different distriutions re used in different ppers, the solute vlue of SR mens nothing, nd only the omprison with the sme networ dt mes sense In order to evlute the solute performne of MEFPA nd void the ove prolem, we propose the method of unnown-re proportion, whih is independent of QoS requests We te the unnown re in Fig 3 s the ineffiieny of MEFPA nd nlyze the proportion of this re to the whole re The unnown re is M in theory, ut for the limited numer of disrete oeffiients, the unnown re will extend to M s shown in Fig 6 The reson is tht we nnot gurntee the nonexistene of lest energy pths in tringle A nd B M is limited spe while nd M re unlimited spes For given (s,t pir, we use Dijstr's lgorithm to onstrut the shortest pth p l with respet to w l The unnown-re proportion is defined s Pr M w( p B M w( p w( p M = M /( M w( p + + M (10 in the suset {w w l 2 wl ( pl } shown s the retngle enlosed y the gry dshed line in Fig 6 Fig 7 shows the verge unnown-re proportion Pr with the 95% onfidene intervl for 10 rndom grphs The X-oordinte is the numer of the degrees to re eh weight nd the Y-oordinte is Pr As shown in this figure, (1 the proportion of unnown re is smll (2 Pr dereses rpidly to onstnt vlue when inreses This shows tht in prtie, to ensure high performne, we only need few uniform oeffiients to find enough pths of different hrteristis, eg when =2, B==7 With lrger, most of M w( p inrese of unnown re the re we onsider Figure 6 Unnown re proportion 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

Pr 0028 0025 0022 0019 0016 0013 001 Pr =2 =3 =4 6 12 18 24 30 36 42 48 Pr 005 004 003 002 =2 =3 =4 001 2 3 4 5 7 10 20 50 N=50 N=100 Pr 008 =2 007 =3 =4 006 005 004 003 002 0 2 3 4 5 7 10 20 50 2 3 4 5 7 10 20 50 N=200 N=500 Figure 7 The solute performne of LEPFA 012 009 006 003 =2 =3 =4 the newly found pths re reduplitions, whih nnot derese Pr furthermore This is onsistent with the onlusion in [22] (3 In lrge-sle networs, Pr dereses when the numer of weights inreses, nd the lrger the, the smller the hnge tht Pr dereses with inresing This shows tht in multiple dimensions, when is smll, we 1 use numer (B= C + 2 of oeffiients Therefore, MEFPA performs well with smll in multiple dimensions (4 With the inrese of node numer N, plys more signifint role to the performne The reson is tht in lrger networs, more pths re ville etween prtiulr (s,t pir nd the possiility to optimize prtiulr weight inreses We hve demonstrted tht the unnown-re proportion Pr is smll, nd now we demonstrte tht the fesiility for request M is lso smll First, we generte some onstrints within the unnown re in Fig 6 rndomly Then, we use H_MCOP [20] to see fesile pths for eh request Fig 8 shows the SR with =2, ie 2-onstrined routing When =7, the suess rtio is less thn 5% so tht most requests my e inherently infesile Hving estlished tht (1 the unnown-re proportion is SR 045 04 035 03 025 02 015 01 005 0 N=500 N=200 N=100 N=50 2 3 5 7 10 15 20 Figure 8 The unfesiility of requests in the unnown re smll, nd (2 request within the unnown re hs low fesiility, MEFPA n refuse the requests within the unnown re with smll proility of misjudgment (refuse fesile requests As result, MEFPA hieves high solute performne Sine the proility of misjudgment is smll, the misjudgment is no longer the mjor ftor tht dereses the performne Insted, the inherent stleness of networ-stte informtion sed on whih QoSR opertes my e the mjor ftor in prtie [32] B Performne omprison sed on rndom onstrints From the relted wor in setion II, it is seen tht the urrent preomputtion lgorithms for QoSR re not effiient enough Some of them tend to hve the prohiitive omputtion omplexity or low performne, nd some re sed on distne vetors, so they re not fit for lrge-sle networs In order to show the performne of MEFPA, we ompre MEFPA with H_MCOP [20], whih is lso sed on Dijstr's lgorithm We use the method of generting onstrints for requests in [20] to ompre the two lgorithms first For given ( s, t pir, we use Dijstr's lgorithm to lulte the shortest pth p l with the eyword w l, respetively We then (, 1 2 generte the onstrints rndomly for eh ( s, t pir: p 1 l+1 ~ uniform [08 w l + 1( pl, 12 w l + 1( pl ] The shdowed re in Fig 9 shows the p 2 onstrints generted in two Figure 9 Rndom onstrints dimensions 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

SR 093 0925 092 0915 091 0905 09 0895 089 0885 088 Figure 10 B=5 B=7 B=10 B=20 B=50 H_MCOP 50 100 200 500 N Performne evelution with rndom onstrints ( two onstrints Fig 10 shows the routing suess rtio in two dimensions, ie for 2-onstrined routing When =7, the SR of MEFPA is higher thn tht of H_MCOP Tle I shows the ses in multiple dimensions, ie -onstrined routing With the inrese of weight numer, euse more requests generted y this rndom method re inherently infesile, the performne of oth lgorithms dereses rpidly However, from the omprison, MEFPA performs well when is smll with reltively lrge (eg, when =5, we hoose =3 This further onfirms the onlusion of the solute performne evlution C Performne omprison sed on simulted onstrints Beuse the performne depends on the distriution or sle tht the generted QoS onstrints oey, experimentl results in different ppers nnot e ompred diretly with others However, the Internet does not hve typil topologies or trffi models [34], nd we re even more short of nowledge out the onstrints of the upoming QoS pplitions Therefore, it is diffiult to give resonle model nd distriution of the onstrints Nevertheless, we now tht most QoS pplitions re different weights to different degrees For exmple, file trnsfer pplitions my re loss rte to muh higher degree thn dely, nd multimedi pplitions my te dely s the most importnt prmeter The distriution of onstrints in Fig 9 does not onsider this spet, nd it n generte onstrints only round the middle etween the w 1 xis nd xis For exmple, it nnot generte onstrint pir, in whih is s importnt s three times w 2 Bsed on the normlized weights in the whole grph, we use the method of weight rtio simultion to generte the TABLE I PERFORMANCE EVALUATION WITH RANDOM CONSTRAINTS (MULTIPLE CONSTRAINTS SR(% =2 =3 =4 =5 =3 918 797 644 557 N=50 =7 928 805 647 558 H_MCOP 925 800 642 556 =3 910 730 618 516 N=100 =7 922 737 627 523 H_MCOP 919 735 620 508 =3 882 734 594 478 N=200 =7 900 750 609 499 H_MCOP 899 738 599 482 onstrints for given pir ( s, t First, we ssume tht eh QoS pplition hs oeffiient vetor The normlized l ( 1 +L + presents the degree, to whih this pplition res the weight w l Bsed on this ssumption, we use the LEF g (Definition 2 to onstrut the onstrints for request For given ( s, t, we use Dijstr's lgorithm to lulte the lest energy pth p ( s, t nd te the weights vetor w( p ( s, t s the QoS onstrint of ( s, t, ie ( s, t = w( p ( s, t Beuse the request with suh simulted onstrints must e fesile, SR n e extended to the solute performne evlution of lgorithms For eh pir ( s, t, we te l ~ uniform(0,1 in prtie Fig 11 shows SR of these QoS requests we simulte The experiment shows tht MEFPA overmthes H_MCOP, nd MEFPA hs good slility euse it is insensitive not only to the networ sle, ut lso to the onstrint numer D Running time omprison Our extensive simultions show tht MEFPA hs higher SR thn H_MCOP, lthough H_MCOP hs to lulte the fesile pth for eh individul request respetively On the spet of omputtion omplexity, H_MCOP lgorithm is O( m log n + n log n + ( 2 + 1 m with -shortest pth lgorithm [33], while MEFPA is O ( B( m + nlog n + n For given B = 7 when =2, MEFPA is etter thn H_MCOP, not only on omputtion omplexity ut lso on the running time in prtil experiments For exmple, the running time of MEFPA is 518 milliseond in 500-node grph, while tht of H_MCOP is 153 milliseond Considering the differene etween preomputtion nd on-line omputtion furthermore, MEFPA hs muh less omplexity For exmple, if on given 102 100 98 96 94 92 90 88 SR =3 =5 =7 H_MCOP 2 3 4 5 6 102 100 98 96 94 92 90 88 SR =3 =5 =7 H_MCOP 2 3 4 5 6 102 100 98 96 94 92 90 88 SR =3 =5 =7 H_MCOP 2 3 4 5 6 N=50 N=100 N=200 Figure 11 Comprison with simulted onstrints 0-7803-7753-2/03/$1700 (C 2003 IEEE IEEE INFOCOM 2003

soure node, there re (N-1 requests to the other (N-1 nodes in the networ, the running time of MEFPA eeps fixed while tht of H_MCOP inreses (N-1 times VI CONCLUSION For the NP-omplete prolem of multi-onstrined QoS routing, we propose novel preomputing lgorithm, MEFPA, sed on the theoretil nlysis of liner energy funtions With this lgorithm, router onstruts numer (B of uniform oeffiients to onstrut B liner energy funtions It then lultes B lest energy trees to preompute the QoS routing tle with the omputtion omplexity O ( B( m + n log n + n Compred with the urrent Internet routing sheme, the router only needs to reple the urrent ost y the energy vlue in SPT omputtion Thus, the preomputtion omplexity of MEFPA is only B times tht of the urrent lgorithm with single ost When QoS requests rrive, the router n loo up fesile pth in the QoS routing tle The size of the QoS routing tle is less thn or equl to B times tht of the urrent routing tle with single ost, so tht the present tehniques of routing tle loo-up is ompetent Beuse of the smll omplexity nd the preomputtion sheme, MEFPA hs good slility in the numer of QoS onstrints, the networ sle nd lso the high-speed rrivl of pets in next-genertion networs It is lso onsistent with the routing rhiteture of the urrent Internet From the performne evlution of routing suess rtio, we find tht the performnes of heuristis re different with different distriutions or sles tht the generted QoS onstrints oey In order to evlute the solute performne, we propose novel pproh, the unnown-re proportion, whih shows tht our MEFPA hieves high solute performne Additionlly, we generte the onstrints of QoS requests y oth of the rndom method nd weight-rtio simultion, respetively Extensive simultions lso show tht our MEFPA performs ompetitively gret As onlusion, following the urrent preomputtion routing rhiteture, we elieve tht the MEFPA is promising QoSR lgorithm for next-genertion high-speed networs euse of its high slility, performne nd simpliity Furthermore, MEFPA n e esily extended to resolve multi-onstrined optiml ost prolem For exmple, ting the ost s the (+1 th weight, this prolem n e trnsferred to (+1-onstrined prolem nd MEFPA n generte the QoS routing tle with (+1 weights When QoS requests rrive, we n loo up n optiml fesile pth, whih stisfies the onstrints nd hs lest ost in the routing tle ACKNOWLEDGMENTS The uthors would lie to thn Ruiing Ho for his reful revision REFERENCES [1] X Xio nd L M Ni, Internet QoS: ig piture, IEEE Networ, vol 13, no 2, pp 8 18, Mr-Apr 1999 [2] Y Cui, J P Wu, K Xu, et l Reserh on internetwor QoS routing lgorithms: survey Chinese Journl of Softwre, vol 13, no 11, 2065-2076, 2002 [3] JC Oliveir, C Soglio, IFAyildiz, et l A new preemption poliy for diffserv-wre trffi engineering to minimize rerouting, IEEE INFOCOM'02, 2002 [4] J Wng nd K Nhrstedt Hop-y-hop routing lgorithms for premium-lss trffi in diffserv networs IEEE INFOCOM'02, 2002 [5] IETF Integrted Servies (diffserv woring group, 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