Efficien Algorihm for Compuing Dijoin QoS Pah Ariel Orda and Alexander Sprinon 1 Deparmen of Elecrical Engineering, Technion Irael Iniue of Technology, Haifa, Irael 32000 Email: ariel@eeechnionacil Parallel and Diribued Compuing Group, California Iniue of Technology, Paadena, California, USA 91125 Email: palex@calechedu Abrac Newor are expeced o mee a growing volume of requiremen impoed by new applicaion uch a mulimedia reaming and video conferencing Two eenial requiremen are uppor of Qualiy of Service (QoS) and reilience o failure In order o aify hee requiremen, a common approach i o ue wo dijoin pah beween he ource and he deinaion node, he fir erving a a primary pah and he econd a a reoraion pah Such approach, referred o a pah reoraion, ha everal advanage, he major one being he abiliy o wich promply from one pah o anoher in he even of a failure A major iue in hi conex i how o idenify wo pah ha aify he QoS conrain impoed by newor applicaion Since newor reource, eg, bandwidh, are allocaed along boh primary and reoraion pah, we need o conider alo he overall newor performance Accordingly, in hi paper we udy he fundamenal problem of finding wo dijoin pah ha aify he QoS conrain a minimum co We preen approximaion algorihm wih provable performance guaranee for hi fundamenal newor problem Index Term: Rouing, Reoraion, Dijoin Pah, Qualiy of Service I INTRODUCTION Newor are expeced o mee Qualiy of Service (QoS) requiremen impoed by new applicaion, uch a mulimedia reaming and video conferencing Thi i faciliaed by curren effor o provide reource reervaion and explici pah rouing, eg, MuliProocol Label Swiching (MPLS) On he oher hand, phyical newor infrarucure may be prone o failure Therefore, a major challenge in hi conex i o develop adequae newor mechanim for eablihing connecion ha aify QoS requiremen and are alo reilien o failure I ha been recognized ha, for many pracical eing, he peed and capaciy of lin do no allow o proviion reoraion pah afer he failure Thu, he reoraion pah mu be proviioned in advance, ie, before a failure occur Thi goal can be achieved by proviioning wo dijoin QoS pah beween he ource and deinaion node Thi approach i widely ued becaue of i abiliy o wich promply from one pah o anoher in he even of a failure The dijoin pah raegy ha many addiional advanage Fir, i allow o ue variou proecion cheme, uch a 1+1 proecion or 1:1 1 Par of hi wor wa done while A Sprinon wa wih he Deparmen of Elecrical Enginnering, Technion proecion [1] Wih 1+1 proecion, raffic i imulaneouly ranmied on boh pah, which allow inananeou recovery from lin failure Alernaively, wih 1:1 proecion, raffic i ranmied along a primary pah, and, upon a failure of one of i lin, he raffic i wiched o a reoraion pah Second, he dijoin pah raegy require minimal newor uppor, becaue failure deecion and reoraion can be implemened a he applicaion level of he ource Finally, he dijoin pah raegy provide a greaer flexibiliy o applicaion deigner, a hey can chooe a proecion cheme (eg, 1+1 or 1:1) ha i mo adequae for each paricular applicaion To faciliae eamle recovery o a reoraion pah in he even of a failure, i i neceary o reerve newor reource (eg, bandwidh) on boh he primary and reoraion pah Such reource hould be conumed in a neworwide efficien manner A common way for modelling he impac of uch reource conumpion on each lin i by aociaing co wih he lin Accordingly, a major problem i o find wo dijoin pah beween ource and deinaion node ha aify end-o-end QoS conrain a minimum co Thi problem i he ubjec of hi udy QoS conrain occur naurally in a number of pracical eing, involving bandwidh and delay-eniive applicaion, uch a voice over IP, audio and video conferencing, mulimedia reaming, ec QoS conrain can be divided ino bolenec conrain, uch a bandwidh, and addiive conrain, uch a delay or jier Bolenec QoS conrain can be efficienly handled by pruning lin ha do no aify hem The problem i hen effecively reduced o finding wo dijoin pah of minimum co; hi problem wa exenively inveigaed in he lieraure [2] Accordingly, in hi udy we focu on addiive QoS conrain, which are more difficul o handle QoS rouing ha been he ubjec of everal recen udie and propoal (ee [3], [4] for comprehenive urvey) However, he problem of finding wo dijoin QoS pah go lile aenion Similarly, pah reoraion and rouing over alernae pah ha alo araced a large body of reearch (ee, eg, [5], [6]) Mo of he propoed oluion, however, conidered only bolenec QoS conrain The few udie ha did conider addiive conrain (eg, [7]), focued on heuriic approache and did no provide proven performance guaranee To he be of our nowledge, hi i he fir udy o pro-
vide a oluion wih provable performance guaranee for hi fundamenal newor problem The problem i clearly NPhard ince even he baic problem of finding a ingle opimal pah ha aifie an addiive QoS conrain i inracable [8] Furhermore, i urned ou ha a pecial cae of our problem wih no co minimizaion, ie, finding wo dijoin pah ha (boh) aify an addiive QoS conrain, i NP-hard Thu, any pracical cheme i necearily ub-opimal and incur ome violaion of he QoS conrain In hi paper we preen oluion ha incur a mall violaion of he QoS conrain and whoe co guaraneed o be wihin a cerain facor away from he opimum Our paper mae he following conribuion Fir, we inroduce he minimum conrained flow (MCF) problem, which i an elaboraed varian of he minimum-co flow problem, and relae i o our problem Specifically, we how ha, by olving he MCF problem, we can obain a oluion o our dijoin-pah problem wih he malle poible violaion of he QoS conrain Second, hi relaion beween our problem and newor flow problem allow o employ mehod and echnique from he heory of newor flow, uch a pah augmenaion and cycle-cancellaion While he pah augmenaion mehod i widely ued for finding dijoin pah, our udy i he fir o enhance i wih he cycle-cancellaion mehod, hu improving he performance of our dijoin QoS pah algorihm Third, we inveigae he fundamenal radeoff beween he co of he idenified oluion, he violaion of he QoS conrain and he compuaional complexiy of he algorihm Finally, we preen a family of algorihm ha allow o find a oluion ha i adequae for any paricular eing, uch a a oluion wih minimum violaion of he QoS conrain, minimum co, ec Due o he fundamenal naure of he conidered problem, our reul can be ued in a variey of pracical applicaion For example, in MPLS newor, here i a requiremen o proec Label Swiched Pah (LSP) [9] Accordingly, our mehod can be ued for idenificaion of dijoin LSP ha aify QoS conrain In ATM newor, alernae pah can be ued upon a cranbac [10] In he Differenial Service framewor [11], a bandwidh broer i reponible for eablihing uiable pah ha aify ervice level agreemen (SLA) Here oo, i i deirable o compue everal dijoin QoS pah in order o faciliae failure reilience and preven congeion We noe ha dijoin pah can be ued for oher purpoe, beyond pah proecion Fir, ending daa on dijoin pah improve newor uilizaion and reduce congeion In fac, ending daa on divere pah i a major ool of raffic engineering Second, rouer may ue a precompuaion approach in order o improve repone ime [12] The ey idea i o compue everal QoS pah in advance and ore hem in a daabae Upon arrival of a connecion reque, a uiable pah i eleced hrough a imple, fa procedure Since newor opology can change, precompuing dijoin pah increae he probabiliy ha a lea one pah i valid Finally, dijoin QoS pah can be ued in he conex of mulipah rouing Wih mulipah rouing, raffic i en along muliple pah in order o increae bandwidh and he probabiliy of delivery Mulipah rouing can be ueful in wired [13] and wirele (Ad hoc) Newor [14] The remainder of he paper i organized a follow In Secion II, we preen he newor model and formulae he problem conidered in hi paper In Secion III, we preen baic concep of newor flow and eablih a relaion beween he conidered problem and newor flow problem In Secion IV, we preen a imple approximaion algorihm for our problem In Secion V, we preen a cycle-cancellaion approach and how how o ue i in order o minimize delay violaion In Secion VI and VII, we how how o improve he compuaional complexiy of ha cheme In Secion VIII, we eablih a lower bound for he problem Finally, concluion are preened in Secion IX II MODEL AND PROBLEM FORMULATION In hi ecion, we decribe he newor model and he main problem addreed in hi paper For impliciy of expoiion, we ue he erm delay requiremen in order o generically refer o addiive QoS conrain A The Newor Model We repreen he newor by a direced graph G(V,E), where V i he e of node and E i he e of lin We denoe by N and M he number of newor node and lin, repecively, ie, N = V and M = E An(, )-pah i a finie equence of diinc node P =( = v 0,v 1,,= v n ), uch ha, for 0 i n 1, (v i,v i+1 ) E Here, n = P i he hop coun of P The ubpah of P ha exend from v i o v j i denoed by P (vi,v j) Acycle i a pah whoe ource and deinaion node are idenical Each lin l E offer a delay guaranee d l The delay D(P ) of a pah P i he um of he delay of i lin, ie, D(P )= l P d l In order o aify QoS conrain, cerain reource uch a bandwidh and buffer pace mu be reerved along QoS pah In order o opimize he global reource uilizaion, we need o idenify QoS pah ha conume a few newor reource a poible Accordingly, we aociae wih each lin l a nonnegaive co c l, which eimae he qualiy of he lin in erm of reource uilizaion The lin co may depend on variou facor, eg, he lin available bandwidh and i locaion The co C(P ) of a pah P i defined o be he um of he co of i lin, ie, C(P )= l P c lwe hall aume ha all parameer (boh delay guaranie and co) are poiive ineger B QoS pah A fundamenal problem in QoS rouing i o idenify a minimum co pah beween a ource and a deinaion ha aifie ome delay and bandwidh conrain Bolenec QoS conrain, uch a bandwidh, can be efficienly handled by imply pruning lin ha do no aify he QoS conrain Thu, in he re of he paper, we only conider delay (ie, addiive) conrain Accordingly, he fundamenal problem
i o find a minimum co pah ha aifie a given delay conrain Thi can be formulaed a a Rericed Shore Pah problem Problem RSP (Rericed Shore Pah) : Given a ource node, a deinaion node and a delay conrain D, find an (, )-pah P uch ha 1) D(P ) D, and 2) C(P ) C( ˆP ) for any oher (, )-pah ˆP ha aifie D( ˆP ) D In general, Problem RSP i inracable, ie, NP-hard [8] However, here exi peudo-polynomial oluion, which give rie o fully polynomial approximaion cheme 2 (FPAS), whoe compuaional complexiy i reaonable (ee [15] [17]) The mo efficien cheme, preened in [17], ha a compuaional complexiy of O(MN( 1 + log log N)), and compue a pah wih delay of a mo D and co of a mo (1 + ) ime he opimum We hall refer o ha cheme a Algorihm RSP C Problem Saemen We are now ready o formulae he problem ha we conider in hi udy The fir problem ee o idenify wo dijoin QoS pah of minimum oal co Problem 2DP (2-Rericed Lin Dijoin Pah) : Given a ource node, a deinaion node and a QoS requiremen D, find wo lin-dijoin (, )-pah P 1 and P 2 uch ha: 1) D(P 1 ) D and D(P 2 ) D; 2) C(P 1 )+C(P 2 ) C( ˆP 1 )+C( ˆP 2 ) for every oher pair of lin-dijoin (, )-pah ˆP 1 and ˆP 2 ha aify D( ˆP 1 ) D and D( ˆP 2 ) D We denoe by OPT he co of an opimal oluion o Problem 2DP for (G,,, D) Problem 2DP include Problem RSP a a pecial cae; hence, i i NP-hard In addiion, a dicued below (in Secion VIII), i i inracable o find a oluion ha doe no violae he delay conrain of a lea one of he pah Furhermore, in mo cae, we canno provide an efficien oluion wihou violaing he delay conrain in boh primary and reoraion pah Accordingly, we inroduce he following definiion of (α, β)-approximaion Definiion 1 ((α, β)-approximaion) : Given an inance (G,,, D) of Problem 2DP, an (α, β)-approximae oluion (P 1,P 2 ) o Problem 2DP i a oluion for which: 1) D(P 1 )+D(P 2 ) 2αD; 2) he oal co of wo pah i a mo β ime more han ha of he opimal oluion, ie, C(P 1 )+C(P 2 ) βopt In general, he pah wih minimum delay among P 1 and P 2 erve a a primary pah Thu, he primary and reoraion pah violae he delay conrain by facor of a mo α/2 and α, repecively, ie, D(P 1 ) αd and D(P 2 ) 2αD 2 A Fully Polynomial Approximaion cheme (FPAS) provide a oluion whoe co i a mo (1 + ) ime more han he opimum wih a ime complexiy ha i polynomial in he ize of he inpu and 1/ Alg Approx Raio Complexiy 2DP-1 (15, 15(1 + )) O(MN( 1 +loglogn)) 2DP-2 (1 + 1,(1 + γ)) O(MN OPT log log(cd)) 2DP-3 (1 + 1 MN3,(1 + γ)(1 + )) O( log(cd)) 2DP-4 (1 + 1,(1 + γ)(1 + )) O( MN2 2 log log(cd)) TABLE I PERFORMANCE CHARACTERISTICS OF PRESENTED ALGORITHMS D Our reul We inroduce four approximaion algorihm for Problem 2DP Table I how he approximaion raio achieved by each algorihm and i complexiy The parameer and capure he rade-off beween he violaion of he delay conrain, he co of he approximaion and he compuaional complexiy of he algorihm For example, Algorihm 2DP-2 achieve an approximaion raio of (1 + 1,(1 + γ)) for a poiive ineger, where γ i a 2(log +1) mall value bounded by Thu, chooing = 4 yield a (225, 55)-approximaion oluion o Problem 2DP In general, maller value of yield oluion wih lower delay violaion a he expene of higher co and running ime Algorihm 2DP-4 i he main conribuion of hi paper The algorihm achieve, for fixed >0 and any ineger >0, he approximaion raio of (1 + 1,(1 + γ)(1 + )), ie, he primary and reoraion pah violae he delay conrain by a facor of (1+ 1 ) and 2(1+ 1 ), repecively Thu, he violaion of delay by he primary pah can be minimized by chooing ufficienly large value of III PRELIMINARIES: NETWORK FLOWS In order o eablih an efficien oluion o Problem 2DP, we employ idea and echnique from he heory of newor flow The oluion o our problem, ha i wo dijoin pah, can be convenienly repreened a a flow Accordingly, in hi ecion we briefly preen he concep of newor flow A comprehenive urvey on he heory of newor flow can be found in [18] We conider flow newor in which each lin i aociaed wih a nonnegaive capaciy We aume ha for any pair of node u and v, he flow newor doe no conain wo lin in oppoie direcion ((v, u) and (u, v)) We noe ha hi aumpion doe no impoe any lo of generaliy, becaue by a uiable ranformaion we can alway define a newor ha i equivalen o any given newor bu aifie he above aumpion: he ranformaion pli each node v ino wo node v and v correponding o node oupu and inpu lin, and replace each original lin (v, u) by a lin (v,u ) wih he ame capaciy, co and delay; i alo add a lin (v,v ) of zero co and delay and infinie capaciy o each node v We proceed o inroduce he fundamenal concep of newor flow We reric ourelve o unary flow, ie, flow ha ae he value of 0 or 1 in each of he lin
Definiion 2 (Unary Flow) :Aunary (, )-flow f i a binary funcion f : E {0, 1} ha aifie he following wo properie: 1) For all l =(u, v) E, i hold ha f l {0, 1}; 2) For all v V \{, }, i hold ha f (w,v) = f (v,w) w:(w,v) E w:(v,w) E For clariy, we ay ha each lin l G for which f l =1 belong o he flow f and ha f include all lin for which f l =1 Definiion 2 ue he lin repreenaion, ie, he flow i decribed by mean of a funcion aociaed wih each lin of he newor Alernaively, a unary flow can alo be repreened by a e of pah P = {P 1,,P } and cycle W = {W 1,,W x }, uch ha exacly one uni of flow i en along each pah and cycle We refer o hi repreenaion a a pah and cycle repreenaion Noe ha given a pah and cycle repreenaion of a flow f, i i eay o deermine he lin repreenaion: he flow f l on each lin l ha belong o a pah in P or o a cycle in W i 1, while he flow on any oher lin i 0 Similarly, given a lin repreenaion f of a flow, we can deermine i pah and cycle repreenaion by uing he flow decompoiion algorihm [18] The value of a flow f i defined a follow: f = f (,v) (1) v:(,v) E A flow of zero value conain only cycle and no pah Such a flow i referred o a a circulaion The co C(f) of a flow f i defined a follow: C(f) = c (u,v) f (u,v) (2) (u,v) E We inroduce he noion of he delay D(f) of flow f D(f) = d (u,v) f (u,v) (3) (u,v) E Noe ha a flow f wih f =2can be decompoed ino wo dijoin pah whoe oal delay and co i a mo D(f) and C(f), repecively Thu, our goal i o find a flow whoe delay and co are no more han 2αD and βopt, repecively A Minimum Conrained Flow Problem We proceed o inroduce he minimum conrained flow (MCF) problem The problem ee a minimum co (, )- flow f uch ha f =2and D(f) D, where D i a given delay conrain Problem MCF (Minimum Conrained Flow Problem) : Given a graph G, a ource node, a deinaion node and a delay requiremen D, find an (, )-flow f uch ha: 1) f =2; 2) D(f) 2D; 3) C(f) C( ˆf) for any oher flow ˆf ha aifie ˆf =2 and D( ˆf) 2D The co of an opimal oluion o Problem MCF for (G,,, D) i denoed by Noe ha Problem MCF i a relaxaion of Problem 2DP In paricular, inead of impoing a delay conrain for each of he wo pah, Problem MCF require ha he oal delay of wo pah be no more han 2D Thu,if(P 1,P 2 ) i a feaible oluion o Problem 2DP, hen he flow f = {P 1,P 2 } i a feaible oluion o Problem MCF We conclude ha he co of he opimal oluion o Problem MCF i lower han ha of Problem 2DP, ie, OPT IV SIMPLE APPROXIMATION ALGORITHM In hi ecion we preen our fir approximaion algorihm, which achieve an approximaion raio of (15, 15(1 + )) The compuaional complexiy of he algorihm i O(MN( 1 + log log N)), which i idenical o ha of he approximaion cheme for Problem RSP [17] The idea of he algorihm i o idenify a uiable flow f beween and uch ha f =2and hen decompoe i ino wo dijoin pah ˆP 1 and ˆP 2 The algorihm employ he pah augmenaion approach [18], which i a andard approach for newor flow and dijoin pah problem The fir ep of he algorihm i o compue a pah P 1 beween he ource node and deinaion node and ha aifie he delay conrain D The pah P 1 i conruced by applying Algorihm RSP for (G,,, D, ) Thi pah define aflowf = {P 1 } whoe value i one uni The nex ep i o augmen hi flow in order o increae i value o 2 To ha end, we conruc a reidual newor G(f) impoed by he flow f Inuiively, he reidual newor coni of lin ha can admi more flow Definiion 3 (Reidual Newor) : Given a newor G wih uni capaciie and flow f, he reidual newor G(f) i conruced a follow For each lin (u, v) G for which f (u,v) =0, we add o G(f) a lin (u, v) of he ame delay and co a in G For each lin (u, v) G for which f (u,v) =1, we add o G(f) a revere lin (v, u) o G(f) of zero co and zero delay Aflow ˆf in he reidual newor G(f) i referred o a an augmening flow Having idenified he flow ˆf, we can augmen he flow f along he flow ˆf by performing he following ep: 1) Omi from f each lin (v, u) whoe revere lin (u, v) appear in ˆf 2) Add o f each lin (v, u) ˆf whoe revere lin (u, v) doe no appear in f Wih he augmenaion pah approach, he flow f i augmened along flow ha coni of a ingle augmening pah In paricular, our algorihm idenifie an augmening pah P 2 in G(f) ha aifie a delay conrain of 2D To ha end, we apply Algorihm RSP for (G(f),,,2D, ) Then, we augmen he flow f along he pah P 2 For each lin l ha belong o P 2 if f l =0we e f l =1, oherwie we e f l =0 The value of he reuled flow f i 2 The final ep i o decompoe he flow f ino wo pah ˆP 1 and ˆP 2, uch ha D( ˆP 1 ) D( ˆP 2 ) For hi purpoe
Algorihm 2DP-1 (G,,, D, ) inpu: G - he graph - ource node -deinaion node D- he delay conrain - he approximaion raio oupu: ( ˆP 1, ˆP 2 )- An approximae oluion o Problem 2DP 1 Idenify pah P 1 in G uch ha D(P 1 ) D by uing Algorihm RSP 2 f {P 1 } 3 Conruc he reidual newor G(f) of G impoed by f: 4 Add o G(f) each lin in G ha doe no belong o P 1 5 for each lin (u, v) P 1 do 6 Add a lin (v, u) o G(f) wih d (v,u) =0and c (v,u) =0 7 Idenify pah P 2 in G(f) uch ha D(P 2 ) 2D by uing Algorihm RSP 8 Augmen flow f along pah P 2 : 9 for each lin l(u, v) P 2 do 10 if f (v,u) =0hen 11 f (v,u) 1 12 ele 13 f (v,u) 0 14 Decompoe flow f ino wo pah, ˆP1 and ˆP 2 uch ha D( ˆP 1 ) D( ˆP 2 ) 15 reurn ˆP 1 and ˆP 2 Fig 1 Algorihm 2DP-1 we employ he following flow decompoiion algorihm We ar a he ource node and elec a lin (, v) for which f (,v) = 1Ifv i a deinaion node, we op; oherwie, here mu be a lin (v, u) for which f (v,u) =1 Thi proce i repeaed unil we eiher encouner a deinaion node or revii a previouly examined node In he former cae we obain an (, )-pah P and in he laer cae we obain a cycle W If we obain a direced pah P, we redefine f l =0for each lin l in P Similarly, if we obain a cycle W, we redefine f l =0for each lin l in W We repea hi proce ill we dicover wo pah beween and The deailed decripion of he approximaion algorihm, referred o a 2DP-1, appear in Fig 1 The correcne of our algorihm i baed on he following lemma Lemma 1: Le G(f) be he reidual newor of G impoed by f = {P 1 } Then, here exi a pah P 2 G(f) uch ha D(P 2) 2D and C(P 2) OPT Proof: Le Ĝ G be a newor impoed by lin ha belong o P op 1, P op 2 and P 1 LeĜ(f) be he reidual newor of Ĝ impoed by he flow f = {P 1} Clearly, Ĝ(f) G(f) We prove ha here exi a pah P 2 Ĝ(f) ha aifie he condiion of he lemma By way of conradicion, aume ha uch a pah doe no exi Then, by he Pah Augmenaion Theorem [18], he flow f = {P 1 } i a maximum flow in Ĝ However, here exi a flow f = {P op 1,P op 2 } of higher value, reuling in a conradicion P 1 (a) (c) P 2 (e) op P 1 op P 2 op P 1 op P 2 ' P 2 ˆP 2 (d) (b) (f) ˆP 1 op P 1 op P2 op P 1 Fig 2 Execuion of Algorihm 2DP-1(a) An opimal oluion (P op 1,P op 2 ) o Problem 2DP (b) Pah P 1 (c) Reidual newor G(f) of G impoed by flow f = {P 1 } (d) Pah P 2 (e) Pah P 2 (f) Pah ˆP 1 and ˆP 2 P op op P 2 Noe ha pah P 2 include only lin ha belong o P op 1, 2 a well a lin originaed from P 1, whoe delay and co are zero A a reul, D(P 2) D(P op 1 )+D(P op 2 ) 2D and C(P 2) C(P op 1 )+C(P op 2 ) OPT Since Ĝ(f) G(f), i hold ha P 2 G(f) and he lemma follow Le (G,,, D) be an inance of Problem 2DP and le (P op 1,P op 2 ) be an opimal oluion for hi inance, ie, D(P op 1 ) D, D(P op 2 ) D and C(P op 1 )+C(P op 2 )= OPT (ee Fig 2(a)) Noe ha he pah P 1 compued in Sep 1 migh hare lin wih he opimal pah (ee Fig 2(b)) We noe alo ha C(P 1 ) (1 + ) min{c(p op 1 ),C(P op 2 )} (1 + ) OPT 2 Fig 2(c) depic he reidual graph G(f) impoed by he flow f = {P 1 } Each reidual lin l G(f) i aigned zero delay By Lemma 1, here exi a pah P 2 G(f) beween and whoe delay i a mo 2D and whoe co i a mo OPT (ee Fig 2(d)) Thu, Algorihm RSP, invoed for (G(f),,,2D), reurn a pah P 2, whoe co i a mo (1 + ) OPT (ee Fig 2(e)) We conclude ha: and D(P 1 )+D(P 2 ) 3D C(P 1 )+C(P 2 ) 15(1 + )OPT Pah ˆP 1 and ˆP 2 include lin ha belong o P 1 and P 2, excluding lin ha were aigned zero co and delay (ee Fig 2(f)) Hence,
D( ˆP 1 )+D( ˆP 2 ) D(P 1 )+D(P 2 ) 3D V 1 (0,1) (2,0) V 2 V 1 V 2 (0,1) (2,0) and (1,0) (1,0) C( ˆP 1 )+C( ˆP 2 ) C(P 1 )+C(P 2 ) 15(1 + )OPT (2,0) (0,1) (2,0) (0,1) Chooing he minimum delay pah among ˆP 1 and ˆP 2 a a primary pah, reul in a (15, 15(1 + )) approximaion algorihm for Problem 2DP The algorihm invoe Algorihm RSP wice, hence i compuaional complexiy i O(MN( 1 + log log N)) We ummarize our dicuion by he following heorem Theorem 1: Algorihm 2DP-1 compue, in O(MN( 1 + log log N)) ime, a (15, 15(1 + ))-approximae oluion for Problem 2DP V MINIMIZING THE DELAY VIOLATION In he previou ecion we preened Algorihm 2DP-1 ha provide a (15, 15(1 + )) approximae oluion for Problem 2DP Fig 3(a) demonrae an inance (G,,, D) of Problem 2DP, for which algorihm 2DP-1 ha he worcae delay violaion, ie, α =15 ForD =2, he opimal oluion i P op 1 = {, v 1,v 2,} and P op 2 = {, v 3,v 4,} (ee Fig 3(b)) The co OPT of he opimal oluion i 2 We now apply 2DP-1 o he inance (G,,, D) The algorihm elec P 1 = {, v 1,v 2,v 3,v 4,} becaue i i he minimum co pah among all pah in G ha aify he delay conrain 2 (ee Fig 3(c)) Fig 3(d) depic he reidual newor G(f) of G impoed by flow f = {P 1 } The only pah beween and in G(f) i P 2 = {, v 3,v 2,}, wih delay 4 and co 0 The algorihm reurn he pah ˆP 1 = {, v 1,v 2,} and ˆP 2 = {, v 3,v 4,}, a depiced in Fig 3(e) Noe ha D( ˆP 1 ) = D( ˆP 2 ) = 3 Thu, we conclude ha he pah augmenaion raegy alone canno achieve a delay raio (α) beer han 15 The baic idea of he algorihm i o find cycle wih negaive delay and o augmen flow f along hee cycle Thi allow o reduce he delay of he oluion and achieve a maller delay raio For example, Fig 3(f) how he reidual newor G(f) of G impoed by he flow f = { ˆP 1, ˆP 2 }, conruced from G by ubiuing each lin l(u, v) f by a lin l (v, u) and eing d l = d l and c l =0 The reidual newor G(f) conain wo negaive delay cycle: he fir cycle i formed by wo lin beween and v 1, while he econd cycle i formed by wo lin beween v 4 and Each cycle ha delay 1 and co 1 Thu, if we augmen he flow along each of hee cycle, he oal delay of he flow i improved by 1 a co 1 By idenifying wo cycle, we find a flow whoe delay and co are 4 and 2 Thi flow can be decompoed ino wo pah whoe oal delay i a mo 4, achieving he approximaion raio of (2, 1) Algorihm 2DP-2 ge a inpu he newor G, he ource and deinaion node and, a delay conrain D and approximaion parameer The algorihm include he following ep Fir, we invoe Algorihm 2DP-1 for (G,,, D, 1 N ), which idenifie wo pah P 1 and P 2 Thee pah impoe (2,0) V V 3 4 V V 3 4 (a) (b) V 1 (1,0) (1,0) V 2 V V 3 4 (c) V 1 V 2 (1,0) (1,0) V 3 V 4 (e) (2,0) V 2 V 1 V 2 (0,1) (2,0) (2,0) (-2,0) V 3 V (d) 4 (f) (0,1) V 1 V 2 (0,1) (-2,0) (-1,0) (-1,0) V 3 V 4 Fig 3 Execuion of Algorihm 2DP-1 Aociaed wih each lin are i delay and co (a) The original newor (b) The opimal oluion pah P op 1 = {, v 1,v 2,} and P op 2 = {, v 3,v 4,} (c) Pah P 1 = {, v 1,v 2,v 3,v 4,} (d) The reidual newor induced by flow f (e) Pah ˆP 1 and ˆP 2 reurned by Algorihm 2DP-1 (f) The reidual newor G(f) of G impoed by he flow f = { ˆP 1, ˆP 2 } aflowf 0 = {P 1,P 2 } If D(f 0 ) 2D(1 + 1 ), hen he algorihm hal and reurn pah P 1 and P 2 Oherwie, we idenify a negaive delay cycle in he reidual graph G(f 0 ) of f 0 and augmen flow f 0 along hi cycle We repea hi proce unil he delay D(f) of he reuled flow f i lower or equal o 2D(1 + 1 ) Finally, we decompoe he flow f ino wo dijoin pah ˆP 1 and ˆP 2 More pecifically, we inroduce he following Procedure IM- PROVEFLOW The procedure ge a inpu flow f 0 and an approximaion parameer We begin by eing f = f 0 and conrucing he reidual graph G(f) of G impoed by flow f The reidual graph i conruced according o Definiion 3, wih he following excepion: he delay of a revere lin (v, u) i e o d (u,v), where (u, v) i he lin in he original newor ha correpond o (v, u) Nex, we find a cycle W in G(f) ha minimize he delay/co raio, ie, he cycle for which D(W ) C(W ) i minimal among all cycle in G(f) Such a cycle i deermined by uing he minimum co-o-ime raio cycle algorihm, preened in [18] Nex, we augmen flow f along W Thi proce (ie, finding cycle and circulaion augmenaion) erminae when he delay D(f) of f i lower or equal o 2D(1 + 1 ) Finally, flow f i decompoed ino wo pah, ˆP1 and ˆP 2, uch ha D( ˆP 1 ) D( ˆP 2 ) The formal decripion of Algorihm 2DP-2 appear in Fig 4 The following heorem eablihe he main properie of (0,1)
Algorihm 2DP-2 (G,,, D, ) 1 (P 1,P 2 ) 2DP-1(G,,, D, 1 N ) 2 f 0 {P 1,P 2 }; 3 if D(f 0 ) 2D(1 + 1 ) hen 4 reurn P 1 and P 2 5 f IMPROVEFLOW(G, f 0,D,) 6 Decompoe flow f ino wo pah, ˆP1 and ˆP 2, uch ha D( ˆP 1 ) D( ˆP 2 ) 7 reurn ˆP 1 and ˆP 2 Procedure IMPROVEFLOW(G, f 0,D,) 1 f f 0 2 while D(f) > 2D(1 + 1 ) do 3 Conruc he reidual newor G(f) of G impoed by f: 4 Add o G(f) each lin l in G for which f l =0 5 for each lin (u, v) in G for which f (u,v) =1do 6 Add a lin (v, u) o G(f) wih d (v,u) = d (u,v) and c (v,u) =0 7 Find a cycle W in G(f) ha minimize D(W ) C(W ) 8 Augmen flow f along W 9 reurn f Fig 4 Algorihm 2DP-2 Algorihm 2DP-2 I proof appear in he nex ecion Theorem 2: Algorihm 2DP-2 compue, in O(MN OPT log log(cd)) ime, a (1 + 1,(1 + γ)) approximae oluion for Problem 2DP, where γ 2(log +1) A Analyi of Algorihm 2DP-2 In hi ecion we analyze he performance of Algorihm 2DP-2 and eablih i compuaional complexiy We ue he following varian of he Augmening Cycle Theorem, aen from [18] Theorem 3: (Augmening Cycle Theorem, [18]) Le f and f 0 be any wo feaible newor flow uch ha f 0 = f Then, f equal f 0 plu he circulaion f in G(f) Furhermore, D(f) =D(f 0 )+D( f) Proof: The proof follow he ame line a in [18], ubiuing he co c l and C(f) by d l and D(f) for each lin l and each flow f, repecively Recall ha i he co of an opimal oluion o Problem MCF We prove ha here exi a circulaion f in G(f 0 ) uch ha D( f) (D(f) 2D), C( f) Lemma 2: Le f be an (, )-flow in G uch ha f =2 and D(f) 2D, and le G(f) be he reidual newor of G impoed by f Then, here exi a circulaion f in G(f), uch ha D( f) (D(f) 2D), C( f) Proof: Le f be he opimal oluion o Problem MCF Noe ha D(f ) 2D Le Ĝ be a ubgraph of G ha include only lin l for which hold eiher f l =1or fl =1 By Theorem 3, here exi a circulaion f in he reidual graph Ĝ(f) of Ĝ impoed by f uch ha D(f )=D(f)+D( f) Since D(f ) 2D, we have D( f) 2D D(f) = (D(f) 2D) We oberve ha all lin wih poiive co in Ĝ(f) belong o f Thu, he co of f i a mo We alo oberve ha Ĝ(f) G(f) Thu, f belong o G(f) and he lemma follow Lemma 2 implie ha here exi a circulaion f in G(f), uch ha D( f) (D(f) 2D) and C( f) Since he delay D( f) of circulaion f i negaive, i hold ha D( f) C( f) D( f) D(f) 2D A circulaion f include a number of cycle, here mu be a cycle W f for which i hold ha D(W ) C(W ) (D(f) 2D) Corollary 1: Le f be an (, )-flow in G uch ha f =2 and D(f) 2D, and le G(f) be he reidual newor of G impoed by f Then, here exi a cycle W in G(f) for which i hold ha D(W ) C(W ) (D(f) 2D) Proof: By Lemma 2, here exi a circulaion f in G(f), uch ha D( f) (D(f) 2D) and C( f) Such a circulaion i a e of cycle and he average raio D(W ) D( f) C( C(W ) for he cycle in hi e i D(f) 2D f) Hence, here exi a lea one cycle uch ha D(W ) C(W ) D(f) 2D In he nex lemma we prove ha he co of each negaive delay cycle idenified by Procedure IMPROVEFLOW i a lea 1 and a mo (2 +1) Lemma 3: Le W be a cycle idenified in he line 7 of Procedure IMPROVEFLOW Then, i hold ha 1 C(W ) ( +1) Proof: Le W be an augmening cycle in G(f) I i eay o verify ha flow f conain no cycle, ie, f include wo dijoin (, )-pah I follow ha any cycle in G(f) mu include a lea one lin ha doe no belong o f Since all lin in W have non-negaive co, i follow ha he oal co of W i a lea 1 We proceed o prove ha C(W ) (2 +1) By Corollary 1, D(W ) D(f) 2D D(W ) C(W ) D(f) 2D, hence C(W ) We noe ha D(W ) D(f), oherwie augmenaion of f by W would reul in an (, )-flow in G whoe delay i negaive, which conradic he fac ha all lin in G have poiive delay Thu, i follow ha C(W ) D(f) D(f) 2D 1 2D/D(f) Since D(f) 2D(1 + 1 ),wehave C(W ) (D+D/) D/ =( +1) Lemma 4: Le f be a flow in G, G(f) be he reidual newor of G impoed by f LeW be a cycle in G(f) and f be a flow reuling from augmening f along W Then, D(f )=D(f)+D(W ) and C(f ) C(f)+C(W ) Proof: Each lin wih poiive delay in W i added o f For each lin l in W wih negaive delay d l, we delee from f a lin whoe delay i d l Hence, D(f )=D(f)+D(W ) Since flow f include lin from f and W and ince each lin in W ha poiive co, i follow ha C(f ) C(f)+C(W ) Lemma 4 implie ha an augmenaion of a flow f by a cycle W decreae he delay D(f) of f by D(W ) (ince D(W ) i negaive) and increae i co by C(W ) Inhe following, we prove ha he number of ieraion needed in order o achieve D(f) D(2 + 1 ) i a mo 2 log Our prove ue he Geomeric Improvemen Approach In paricular, we ue he following heorem, aen from [18]
Theorem 4: Suppoe ha an algorihm A ieraively minimize ome value z uch ha z 0 i he iniial value of z, z i i he value of z a he i-h ieraion and z he minimum objecive funcion value Furhermore, uppoe ha he algorihm A guaranee ha, for every ieraion i, z i z i+1 ζ(z i z ) (4) for ome conan ζ wih 0 <ζ<1 Then, wihin 2/ζ conecuive ieraion, i hold ha z z + z0 z 2 Furhermore, he algorihm A erminae afer a mo 2log(z z 0 ) ζ ieraion Proof: See [18] Theorem 5: Procedure IMPROVEFLOW reurn a flow f wih delay D(f) 2D(1 + 1 ) and co C(f) ( + 2 log +1+ 15 N ) The compuaional complexiy of Procedure IMPROVEFLOW i O(MN log(cd) log ) Proof: Conider he main loop of Procedure IMPROVE- FLOW, ie, he loop ha begin a line 2 We denoe by f i he ae of flow f a he beginning of ieraion i (a line 7) Furher, we denoe by W i he cycle ha ha been idenified a ieraion i and by C i he co of W i, ie, C i = C(W i ) Fir, we prove ha he procedure op wihin 2log ieraion, ie, a ome ieraion j 2log OPT i hold ha D(f j ) 2D(1 + 1 ) By Lemma 3, a each ieraion i we idenify a cycle W i whoe co i a lea 1 Corollary 1 implie ha he delay of he cycle i a lea (D(fi) 2D) Thu, by Lemma 4, he delay of flow f i reduced by a lea (D(fi) 2D) a each ieraion By Theorem 4, afer a mo 2log ieraion, i hold ha D(f) 2D + D(f 0 ) 2D 2D(1 + 1 ) Second, we prove ha he procedure reurn a flow f whoe co i a mo C(f 0 )+(2 + 2 log +25) Lej be he la ieraion of he procedure We how ha he oal co of all cycle idenified in he fir j 1 ieraion i a mo 2 log We aume, by way of conradicion, ha j 1 i=1 C(W i) 2 log We conider wo cae In he fir cae all cycle W 1,,W j 1 have co of exacly 1 Then, a each ieraion, he delay of flow f decreae by a lea D(fi) 2D Thu, by Theorem 4, afer j 1 ieraion, he delay of flow f i a mo 2D + D(f0) 2D 2D(1 + 1 ), which conradic he fac ha j 1 i no a la ieraion The la inequaliy follow from he fac ha D(f 0 ) 3D (by Theorem 1) Now we conider he econd cae, in which he co a each W i may be more han 1 In hi cae, a ieraion i, he delay of flow f decreae by a lea D(fi) 2D C(W i )We noe ha augmening of flow f along cycle W i i equivalen (D(f i) 2D) o augmening i along C(W i ) cycle of co 1 and delay Thu, if j 1 i=1 C(W i) 2 log, hen cycle W 1,,W j can be ubiued by 2 log cycle of co 1 Thi implie ha afer ieraion j 1 we have D(f j 1 ) 2D(1 + 1 ), and again we have a conradicion By Lemma 3, he co of he cycle W j idenified a he la ieraion i a mo ( +1) Thu, he oal co of flow f reurned by Procedure IMPROVEFLOW i a mo (+1+2log ) +C(f 0 ) (+2 log +1+ 15 N ) Finally, we analyze he compuaional complexiy of Procedure IMPROVEFLOW We proved ha he procedure perform a mo 2 log ieraion The running ime of each ieraion i dominaed by he ime required o idenify a cycle ha minimize D(W ) C(W ) Such a cycle i idenified by invoing he minimum co-o-ime raio cycle algorihm [18], which incur O(MN log(cd)) ime We conclude ha he compuaional complexiy of he procedure i O(MN log log(cd)) We are ready now o prove Theorem 2 Theorem 2: Algorihm 2DP-2 compue, in O(MN OPT log log(cd)) ime, a (1 + 1,(1 + γ)) approximae 2(log +1) oluion for Problem 2DP, where γ Proof: The compuaional complexiy of Algorihm 2DP-2 i dominaed by he ime required o idenify pah P 1 and P 2 in (line 1) and he running ime of Procedure IMPROVEFLOW By Theorem 1 The compuaional complexiy of Algorihm 2DP-1 i O(MN 2 ) (by Theorem 1) By Theorem 5, Procedure IMPROVEFLOW require O(M N log log(cd)) ime Since N OPT and OPT, he compuaional complexiy of Algorihm 2DP-2 i O(MN OPT log log(cd)) By Theorem 5, Procedure IMPROVEFLOW reurn a flow f wih delay D(f) 2D(1+ 1 ) and co C(f) (+2 log + 1+ 15 N )OPT 2(log +1) 1 (1 + γ)opt, where γ We conclude ha Algorihm 2DP-2 compue a (1 + 1,(1 + γ)) approximae oluion for Problem 2DP VI MINIMIZING THE COMPUTATIONAL COMPLEXITY While Algorihm 2DP-2 provide a good approximae oluion for Problem 2DP, i compuaional complexiy i proporional o he co OPT of he opimal oluion The algorihm can be ued in eing in which he co of each lin i a relaively mall value However, for eing wih high co value, i compuaional complexiy may be prohibiive Accordingly, in hi ecion we preen Algorihm 2DP-3 whoe compuaional complexiy doe no depend on he value of lin co The algorihm ue he ame idea a Algorihm 2DP-2 and, in addiion, employ he co caling approach [16] in order o reduce he compuaional complexiy Algorihm 2DP-3 begin by invoing Algorihm 2DP-1, which idenifie wo pah P 1 and P 2 Thee pah form a flow f 0 = {P 1,P 2 } and we conruc he reidual newor G(f 0 ) of G impoed by flow f 0 In he previou ecion we howed ha here exi a circulaion f in G(f 0 ), uch ha D( f) (D(f) 2D), C( f) (Lemma 2) Algorihm 2DP-2 ue hi fac o reduce he delay of he flow f 0 In paricular, i invoe Procedure IMPROVEFLOW, whoe running ime depend on he co of he circulaion, ie, The baic idea of Algorihm 2DP-3 i o reduce he co of each lin in G by a cerain facor The co of he circulaion f in he reuling graph, and in urn, he compuaional complexiy of Procedure IMPROVEFLOW, are much maller A ey requiremen in he caling approach i o ge ufficienly igh upper and lower bound L and U on he co of
he opimal oluion o Problem MCF We preen an efficien echnique for obaining hee bound in Secion VI-A We proceed o decribe co caling in more deail We cale he co c l of each lin l in G, replacing i by c l, a follow: c cl l = +1, (5) where = L 2N Le G (f 0 ) be he reidual graph of G impoed by flow f 0 wih caled lin co We how ha here exi a circulaion f in G (f 0 ), uch ha D( f) (D(f) 2D) and C( f) 2N U L +2N Lefop be he opimal oluion for Problem 2DP I i eay o verify ha flow f op conain a mo 2N lin Thu, he co of f wih repec o he caled lin co i a mo OPT 1 = 2N OPT L 1 +2N I can be hown, in he ame way a in Lemma 2, ha here exi a circulaion f in G (f 0 ), uch ha D( f) (D(f) 2D) and whoe co i a mo 2N OPT L 1 +2N Thu, wih caled lin co Procedure IMPROVEFLOW perform ju O ( ) N U L ieraion For ufficienly igh lower and upper bound, L and U, he number of ieraion i mall Scaling allow o reduce he compuaional complexiy of he algorihm, bu i incur ome penaly in erm of he oluion co Le f we a circulaion whoe co wih repec o he caled lin co i a mo OPT 1 The co of ha circulaion wih repec o original lin co i a mo OPT 1 + L (1 + ) Lower value of yield beer approximae oluion on he expene of he running ime of he algorihm The deailed decripion of he algorihm appear in Fig 5 A Compuing Lower and Upper Bound, L and U In hi ubecion we preen Procedure BOUND (ee Fig 5), which idenifie lower and upper bound L, U on uch ha U/L 2N We ue he echnique preened in [17] We denoe by c 1 <c 2 < <c r he diinc co value of he lin Our goal i o find he maximum co value c {c i } uch ha he graph G derived from G by omiing all lin whoe co i greaer han c, doe no conain a feaible flow f, ie, a flow wih value f =2and wih delay D(f) 2D Clearly, a feaible flow conain a lea one lin whoe co i c or more, hence c i a lower bound on In addiion, here exi a feaible flow ha comprie lin whoe co i c or le Since an opimal flow f include a mo wo pah, i include a mo 2N lin We conclude ha 2N c i an upper bound on Procedure BOUND perform a binary earch on he value c 1,c 2,,c r A each ieraion, we need o chec wheher c c, where c i he curren eimae of c For hi purpoe, we remove from G all lin whoe co i more han c, and aign he uni co o he remaining lin Then, we find a minimum delay flow in he reuling graph For hi purpoe we employ he minimum co dijoin pah algorihm aen from [2], wih repec o lin delay If hi algorihm reurn a feaible flow, hen c c ; oherwie, c<c Procedure BOUND perform O(log N) ieraion Algorihm 2DP-3 (G,,, D,, ) 1 (P 1,P 2 ) 2DP-1(G,,, D, 1 N ) 2 f 0 {P 1,P 2 } 3 if D(f 0 ) 2D(1 + 1 ) hen 4 reurn P 1 and P 2 5 L, U BOUND(G,,, f, D) 6 L 2N 7 for each lin l E do 8 c l c l +1 9 f IMPROVEFLOW(G, f 0,D,) 10 Decompoe flow f ino wo pah, ˆP1 and ˆP 2, uch ha D( ˆP 1 ) D( ˆP 2 ) 11 reurn ˆP 1 and ˆP 2 Procedure BOUND (G(V,E),,,D) 1 le c 1 <c 2 < <c r he diinc co value of he lin 2 low 1; high r 3 while low < high 1 4 j (high + low)/2 5 E {l c l c j } 6 find a minimum delay flow f beween and in (G (V,E ) uch ha f =2 7 if D(f) > 2D hen 8 high j 9 ele 10 low j 11 U 2N c high ; L c high 12 reurn L, U Fig 5 B Analyi of Algorihm 2DP-3 Algorihm 2DP-3 The following heorem eablihe he correcne of Algorihm 2DP-3 Theorem 6: Algorihm 2DP-3 compue, in log M N 3 O( log(cd)) ime, a (1 + 1,(1 + γ)(1 + ))- 2(log +1) approximae oluion for Problem 2DP, where γ Proof: Le f op be an opimum oluion o Problem MCF, ie, C(f op ) = Noe ha f op can be repreened by wo dijoin (, )-pah Indeed, if f op conain a cycle, hi cycle can be eliminaed, reuling in a flow whoe co i le han ha of f op We conclude ha f op include a mo 2N lin For each lin l G i hold ha c l c l c l +1 Thu, he co of f op wih repec o caled lin co i a mo OPT 1 = OPT 1 +2N By Theorem 5, Procedure IMPROVEFLOW reurn a flow f wih delay D(f) 2D(1 + 1 15 ) and co C(f) ( + 2 log +1+ N )OPT 1 = (1+γ)OPT 2(log +1) 1, where γ The co of flow f wih repec o original lin co i C(f) (1 + γ) OPT 1 (1 + γ)( + L) (1 + γ)(1 + )OPT We proceed o analyze he compuaional complexiy of Algorihm 2DP-3 We begin wih Procedure BOUND A dicued above, he procedure execue O(log N) ieraion of he main loop (ie, he loop ha begin a line 3 of
he procedure) A each ieraion, he procedure invoe he minimum co dijoin pah algorihm [2], whoe running ime i O(M + N log N) We conclude ha he compuaional complexiy of Procedure BOUND i O((M +N log N) log N) Since Procedure IMPROVEFLOW i invoed on he graph wih caled lin co, i compuaional complexiy i O(MN OPT 1 log(cd) log ) Since OPT 1 2N U +2N L 2N 2 he compuaional complexiy of Algorihm 2DP-3 i O( log(cd)) log M N 3 VII FURTHER IMPROVEMENTS In hi ecion we furher improve he compuaional complexiy of our oluion To ha end we exend Algorihm 2DP- 3, by inroducing he following change Fir, we modify Procedure IMPROVEFLOW, in order o enure ha i running ime i bounded by a cerain value Second, we inroduce an addiional procedure, referred o a Procedure IMPROVEBOUNDS, whoe purpoe i o obain igher lower and upper bound on he co of he opimal oluion o Problem MCF More pecifically, we add a new parameer Û o Procedure IMPROVEFLOW in order o enure ha he co of he reurned flow f i no more han 2Û Wih hi modificaion, Procedure IMPROVEFLOW ha he following properie Lemma 5: 1) If Û, hen Procedure IMPROVEFLOW reurn aflowf wih delay D(f) 2D(1 + 1 ) 2) If Procedure IMPROVEFLOW doe no fail, i reurn a flow f wih delay D(f) 2D(1 + 1 ) and co C(f) 15 min{2û,( + 2 log +1+ N )} 3) The compuaional complexiy of Procedure IMPROVE- FLOW i O(MN min{, Û} log(cd) log ) Proof: The proof follow he ame line a ha of Theorem 5 We proceed o decribe Procedure IMPROVEBOUNDS The procedure obain upper and lower bound, L and U, uch ha: L i a lower bound on he co of an opimal oluion o Problem MCF, ie, for each flow in G wih delay D(f) 2D, i hold ha C(f) L U i an upper bound on he co of a flow in G wih delay D(f) 2D(1 + 1 ), ie, for each flow in G wih delay D(f) 2D(1 + 1 ), i hold ha C(f) U Lower and upper bound L and U for which i hold U ha L 2N can be obained hrough Procedure BOUND The goal of Procedure IMPROVEFLOW i o improve hee bound uch ha U L 162 A each ieraion we compare a e value B = L U wih he co of he opimal oluion and updae he bound L and U accordingly More pecifically, we cale he lin co by a facor = B 2N and invoe Procedure IMPROVEFLOW wih Û =4N on he graph wih caled lin co A we how in Lemma 6, if Procedure IMPROVEFLOW fail, hen >B, hence we e L = B Oherwie, Procedure IMPROVEFLOW reurn a Algorihm 2DP-4 (G,,, D,, ) 1 (P 1,P 2 ) 2DP-1(G,,, D, 1 N ) 2 f 0 {P 1,P 2 } 3 if D(f 0 ) 2D(1 + 1 ) hen 4 reurn P 1 and P 2 5 L, U BOUND(G,,, D) 6 L, U IMPROVEBOUNDS(G,,, f 0,D,L,U) 7 L 2N 8 for each lin l E do 9 c l c l +1 10 Û U +2N +1 11 f IMPROVEFLOW(G, f 0,D,,Û) 12 Decompoe flow f ino wo pah, ˆP1 and ˆP 2, uch ha D( ˆP 1 ) D( ˆP 2 ) 13 reurn ˆP 1 and ˆP 2 Procedure IMPROVEBOUNDS (G(V,E),,,f,D,L,U) 1 while U/L > 16 2 do 2 B L U 3 B 2N 4 for each lin l E do 5 c l c l +1 6 Û 4N 7 f IMPROVEFLOW(G, f 0,D,,Û) 8 if Procedure IMPROVEFLOW reurned FAIL hen 9 L B 10 ele 11 U 4B 12 reurn L, U Procedure IMPROVEFLOW(G, f 0,D,,Û) 1 f f 0 2 while C(f) 2Û do 3 G(f) reidual newor of G impoed by f 4 find a cycle W in G(f) ha minimize D(W ) C(W ) 5 augmen flow f along W 6 if D(f) D(2 + 1 ) hen 7 reurn f 8 reurn FAIL Fig 6 Algorihm 2DP-4 flow whoe delay and co are a mo 2D(1 + 1 ) and 4B, repecively, hence U 4B Accordingly, in hi cae we e U =4B Noe ha, if he raio U/L i equal o x a he beginning of an ieraion, hen a he end of he ieraion we have U L 4 x Thu, i can be hown ha, afer O(log log N) ieraion, he raio U/L i a mo 16 2 We hen ue hee igh upper and lower bound in order o efficienly find an approximae oluion o Problem 2DP The deailed decripion of he algorihm, referred o a Algorihm 2DP-4, appear in Fig 5 A Analyi of Algorihm 2DP-4 Lemma 6: Procedure IMPROVEBOUNDS reurn valid upper and lower bound L and U Proof: We noe ha Procedure BOUND reurn valid upper and lower bound L and U Thi follow from he fac
ha he minimum co of a flow whoe delay i a mo 2D i lower han he minimum co of a flow whoe delay i a mo 2D(1 + 1 ) We how ha he lower and upper bound remain valid during he execuion of he procedure Fir, we prove ha if Procedure IMPROVEFLOW fail hen > B By way of conradicion, uppoe ha hi i no he cae, ie, B Procedure IMPROVEFLOW i applied o he graph whoe co were caled by facor of = B 2N In he reuling graph, he co OPT 1 of he opimal oluion o Problem MCF i a mo 2N OPT B 1 +2N, which i le han 4N Thu, by Lemma 5, par 1, Procedure IMPROVEFLOW reurn a flow f wih delay D(f) 2D(1 + 1 ), which conradic he fac he he procedure fail Second, by Lemma 5, par 2, if Procedure IMPROVEFLOW doe no fail, i reurn a flow f whoe co wih repec o he caled lin co i a mo 2Û and delay i a mo 2D(1 + 1 ) The co of flow f wih repec o he original co i a mo 2Û =4B In hi cae, he minimum co of a flow wih delay a mo 2D(1+ 1 ) i bounded by 4B, hence U remain o be a valid upper bound ( Theorem 7: Algorihm) 2DP-4 compue, in O MN 2 2 log log(cd) ime, a (1 + 1,(1 + γ)(1 + ))- 2(log +1) approximae oluion for Problem 2DP, where γ Proof: A we proved in Lemma 6, Procedure IMPROVE- BOUNDS reurn valid lower and upper bound L and U Thu, i can be hown ha Algorihm 2DP-4 compue a (1+ 1,(1+γ)(1+))-approximae oluion for Problem 2DP, 2(log +1) where γ (The proof follow he ame line a he one for Theorem 6) We proceed o analyze he compuaional complexiy of he algorihm A dicued above, Procedure IM- PROVEBOUNDS execue O(log log N) ieraion of i main loop (ie, he loop ha begin in line 1) A each ieraion we invoe Procedure IMPROVEFLOW, whoe complexiy i O(MN 2 log(cd) log ) We conclude ha he running ime of Procedure IMPROVEBOUNDS i O(MN 2 log log N log(cd) log ) By Lemma 5, he compuaional complexiy of Procedure IMPROVEFLOW, invoed a line11io(mn Û log(cd) log ) Since he procedure i invoed wih Û = U +2N+1 = O( 2 N ), he compuaional complexiy of he procedure i O( MN2 2 log log(cd)) We 1 aume ha log log N, hence he compuaional complexiy of he algorihm i dominaed by he ime incurred by invocaion of Procedure IMPROVEFLOW in line 11, ie, O( MN2 2 log log(cd)) VIII LOWER BOUND In hi ecion we prove ha finding wo dijoin pah ha aify a given delay conrain D i an NP-hard problem Furhermore, we how ha no polynomial algorihm can approximae hi problem by a facor α uch ha 1 α<2 Tha i, finding wo dijoin pah uch ha he delay of each pah i a mo αd i an inracable problem for any 1 α<2 Specifically, we how a reducion from he problem Fig 7 1 2 G Conrucion of auxiliary graph Ĝ of finding wo lin dijoin pah in a direced newor, one beween 1 and 1 and he oher beween 2 and 2 Thi problem i nown o by NP-hard [19] Our proof follow he ame line a he proof preened in [20] for a relaed problem By way of conradicion, aume ha here exi an algorihm, ay Algorihm A, which idenifie, in polynomial ime, wo dijoin (, )-pah uch ha he delay of each pah i a mo αd We how ha hi algorihm can find wo dijoin pah beween 1 and 1 and beween 2 and 2 in a direced graph G We build an auxiliary graph Ĝ formed from G by adding wo node and and four lin (, 1 ), (, 2 ), ( 1,) and ( 2,) We aign delay o hee four lin a follow: d (,1) = d (2,) =1and d (,2) = d (1,) =0 All oher lin in Ĝ are aigned zero delay Fig 7 depic he conrucion of an auxiliary graph Ĝ By employing Algorihm A (for D =1) we can find wo dijoin pah P 1 and P 2 beween and uch ha D(P 1 ) α and D(P 2 ) α However, if α<2, hen one of he pah mu connec 1 and 1 and he oher mu connec 2 and 2 Thi conradic he fac ha he problem of finding uch wo dijoin pah in a direced newor i NP-hard IX CONCLUSION In hi paper we inveigaed he fundamenal problem of proviioning dijoin QoS pah We preened a comprehenive analyi of he problem, by uing he framewor of newor flow We howed ha any polynomial algorihm for hi problem violae he delay conrain by a lea a facor of 2 In addiion, we indicaed rade-off beween violaion of he delay conrain, co and compuaional complexiy More pecifically, he major conribuion of our udy are four approximaion algorihm for he conidered problem The fir algorihm i concepually imple and ha low compuaional complexiy Thi algorihm idenifie a oluion ha violae he delay conrain by facor of 15 and 3 for he primary and reoraion pah, repecively The econd algorihm reduce he delay violaion a he expene of higher co and compuaional complexiy The hird and fourh algorihm, achieve imilar co and delay raio a he econd, bu have ignificanly lower compuaional complexiy In paricular, he algorihm compue, for any fixed >0 and ineger >0 a oluion ha violae he delay conrain by facor of a mo 1+ 1 and 2(1+ 1 ) for he primary and reoraion pah, repecively, and whoe co i a mo (1 + γ)(1 + ) ime 1 2
more han he opimum, where γ i a mall value bounded by 2(log +1) We have indicaion ha our reul can be exended o a broader cla of newor reoraion and newor deign problem In paricular, our mehod, epecially he cyclecancellaion approach, can be ued in order o olve h-dijoin pah problem for any h > 2 In addiion, he echnique eablihed in he udy can be ued alo in he domain of local reoraion [21], where a reoraion opology comprie of a primary pah and everal bridge, each proecing a porion of he primary pah REFERENCES [1] E Mannie and D Papadimiriou (edior), Recovery (Proecion and Reoraion) Terminology for Generalized Muli-Proocol Label Swiching (GMPLS), Inerne draf, Inerne Engineering Ta Force, May 2003 [2] J Suurballe and R Tarjan, A Quic Mehod for Finding Shore Pair of Dijoin Pah, Newor, vol 14, pp 325 336, 1984 [3] S Chen and K Nahred, An Overview of Qualiy-of-Service Rouing for he Nex Generaion High-Speed Newor: Problem and Soluion, IEEE Newor, Special Iue on Tranmiion and Diribuion of Digial Video, vol 12, no 6, pp 64 79, November/December 1998 [4] FA Kuiper, T Kormaz, M Krunz, and P Van Mieghem, A Review of Conrain-Baed Rouing Algorihm, in Technical Repor, Lauanne, Swizerland, June 2002 [5] K Kar, M Kodialam, and T V Lahman, Rouing Reorable Bandwidh Guaraneed Connecion uing Maximum 2-Roue Flow, in Proceeding of IEEE INFOCOM 2002, New Yor, NY, USA, June 2002 [6] G Li, D Wang, C Kalmane, and R Doverpie, Efficien Diribued Pah Selecion for Shared Reoraion Connecion, in Proceeding of IEEE INFOCOM 2002, New Yor, NY, USA, June 2002 [7] N Taf-Ploin, B Bellur, and RG Ogier, Qualiy-of-Service Rouing Uing Maximally Dijoin Pah, in Proceeding IEEE/IFIP IWQoS, London, UK, June 1999 [8] MR Garey and DS Johnon, Compuer and Inracabiliy, Freeman, San Francico, CA, USA, 1979 [9] JP Lang and B Rajagopalan (edior), Generalized MPLS Recovery Funcional Specificaion, Inerne draf, Inerne Engineering Ta Force, January 2003 [10] Privae Newor-Newor Inerface Specificaion v10 (PNNI), ATM Forum Technical Commiee, March 1996 [11] S Blae, An archiecure for Differeniaed Service, RFC No 2475, Inerne Engineering Ta Force, December 1998 [12] A Orda and A Sprinon, Precompuaion Scheme for QoS Rouing, IEEE/ACM Tranacion on Neworing, vol 11, no 4, pp 578 591, Augu 2003 [13] I Cidon, R Rom, and Y Shavi, Analyi of Muli-pah Rouing, IEEE/ACM Tranacion on Neworing, vol 7, no 6, pp 885 896, 1999 [14] S Lee and M Gerla, Spli Mulipah Rouing wih Maximally Dijoin Pah in Ad Hoc Newor, in Proceeding of he IEEE ICC 2001, 2001 [15] F Ergun, R Sinha, and L Zhang, An Improved FPTAS for Rericed Shore Pah, Informaion Proceing Leer, vol 83, no 5, pp 237 293, Sepember 2002 [16] R Hain, Approximaion Scheme for he Rericed Shore Pah Problem, Mahemaic of Operaion Reearch, vol 17, no 1, pp 36 42, February 1992 [17] DH Lorenz and D Raz, A Simple Efficien Approximaion Scheme for he Rericed Shore Pah Problem, Operaion Reearch Leer, vol 28, no 5, pp 213 219, June 2001 [18] R K Ahuja, T L Magnani, and J B Orlin, Newor Flow, Prenice- Hall, NJ, USA, 1993 [19] S Forune, J Hopcrof, and J Wyllie, The Direced Subgraph Homeomorphim Problem, Theoreical Compuer Science, vol 10, no 2, pp 111 121, 1980 [20] CL Li, T McCormic, and D Simchi-Levi, The Complexiy of Finding Two Dijoin Pah wih Min-Max Objecive Funcion, Dicree Applied Mahemaic, vol 26, pp 105 115, 1990 [21] Y Bejerano, Y Breibar, A Orda, R Raogi, and A Sprinon, Algorihm for Compuing QoS Pah wih Reoraion, in Proceeding of IEEE INFOCOM 2003, San Francico, CA, USA, April 2003