Quantum-Inspired Evolutionary Multicast Algorithm

Transcription

1 Proceedngs of he 009 IEEE Inernaonal Conference on Sysems, Man, and Cybernecs San Anono, X, USA - Ocober 009 Quanum-Inspred Evoluonary Mulcas Algorhm Yangyang L yyl@xdan.edu.cn Lcheng Jao lchjao@mal.xdan.edu.cn Jngjng Zhao lasemple@6.com Quy Wu dwudwu@6.com Absrac As a global opmzng algorhm, genec algorhm (GA s appled o solve he problem of mulcas more and more. GA has more powerful searchng ably han radonal algorhm, however s propery of premaury makes dffcul o ge a good mulcas ree. A quanum-nspred evoluonary algorhm (QEA o deal wh mulcas roung problem s presened n hs paper, whch salenly solves he premaury problem n Genec based mulcas algorhm. Furhermore, n QEA, he ndvduals n a populaon are represened by mulsae gene quanum bs and hs represenaon has a beer characersc of generang dversy n populaon han any oher represenaons. In he ndvdual s updang, he quanum roaon gae sraegy s appled o accelerae convergence. he algorhm has he propery of smple realzaon and flexble conrol. he smulaon resuls show ha QEA has a beer performance han CS and convenonal GA. Keywords genec algorhm, mulcas, quanum-nspred evoluonary algorhm I. INRODUCION Wh he developmen of nework and communcaon echnologes, mulcas roung servce s becomng a key requremen of nework mulmeda applcaons whch requre ceran qualy of servce (QoS. Mulcasng nvolves he ranspor of he same nformaon from a ser o mulple recevers along a mulcas ree. In he pas, much work has been focused on algorhms for compung low-cos mulcas rees so ha nework resources can be effcenly managed. Snce he problem of compung mnmum-cos mulcas rees (call Sener rees n a nework [] s NP-complee, mos of he hs work was suppored by he Naonal Hgh echnology Research and Developmen Program (863 Program of Chna (No. 009AAZ0, he Naonal Naural Scence Foundaon of Chna (Nos , and , he Provncal Naural Scence Foundaon of Shaanx of Chna (No. 007F3, he Chna Posdocoral Scence Foundaon funded projec (No , he Chna Posdocoral Scence Foundaon Specal funded projec (No and he Program for Cheung Kong Scholars and Innovave Research eam n Unversy (No. IR0645. proposed algorhms are heursc ones []-[4]. Besdes some convenonal heursc algorhms, evoluonary algorhms [5] [6] are promsng for solvng hs problem. Despe he successes of genec algorhms, hey have he same defecs of pre-maury and sagnaon. Arfcal mmune algorhms are new compuaonal nellgence mehods nspred by heorecal mmunology. hey have also been appled o QoS mulcas roung problem and acheved good performances [7] [8]. In parcular, superor o convenonal genec algorhms (CGAs and bounded shores mulcas algorhm (BSMA [] whch s he bes heursc consraned mulcas algorhm, A Clonal Sraeges (CS based mulcas algorhm [8] manans good dversy and less lkely o be rapped n local opma. Unforunaely, CS obans a good local searchng ably a a cos of addng he scale of he populaon by he clone operaor; he characerscs of populaon dversy and selecve pressure are no easy o be mplemened n mmune clonal algorhm [9]. he quanum-nspred evoluonary algorhm (QEA recenly proposed n [0] can rea he balance beween exploraon and exploaon more easly when compared wh CGAs. In hs paper, QEA can explore he search space wh a smaller number of ndvduals and explo he search space for a global soluon whn a shor span of me. QEA s based on he concep and prncples of quanum compung, such as he quanum b and he superposon of saes. Wh he scale of he nework ncreased, QEA can fnd beer mulcas rees han GA and CS. II. HE MULICAS ROUING PROBLEM DEFINIION he communcaon nework s modeled as a graph GV (, E, where V s he se of nodes and E s he se of edges. On he graph G, we defne he funcons c(x,y and d(x,y, where c(x,y s he cos of usng edge ( xy, Eand d(x,y s he delay along edge ( xy, E. Le P(a,b denoe he /09/$ IEEE 54

2 pah from a o b, hen delay funcon Delay( a, b and cos funcon Cos( a, b can be expressed as follow: Delay( a, b = d( x, y, ( ( xy, Pab (, Cos( a, b = c( x, y. ( ( xy, Pab (, Supposes s s he source node, D V {s} denoes a se of desnaon nodes. he mulcas roung problem searches for a ree = ( V, E ( G, whch subjecs o he followng consrans: Cos( = mn( c( x, y, (3 ( xy, P ( xy, ( xy, P ( xy, d( x, y Δ, y D, (4 where Δ s a delay olerance and P ( x, y denoes he unque pah from x o y D n he mulcas ree. hroughou hs paper, we le n = V, m= D. III. QEA A. Represenaon In hs sudy, an ndvdual represens a search pon n decson space. In [9], we gve a new represenaon, a qub for he probablsc represenaon ha can represen a lnear superposon of saes and has a beer characersc of populaon dversy han oher represenaons [0], whch are defned below. Defnon : he probably amplude of one qub s defned wh a par of numbers ( α, β as: sasfyng where α β, (5 α + β =, (6 α denoes he probably ha he qub wll be found n he 0 sae and β denoes he probably ha he qub wll be found n he sae. Defnon : A qub ndvdual as a srng of N qubs s defned as: α α... αn β β... βn, (7 where αl + βl =,( l =,,..., N. B. Descrpon of he Algorhm QEA s a probablsc algorhm ha s smlar o EA. I manans a quanum populaon Q ( = { q, q,..., q M } a he - h generaon where M s he sze of populaon, and N s he lengh of he qub ndvdual q whch s defned as: α α... α N q =, =,,..., M. β β... βn he man loop of QEA s as follows. Algorhm : quanum-nspred evoluonary algorhm Q( (populaon of qub ndvduals a he -h generaon P( (populaon of classcal b ndvduals a he -h generaon b (bes classcal b ndvdual a he curren generaon Sep: Inalze Q ( and b, =0. Sep: Produce P( by observng he updaed Q (. Sep3: Evaluae he fness of P( soluons among P( o b. Sep4: Updae Q ( by quanum muaon. and sore he bes Sep5: Judge he ermnaon condon, f s sasfed, oupu he bes soluon and he process, oherwse =+. Sep6: Go o sep. IV. QUANUM-INSPIRED EVOLUIONARY MULICAS ALGORIHM A. Consrucon of canddae pah se Fgure. Flow char of searchng he canddae pah se. 543

3 Assume Graphc s a complee map ha has n nodes, ha s, each node n he nework has a degree of n-. he number of he feasble pahs from a node o a desgnaed node n he 3 nework s PahCoun = + P n n Pn Pn Pn > n!. If we consruc a mulcas ree n such a nework, he number of possble combnaons s descrbed as: PahCoun PahCoun PahCoun, (8 m where m s he number of he desnaon nodes. For an Unconsraned nework-wde connecvy, s canddae pah se s very large. Bu afer addng he delay consran, many canddae pahs wll be removed from he orgnal pah se. hen he scale of he problem s reduced. he more ghly he me delay bound, he smaller he canddae pah se s. For each d D ( =,,..., m, we frs compue all he pahs P( s, d sasfyng a delay consran ha s equal o or less han Δ. And he canddae pah se s arranged n ascng cos order. Fgure shows he process of searchng he canddae pah se. B. Encoder based on Ele sraegy In he vew of sasc, he Sener ree consss of pahs wh smaller cos n canddae pah se, and he fac can be found n boh Prm s and Kruskal s algorhms whch are based on he sraegy of greedy []. ha s, successve spannng ree wll have o jon he shores sde of he node ree. hen n a Sener ree, he cos of pah P (, s d,( =,,..., m s smaller, no necessary he smalles one n all he pahs. So s essenal o reduce he number of canddae pahs o speed up he convergence rae and o narrow he search space. hen we can assume ha a Sener ree mus conss of some shores pahs o her own desnaon. herefore, we encode he canddae pah wh ele sraegy. ha s, we collec he frs N pahs for every desnaon and make all he m N pahs o be an ele group. Here N denoes he number of qubs used o represen a canddae pah o a desnaon node. So he lengh of he qub ndvdual s D N, whch defned as α α... α... α α... α N m m mn q =, where β β... βn... βm βm... βmn α + β =,( =,,..., m, j =,,..., N. C. Observng operaor In order o evaluae he populaon wh fness value, observng operaor s a process of decdng he populaon by observng he amplude of each gene n he qub ndvdual populaon. ha s, n he ac of observng a quanum sae, collapses o a sngle sae (namely classcal represenaon. For example, we observe he qub ndvdual q a he -h generaon and produce bnary srngs soluon P ( = { x, x,..., x N,..., xm, xm,..., x mn}, where x ( =,,..., m, j =,,..., N s numerc srngs of lengh m N derved from he ampludeα or β ( =,,..., m, j =,,..., N. he process s descrbed as follows: (a generae a random number r [0,] ; β (b f s smaller han, he correspondng b n P ( akes 0, oherwse akes. And he specfc process s shown n Fgure. Procedure: begn j:=0 whle ( j < m N do begn j:=j+ f random[0,]< hen p = 0 else p = β Fgure. Observng operaor. hen we decode, for example, 0 denoe he 7h pah of he desnaon node. When he number of he pahs o a desnaon node s 7, boh 0 and denoe he 7h pah, and hen we evaluae mulcas ree cos. D. Quanum muaon In he quanum heory, he ransform of saes s fulflled by he quanum ransformaon marx. For example, he updang operaor can be denoed by he quanum roaon gaes. Also, he bes anbody n he curren generaon s mporan,.e., he communcaon among anbodes s needed o speed up he convergence. Quanum muaon focuses on a wse gudance by he curren bes anbody n subpopulaon. A quanum roaon gaeu s cos( Δθ sn( Δθ U ( Δ θ = sn( Δθ cos( Δθ, (9 where Δ θ s he roae angle whch conrols he convergence of QEA and Δθ s defned as Δ θ = δ s( α, β, (0 where δ s a coeffcen deermnng he speed of convergence. If he value of δ s oo small, he speed of convergence wll slow down, whle f he value of δ s oo large, dvergence or premaure convergence may happen. So a dynamc regulaon sraegy of roang angle s used, whch regulaes he value of δ beween 0.π and 0.005π accordng o he nherance generaon. he followng lookup able can be used as a sraegy for convergence. I should be ndcaed ha s a general way of muaon for dfferen problems. 544

4 x ABLE I. HE LOOK UP ABLE OF Δθ f( x > s( α, β b Δθ f( b αβ > 0 αβ < 0 α = 0 β = false rue false In able I, b and x ( =,, m N are he -h b of he bes soluon b and bnary soluon x respecvely, f(x s he fness of x, s( α, β s he sgn of he angle ha conrols he drecon of roaon. Why can converge o a beer soluon? Lookng no Fgure 3, you may ge some deas. 0 α β Fgure 3. Quanum Roaon gae. For example, f x = 0, b =, f(x>f(b. he probably of curren soluon x should be larger o ge a beer ndvdual,.e. β should be larger, so f ( α, β n he frs or hrd quadran, θ should roae clockwse or else roae counerclockwse. he updae procedure can be descrbed as + α α ( * + U θ,(,,..., N m β = = β ( E. Evaluaon of Fness Funcon Somemes we should also ake no accoun he number of edges ha may affec he cos of he mulcas ree. If we use as many publc pahs as we can, he cos could be decreased, as showed n Fgure 4.here are wo pah ( 4+5,6+6 from pon a o pon b and also here are wo pah ( 5+8,6+8 from pon a o pon c and ( 6+7,8+4 from pon a o pon d respecvely. I s obvously ha he cos of he pah conss of real lne s less han he doed lne. Bu he cos of mulcas ree ha conss of real lne s more han ha of doed lne. he wo mulcas ree coss are =9 and =7. he doed lne a-e denoes he publc pah. β 0 0 rue δ - + ± 0 0 false δ - + ± 0 0 rue δ ± 0 0 false δ ± 0 0 rue δ ± Fgure 4 shows he publc pah s used o decrease he cos of he mulcas ree. Fgure 4. An example for fndng he publc pah. herefore, he fness funcon s defned as: Fness = alpha 0.0 / MulreeCos +, ( (.0 - alpha.0 / reeedgescoun where alpha [,.0]. When he fness of offsprng s greaer han ha of parens, he offsprng Q ( wll be kep o lead he followng muaon. If he bes ndvdual fness has no been mproved afer successve 0 generaons, we wll randomly selec a non-zero b x ( = m N n bnary srngs soluon P ( and se o be 0. hen f has mproved he fness funcon, we sore he new x, oherwse we keep he non-zero x. A he same me, we updae q by roang gae. F. he ermnaon condon he ermnaon condon s he gven number of generaon. G. Compuaonal Complexy In hs secon, we only consder populaon sze n compuaonal complexy. Assumng ha he sze of nework nodes s n, he sze of desnaon nodes s m, he sze of canddae pah se s P, he sze of populaon s M, he maxmum sze of generaons s g and hen he me complexy for he algorhm can be calculaed as follows: he me complexy for searchng he canddae pah se s On ( ; he me complexy for sorng he canddae pah se s O(log( P ; he me complexy for fndng he opmal mulcas ree s Og ( M m. So he wors oal me complexy for he proposed algorhm s Og ( M m + O(log( P + On (. (3 If we correlae lnearly g and M wh n, and accordng o he operaonal rules of he symbol O, he wors me complexy for he proposed algorhm can be smplfed as On ( m (4 3 Bu he wors me complexy for BSMA s On ( log n [], so he me complexy for QEA used o solve mulcas roung problem s superor o hose of BSMA. 545

5 V. CONVERGENCE OF HE ALGORIHM n Defnon 3 le X ( = ( x(, x(,, x n ( n S be he populaon a me and for X, ( defned: M = { X f( X = max{ f( X (, n}}, (5 n M = { X f( X = max{ f( X, X S }}. (6 M s called he sasfed se of populaon X and M s defned as he global sasfed se of sae S n. Defnon 4 supposes arbrary nal dsrbuon, he followng equaon sasfes: lm PM { M } hen we call he algorhm s convergen. =. (7 heorem he populaon seres of QEA { Q, 0} s fne homogeneous Markov chan. Proof: Lke he evoluonary algorhms, he sae ransfer of QEA are processed on he fne space, herefore, populaon s fne, snce Q ( + = Q ( ( = s u Q (. (8 s and u ndcae he selecon operaor and he updae operaor respecvely. Noe ha s and u have no relaon wh, so Q ( + only relaes wh Q (. Namely, { Q, 0} s fne homogeneous Markov chan. heorem he M of Markov chan of QEA s monoone ncreasng, namely, 0, f( Q+ f( Q. Proof: Apparenly, he ndvdual of QEA does no degenerae for our adopng holdng bes sraegy n he algorhm. heorem 3 he quanum-nspred evoluonary algorhm (QEA s convergen. Proof: For heorem and heorem, he QEA s convergen wh he probably. VI. PERFORMANCE COMPARISON In hs secon, we compare convenonal GA and CS [8] wh QEA n solvng he mulcas roung problems ha were frequenly used as benchmark problems. he neworks wh dfferen scales we used n our expermens are produced by he Waxman model []. For each random nework, he coordnae area s 4000km 4000km, he mean degree of a nodes s 4. he delay of lnk (x,y s defned as d(x, y = 000 ds(x, y/c, n whch ds(x, y s he dsance beween x and y, c s he lgh speed. Delay consran =00. We apply he followng merc o evaluae he hree algorhms performance, where s he number of ndepen runs, Cos s he solved opmal mulcas ree cos and me s he runnng me for each algorhm. Obvously, he smaller R s, he beer he performance of QEA s, compared wh GA and CS. R R cos = = me = = Cos( QEA, Cos( (GAor(CS me( QEA. me( (GAor(CS (9 In he followng expermens, we performed 30 ndepen runs on each es problem. All expermens are execued on a.33ghz Penum IV PC wh G RAM by programmng wh C++. For GA, we use a populaon of sze 0, pc = and p m = 0.0. For CS, he populaon sze s 0, he clone sze s 0 and p m =. For QEA, he sze of populaon M = 0, where N denoes he number of qubs used o represen a desnaon node n he canddae pah. Comparson beween QEA and anoher wo algorhms- GA and CS wh nework scale ncreased. o make a far comparson, he number of generaons s kep 00 when m/n= (m s he number of desnaon nodes, and n s he scale of he nework for all he hree algorhms n Fgure 5 o Fgure 8. he nework scale s ncreased from 00 o 000 and sampled a nervals of 00 n Fgure 5 o Fgure 8. Rao of Mulcas ree Cos N=3,k=0 R(QEA/GA R(QEA/CS Nework Scale Fgure 5. Performances of mulcas cos versus nework scale wh N=3 and k=0,where k represens ha n boh GA algorhm and CS algorhm ha we only ake he frs k% of he canddae pah se for each desnaon node. Rao of Mulcas ree Cos N=,k=0 R(QEA/GA R(QEA/CS Nework Scale Fgure 6. Performances of mulcas cos versus nework scale wh N= and k=0,where k represens ha n boh GA algorhm and CS algorhm ha we only ake he frs k% of he canddae pah se for each desnaon node. R s less han and s sable under he dfferen parameers seng n Fgure 5 and Fgure 6. I s shown ha QEA s beer expansbly and s beer performance a complex and massve problems compared wh GA and CS. 546

6 Rao of me Cos N=3,k=0 R(QEA/GA R(QEA/CS populaon o accelerae convergence of he proposed algorhm. As a resul, he proposed QEA has he auomac balance ably beween exploraon and exploaon. Furhermore, we appled QEA o solvng he mulcas roung problems. When compared wh convenonal GA and CS, QEA s more effecve n erms of convergence and dversy wh he scale of nework ncreased. N=.k= Nework Scale Fgure 7. Performances of me cos versus nework scale wh N=3 and k=0 where k represens ha n boh GA algorhm and CS algorhm ha we only ake he frs k% of he canddae pah se for each desnaon node. Rao of me Cos N=,k=0 R(QEA/GA R(QEA/CS Nework Scale Fgure 8. Performances of me cos versus nework scale wh N= and k=0 where k represens ha n boh GA algorhm and CS algorhm ha we only ake he frs k% of he canddae pah se for each desnaon node. Fgure 7 and Fgure 8 show ha he percenage of me cos of QEA o ha of CS and GA s beween 30 and 70. QEA makes full use of he parallelsm of quanum compung and ses few parameers, compared wh GA and CS. hereby he ulzaon of memory s mproved. Comparson beween QEA and anoher wo algorhms- GA and CS wh he number of desnaon nodes ncreased. We use he Waxman model [] produce a 000 node nework o nspec algorhm s performance a dfferen number of desnaon nodes. he percenage of desnaon nodes o nework nodes s ncreased from 5 o 50 and s sampled a nervals of 5. Fgure 9 shows R s less han under dfferen number of he desnaon nodes. Wh ncreased of desnaon nodes, he rao of mulcas ree cos s declne. ha also ndcaes QEA has sronger search capably han CS and GA. VII. CONCLUSIONS In hs paper, we proposed a new opmzaon algorhmquanum-nspred evoluonary algorhm (QEA o solve he mulcas roung problems, nspred by he concep of quanum compung. In parcular, a qub ndvdual was defned as a srng of quanum bs for he probablsc represenaon, so can mprove dversy of he populaon and avodng premaure convergence. Due o he novel represenaon, we pu forward he quanum muaon operaor whch was used n he Rao of Mulcas ree Cos R(QEA/GA R(QEA/CS Percenage of he Desnaon Nodes o he Nework Nodes Fgure 9. Percenage of QEA o GA and CS versus number of desnaon nodes wh N= and k=0 where k represens ha n boh GA algorhm and CS algorhm ha we only ake he frs k% of he canddae pah se for each desnaon node. REFERENCES [] B.M. Waxman, Roung of Mulpon Connecons, IEEE Journal of Seleced Areas n Communcaons, 6(9:67-6, 988. [] M. Parsa, Q. Zhu, Garca-Luna-Aceves JJ. An Ierave Algorhm for Delay-Consraned Mnmum-Cos Mulcasng, IEEE/ACM rans on Neworkng, 6(4: , 998. [3] V.P. Kompella, J.C. Pasquale, G.C. Polyzos, Mulcasng for Mulmeda Applcaons, In: Elevenh Annual Jon Conference of he IEEE Compuer and Communcaons Socees. Pscaaway, NJ, USA: IEEE Press, , 99. [4] Q. Sun, H. Langorfer, Effcen Mulcas Roung for Delay- Sensve Applcaons, In: Proceedngs of he h Workshop on Proocols for Mulmeda Sysems, 4-458, 995. [5] Q.F. Zhang; Y.W. Leung, An Orhogonal Genec Algorhm for Mulmeda Mulcas Roung, IEEE ransacons on Evoluonary Compuaon, 3(: 53-6, 999. [6] A.. Haghgha, K. Faez, M. Dehghan, A. Mowlae, Y. Ghahreman, GA-Based Heursc Algorhms for Bandwdh-Delay-Consraned Leas-Cos Mulcas Roung, Compuer Communcaon, 7(:- 7, 004. [7] Y. Q, L.C. Jao, F. Lu, Mul-Agen Immune Memory Clone Based Mulcas Roung, Chnese of Journal Elecroncs, 7(:89-9, 008. [8] F. Lu, H. C. Yang, A Mulcas Roung Algorhm Based on Clonal Sraeges, Journal of elecroncs and nformaon echnology, 6(: 85-89, 004. (n Chnese [9] L.C. Jao, Y.Y. L, M.G. Gong and X.R. Zhang, Quanum-nspred mmune clonal algorhm for global opmzaon, IEEE ransacons on Sysem, Man, and Cybernecs, Par B, 008, 38(5: [0] K.-H. Han and J.-H. Km, Quanum-Inspred Evoluonary Algorhms Wh a New ermnaon Creron, H ε Gae, and wo-phase Scheme, IEEE rans. Evol. Compu., vol. 8, pp , Apr