GA-based heuristic algorithms for bandwidth-delayconstrained least-cost multicast routing

Transcription

1 GA-base heurstc algorthms for banwth-elayconstrane least-cost multcast routng A. T. Haghghat ***, K. Faez *, M. Dehghan **, A. Mowlae *, Y. Ghahreman * Dept. of Electrcal Engneerng, Amrkabr Unversty of Technology, Tehran 15914, Iran ** Iran Telecommuncaton research Center (ITRC), Tehran, Iran *** Atomc Energy Organzaton of Iran (AEOI), Tehran, Iran Tel: Fax: Emal :kfaez@atu.ac.r Abstract The banwth-elay-constrane least-cost multcast routng s a challengng problem n hgh-spee multmea networks. Computng such a constrane Stener tree s an NP-complete problem. In ths paper, we propose a novel QoS-base multcast routng algorthm base on the genetc algorthms (GA). In the propose metho, the connectvty matrx of eges s use for genotype representaton. Some novel heurstc algorthms are also propose for mutaton, crossover, an creaton of ranom nvuals. We evaluate the performance an effcency of the propose GA-base algorthm n comparson wth other exstng heurstc an GA-base algorthms by the result of smulaton. Ths propose algorthm has overcome all of the prevous algorthms n the lteratures. Keywors: Qualty of Servce, Multcast routng, Mult-constrane Stener tree, Genetc algorthm 1. Introucton Recently, avances n mea technologes such as optcal fber an swtch technologes such as ATM an MPLS have resulte n a new generaton of ggabt-per-secon we area networks. These networks are expecte to support a we range of communcaton-ntensve real-tme multmea applcatons lke gtal auo an veo. The eployment of hgh-spee networks opens a new menson of research, provng qualty of servce (QoS) such as guarantee throughput for veo-on-eman applcaton, low en-to-en elay for veo conferencng, less than 200 ms en-to-en elay an low cell loss rato for real-tme auo applcaton an hgh transmsson relablty for strbute control applcatons. It s techncally a challengng an complcate problem to elver multmea nformaton n a tmely, smooth, synchronze manner over a ecentralze, share network envronment, especally one that was orgnally esgne for best-effort traffc such as Internet. In the past, most of the applcatons were uncast n nature an none of them ha any QoS requrements. Therefore, the routng algorthms were very smple. However, wth emergng strbute real-tme multmea applcatons such as veo conferencng, stance learnng, an veo on eman, the stuaton s completely fferent now. These applcatons wll nvolve multple user wth ther own fferent QoS requrements n terms of throughput, relablty, an bouns on en-to-en elay, jtter, an packet loss rato. Accorngly, a key ssue n the esgn of broa-ban archtectures s how to effcently manage the resources n orer to meet the QoS requrements of each connecton. The establshment of effcent QoS routng schemes unoubtely, one of the major bulng blocks n such archtectures. Supportng pont to mult-pont connectons for multmea applcatons requres the evelopment of effcent multcast routng algorthms. Multcast employs a tree structure of the network to effcently elver the same ata stream to a group of recevers. In multcast routng, one or more constrants must be apple to the entre tree. Several well-known multcast routng problems have been stue n the lteratures. The Stener tree problem [1] tres to fn the least-cost tree, the tree coverng a group of estnatons wth the mnmum total cost over all the lnks. It s also calle the least-cost multcast routng problem, belongng to the class of tree-optmzaton problems. Fnng ether a Stener tree or a constrane Stener tree s NP (Nonetermnstc Polynomal)-complete [2]. In ths paper, we conser a banwth-elay-constrane leastcost multcast routng. For the purpose of clarty, n ths paper we assume an envronment where a source noe s presente wth a request to establsh a new least-cost tree wth two constrane: banwth constrant n all the lnks of the tree an en-to-en elay constrant from the source noe to each of the Ths work was supporte by ITRC uner grant number

2 estnatons. In other wor we conser the source routng strategy, n whch each noe mantans the complete global state of the network, nclung the network topology an state nformaton of each lnk. Most of the propose algorthms for Stener tree (wthout constrant) are heurstc. Some of the well-known Stener tree heurstcs are the RS heurstc [8], the TM heurstc [9], an the KMB heurstc [7]. Several algorthms base on neural networks [10] an genetc algorthms (GA) [23-27] have been also propose for solvng ths problem. Recently, a lot of elay-constrane least-cost multcast routng heurstcs such as the KPP heurstc [4], the BSMA heurstc [3] an so on ([5], [6], an [11]) have been propose. However, the smulaton results gven by Salama et al. [17] have shown that most of the heurstc algorthms ether work too slowly or can not compute elay-constrane multcast tree wth least cost. The best etermnstc elay constrant lowcost (near optmal) algorthm s BSMA ([17], [28], [32]). Note that the above algorthms have esgne specfcally for real-tme applcatons wth only one QoS constrant wthout mentonng how to exten these algorthms to real-tme applcatons wth two or more QoS constrants. Snce etermnstc heurstc algorthms for QoS multcast routng are usually very slow, methos base on computatonal ntellgence such as neural networks an genetc algorthms may be more sutable. Chotpat et al. [18] have been propose an algorthm base on Hopfel neural network to solve QoS multcast routng. However, the selecton of the coeffcents n energy (or Lyapunov) functon s complex an sometmes may lea to unexpecte wrong soluton. In aton, because of usng a contnuous Hopfel neural network, the QoS routng solutons must be assume to be contnuou whch makes the problem more complex. In the fel of computatonal ntellgence, GA-base algorthms have emerge as powerful tools for solvng NP-complete constrane optmzaton problems. Several GA-base algorthms [23-27] have been propose for solvng Stener tree problem wthout QoS constrants. Also, Sun [28] has extene the algorthm propose n [26] for the least-cost multcast routng problem wth one QoS constrant (elay). For eployng the genotype encong use n [26], [28], another NP-complete sub-problem (a etermnstc elay-constrane least-cost multcast routng algorthm, CKMB [11]) must be solve urng the econg phase. Furthermore, the algorthm assumes the same elay constrants for all estnaton whch greatly restrcts ts applcaton. However, the smulaton results gven by Sun have shown that hs algorthm can acheve trees wth smaller average cost than those of BSMA, n a shorter runnng tme for relatvely large networks. Xang et al. [29] have propose a GA-base algorthm for QoS routng n general case. Ths algorthm aopts an N * N one-mensonal bnary encong scheme, where N represents the number of noes n the graph. However, n ths encong scheme, the transformaton back an forth between genotype an phenotype space s very complcate, especally for large networks. Ravkumar et al. [30] have propose a GA-base algorthm wth novel nterestng approaches for crossover an mutaton operators for the elay-constrane least-cost multcast routng problem. However, they have not efne ther scheme for encong an econg of nvuals. Snce ther algorthm may lea to premature convergence, an approach must be esgne to prevent ths phenomenon [33]. Zhang et al. [31] have propose an effectve orthogonal GA for elay-constrane least-cost multcast routng problem. Ths algorthm also assumes the elay constrants for all estnatons are entcal. Also, Wu et al. [32] have propose a GA-base algorthm for multple QoS constrants multcast routng problem n general case. However, ther propose genotype representaton oes not necessarly represent a tree. On the other han, It s necessary to construct an store a very large amount of possble routes for each pars of noes n the graph usng the K-shortest path algorthm. Wang et al. [33] have propose an effcent GA-base heurstc algorthm for banwthelay-constrane least-cost multcast routng problem. They have use a tree ata structure for genotype representaton, but not clearly efne ther encong an econg schemes. In ths paper, we propose a novel QoS-base multcast routng algorthm base on genetc algorthms (GA). In the propose metho, the connectvty matrx of eges s use for genotype representaton. Some novel heurstc algorthms are also propose for mutaton, crossover, an creaton of ranom nvuals. We evaluate the performance an effcency of the propose GA-base algorthm n comparson wth other exstng heurstc an GA-base algorthms by the result of smulaton. Ths propose algorthm has overcome all of the prevous algorthms n the lteratures. The remaner of ths paper s organze as follows. The problem escrpton an formulaton s gven n secton 2. In Secton 3, we escrbe the propose algorthms. We then evaluate the convergence of the propose GA-base algorthms n Secton 4. Secton 5, gves the performance evaluaton of the propose algorthms an the comparson of them wth other smlar algorthms. Secton 6 conclues ths stuy an scusses future works.

3 2. Problem escrpton an formulaton A network s moele as a recte, connecte graph G = (V, E), where V s a fnte set of vertces (network noes) an E s the set of eges (network lnks) representng connecton of these vertces. Let n = V be the number of network noes an l = E be the number of network lnks. The lnk e = (u, v) from noe u V to noe v V mples the exstence of a lnk e = (v, u) from noe v to noe u. Three non-negatve real value functons are assocate wth each lnk e (e E): cost C(e):E R +, elay D(e):E R +, an avalable banwth B(e):E R +. The lnk cost functon, C(e), may be ether monetary cost or any measure of the resource utlzaton, whch must be optmze. The lnk elay, D(e), s consere to be the sum of swtchng, queung, transmsson, an propagaton elays. The lnk banwth, B(e), s the resual banwth of the physcal or logcal lnk. The lnk elay an banwth functon D(e) an B(e), efne the crtera that must be constrane (boune). Because of the asymmetrc nature of the communcaton network t s often the case that C(e) C(e ), D(e) D(e ), an B(e) B(e ). A multcast tree T( M) s a sub-graph of G spannng the source noe s V an the set of estnaton noes M V-{s}. Let m = M be the number of multcast estnaton noes. We refer to M as the estnaton group an {s} M the multcast group. In aton, T( M) may contan relay noes (Stener noes), that the noes n the multcast tree but not n the multcast group. Let P T ( ) be a unque path n the tree T from the source noe s to a estnaton noe M. The total cost of the tree T( M) s efne as the sum of the cost of all lnks n that tree an can be gven by C( T ( M )) = C( e) The total elay of the path P T ( ) s smply the sum of the elay of all lnks along P T ( ): D( PT( )) = D( e) e PT ( ) The bottleneck banwth of the path P T ( ) s efne as the mnmum avalable resual banwth at any lnk along the path: B( PT( )) = mn{ B( e), e PT( )} Let be the elay constrant an Β the banwth constrant of the estnaton noe. The banwthelay-constrane least-cost multcast problem s efne as mnmzaton of C(T( M)) subject to 3. The propose GA-base algorthms Genetc algorthm as powerful an broaly applcable stochastc search an optmzaton technque are the most wely known types of evolutonary computaton methos toay. In general, a genetc algorthm has fve basc components as follows: 1) An encong metho, that s a genetc representaton (genotype) of solutons to the program. 2) A way to create an ntal populaton of nvuals (chromosomes). 3) An evaluaton functon, ratng solutons n terms of ther ftness an a selecton mechansm. 4) The genetc operators (crossover an mutaton) that alter the genetc composton of offsprng urng reproucton. 5) Values for the parameters of genetc algorthm. A general structure of the genetc algorthms s as follows: Proceure: Genetc Algorthms t := 0; ntalze P(t); {P(t) s the populaton of nvuals n generaton t} evaluate P(t); Whle (not termnaton conton) o recombne P(t) to yel C(t); e T ( M ) D( PT( )), M B( PT( )) Β, M {creaton of offsprng C(t) by means of genetc operators} evaluate C(t); select P(t + 1) from P(t) an C(t); t := t + 1; Fgure 1: General structure of the genetc algorthms

4 3.1. Genotype Let us efne the connectvty matrx of ege Y n*n, such that the value of each element (Y[, j] {0, 1}) tells whether or not a specfc ege connects the par of noes (, j). For convertng the connectvty matrx Y nto a one-mensonal chromosome x, whch conssts of n*(n-1)/2 element we shoul transfer the elements on the top trangle of matrx Y, from the frst row an from left to rght nto the chromosome x as ncate n the Fgure 2. Although, we conser that the network s asymmetrc, It s not necessary to use all elements of the connectvty matrx of eges to represent the Stener tree. In other wor, the top trangle of the connectvty matrx of eges s suffcent to represent the Stener tree. Note that f x[k] = Y[, j], then the nex k s represente as a functon of, j by the followng equaton: n( n 1) ( n )( n + 1) k = + ( j ) 2 2 Proceure: Connectvty matrx econg For I := 1 to n o For j := + 1 to n o Temp[, j] := Y[, j]; Current-vertex := s; {Select the source noe s (root vertex) as the current vertex} k := 1; A s to the k-th (frst) path-lst; Whle (there s any tem equal to one n Temp matrx) o {Check for a successor vertex to the Current-vertex from left to rght} v := 0; For := 1 to Current-vertex - 1 o If (Temp[, Current-vertex] = 1) then v := ; Temp[, Current-vertex] := 0; ext; If (v = 0) then For := Current-vertex + 1 to n o If (Temp[Current-vertex, ] = 1) then v := ; Temp[Current-vertex, ] := 0; ext; If (v 0) then A v to the k-th path-lst; Degree := 0; For := 1 to v - 1 o Degree := Degree + Y[, v]; For := v+1 to n o Degree := Degree + Y[v, ]; If (Degree = 1) then {f v s a leaf } Copy k-th path-lst to (k+1)-th path-lst; k := k+1; Remove the last tem of k-th path-lst; Current-vertex := v; If (Current-vertex s not s) then Remove the last tem of the k-th lnk lst; Current-vertex := Preecessor vertex; {the last tem of k-th lnk lst} k := k -1; {Remove the last lnk lst} Fgure 2: Connectvty matrx econg algorthm

5 3.2. Pre-Processng Phase Before startng the genetc algorthm, we can remove all the lnk whch ther banwth are less than the mnmum of all requre threshols (Mn {B M}). If n the refne graph, the source noe an all the estnaton noes are not n a connecte sub-graph, ths topology oes not meet the banwth constrant. In ths case, the source shoul negotate wth the relate applcaton to relax the banwth boun. On the other han, f the source noe an all the estnaton noes are n a connecte sub-graph, we wll use ths sub-graph as the network topology n our GA-base algorthms Intal populaton The creaton of the ntal populaton n ths stuy s base on the ranomze epth-frst search algorthm [30],[33]. We propose a mofe ranomze epth-frst search algorthms for ths purpose: Ranom nvual creaton algorthm: In ths algorthm, a lnke lst s constructe from the source noe s to one of the estnaton noes. Then, the algorthm contnues from one of the unvste estnatons an at each noe the next unvste noe s ranomly selecte untl one of the noes n the prevous sub-tree (the tree that s constructe n the prevous step) s vste. The algorthm termnates when all estnaton noes have been mounte to the tree. The proceure of creaton the ntal populaton has been shown n Fgure 3. Ths proceure must be calle pop-sze tmes to create the total of ntal populaton. Proceure: ranom nvual creaton n := 1; Frst := True; Whle (n<=number of Destnatons) o Intalze the n-th lnk lst; If (Frst) then Current-noe := Source Current-noe := One of unvste Destnatons; G TM := Temporary matrx of the network graph; A the Current-noe to the n-th lnk lst; Lnk-lst-comp := False; Whle (Not Lnk-lst-comp) o k := Number of connecte noes to the Current-noe n G TM; If (k=0) then Remove the Current-noe n the n-th lnk lst; Remove the lnk between the Current-noe an the prevous noe n G ol; Current-noe := prevous noe n the n-th lnk lst; G TM := G ol := a ranom natural number n nterval [1,k]; A the -th noe to the n-th lnk lst; G ol := G TM; Remove all lnks to the Current-noe n G TM; Current-noe := the -th noe; If (Frst) then If (Current-noe s one of the estnatons) then Lnk-lst-comp := True; Make an nvual by n-th lnk lst; n := n+1; Frst := False; Mark the foun estnaton as a vste estnaton If (the Current-noe s a noe n one of the prevous lnk lsts(for example j-th lnk lst)) then { f the Current-noe has a connecton to the source noe, ths lnk has hgher prorty} n-th lnk lst := j-th lnk lst from the source noe to foun poston + Inverse (n_th lnk lst); Lnk-lst-comp := True;

6 A the n-th lnk lst to the nvual; n := n+1; Mark ths estnaton as a vste estnaton {} {nner whle} {outer whle} {proceure} Fgure 3: A mofe epth-frst search algorthm to create a ranom nvual 3.4. Ftness functon The ftness functon n our stuy s an mprove verson of the scheme propose n [33]. We efne the ftness functon for each nvual, the tree T( M), usng the penalty technque, as follows: F( T ( M )) = 1 φ( z) = γ α C( e) e T ( M ) z 0 z > 0 Where α s a postve real coeffcent, φ(z) s the penalty functon an γ s the egree of penalty (γ s consere equal to 0.5 n our stuy). Wang et al. [33] have assume that the banwth constrants (B ) for all estnatons are entcal Selecton The selecton process use here s base on spnnng the roulette wheel pop-sze tme an each tme a sngle chromosome s selecte as a new offsprng. The probablty P that a parent T s selecte s gven by: Where F(T ) s the ftness of the T nvual. M p φ( D( P( )) F( T ) = pop sze F( j= Crossover Several crossover operators are escrbe n the lteratures [23-33] for Stener tree an constrane Stener tree problems. Some of them have use the tratonal well-known crossover operator such as the followng schemes: One pont crossover operator (e. g. see [28]) One pont crossover operator, wth a fxe probablty P c ( ) (e. g. see [27]) Two pont crossover operator (e. g. see [32]) One pont crossover operator plus "an" an "or" logc operatons wth a fxe probablty P c (see [29]) Unfortunately, accorng to the genotype representaton n these paper the above crossover operators are not sutable for recombnaton of two nvuals (the crossover operaton mostly leas to llegal nvuals). However, Ravkumar et al. [30] have propose a new nterestng approach for crossover of Stener trees an Wang et al. [33] have use the same scheme wth some mofcatons. In ths scheme, two multcast tree T F ( M) an T M ( M), are selecte as parents an the crossover operaton prouces an offsprng T O ( M) by entfyng the lnks that are common to both parents. The operator selects the same lnks of two parents for qucker convergence of the genetc algorthm. However, these common lnks may be n some separate sub-tree an some eges may have to be ae n orer to transform them nto a multcast tree. In ths step, a multcast tree s constructe from these separate sub-trees. Frst, two separate sub-trees among these sub-trees are ranomly selecte, an are connecte them wth the least-elay or the least-cost path (n [30] all sub-trees are connecte to the frst sub-tree). If none of the parents satsfes the elay constrant, the least-elay path s chosen. Otherwse the least-cost path s chosen (n [30] ths conton s checke for all nvuals n the populaton). The path, whch s ae to jon two sub-trees s selecte heurstcally. The two connecte sub-trees s replace wth the new sub-tree n the sub-trees set. Next, conformng to the same rule, a new selecton begns agan. The selecton T j ) ) M φ( B( P( )) B )

7 repeats untl a multcast tree s constructe. Clearly there s no loop n the multcast tree constructe by ths connecton scheme. Fnally, t may be possble that some of the leaf noes of T O are not the source noe or estnaton noes. These noes are elete from the offsprng. However, the frst savantage of ths scheme s the complexty of the heurstc algorthm, whch selects a path to jon the two separate sub-trees. The secon savantage of ths scheme s that the result of ths complex heurstc algorthm s not necessarly a multcast tree nclung the source noe an all estnaton noes. We propose two novel crossover schemes for recombnaton of two nvual whch represent Stener trees: Crossover I: Let {P F ( 1 ), P F ( 2 ),, P F ( m )} be the set of paths from the source noe s to all estnaton noes n T F an {P M ( 1 ), P M ( 2 ),, P M ( m )} be the same set n T M. Snce, we have foun these paths for all nvuals n the current populaton for calculatng the ftness functon of them, the propose algorthm wll not be complex. We efne a ftness functon for the path P( ) base on the total cost, the total elay an the mnmum banwth of the path usng the penalty technque, as follows: α F( P( )) = φ( D( P( C( e) e P ( ) 1 z 0 φ( z) = γ z > 0 )) ) φ( B( P( )) B Where α s a postve real coeffcent, φ(z) s the penalty functon an γ s the egree of penalty (γ s consere equal to 0.5 n our stuy). Accorng to the crossover probablty of P c, two multcast trees T F ( M) an T M ( M) are selecte as parents an the crossover operaton prouce an offsprng T O ( M). Each nvual may be recombne wth ts rght nvual an ts left nvual through the crossover operator. For each estnaton noe, we compute the ftness of P M ( ) an P F ( ) an select the better path. Fnally, we compose all selecte paths an construct a new Stener tree (see Fgure 4). ) Proceure: The crossover operator For :=1 to m o { m s the number of estnaton noes} If F(P M( )) > F(P F( )) then P O( ) := P M( ) P O( ) := P F( ); Current-tree := P O( 1); For :=2 to m o Prevous-noe := s; Start-noe := s; Current-noe := The secon noe n the P O( ); New-lnk := False; Whle (Prevous-noe <> ) o If the Current-noe oes not exst n the current-tree then A the lnk between the Current-noe an the Prevous noe to the current-tree; New-lnk := True; If the New-lnk = True then Remove all lnk from Start-noe to the Prevous-noe n P O( ) n the current-tree; Start-noe := Current-noe New-lnk := False; Prevous-noe := Current-noe; If there s another noe n P O( ) then Current-noe := the next noe n the P O( ) Fgure 4: The propose heurstc crossover I operator

8 Crossover II: In ths scheme, we frst use a smple one-pont crossover operator, wth a fxe probablty P c. The constructe offsprng o not necessarly represent Stener trees. Then, the effectve an fast check an recovery algorthm propose n [31] s use to connect the separate sub-trees n the offsprng an also connectng the absent noes of multcast group to the fnal tree Mutaton Many of propose GA-base algorthms for multcast routng such as [27], [28], [29], an [32] have use the bt-flp mutaton wth a fxe small probablty P m ( ). Unfortunately, accorng to the genotype representaton n these paper the bt mutaton generates llegal nvuals an ecreases the performance of them. However, Ravkumar et al. [30] have propose a new scheme for mutaton of Stener trees an Wang et al. [33] have use the mprove verson of t n ther stuy. In ths scheme [33], accorng to the mutaton probablty P m, the mutaton proceure ranomly selects a subset of noes an breaks the multcast tree nto some separate sub-trees by removng all the lnks that are ncent to the selecte noes. Then, t re-connects those separate sub-trees nto a new multcast tree by ranomly selectng the least-elay or the least-cost paths between them. However, the result of ths complex heurstc algorthm s not necessarly a multcast tree nclung the source noe an all estnaton noes. In ths paper, we propose two followng algorthms for mutaton operator: Mutaton I: Frst, we propose an mprove verson of the scheme presente n [33]. The mutaton proceure ranomly selects a subset of noes an breaks the multcast tree nto some separate sub-trees by removng all the lnks that are ncent to the selecte noes. Then, the effectve an fast check an recovery algorthm propose n [31] s use to connect the separate sub-trees an also connectng the absent noes of multcast group to the fnal tree. Mutaton II: Accorng to the mutaton probablty P m, the mutaton proceure ranomly selects an nfeasble chromosome from one of the followng class (If the frst class s empty, a chromosome s selecte from the secon class an so on) Class 1: The chromosome whch o not satsfy the elay an the banwth constrants. Class 2: The chromosome whch o not satsfy the elay constrant. Class 3: The chromosome whch o not satsfy the banwth constrant. If all chromosomes n the current populaton satsfy both of the QoS constrant we ext from the mutaton proceure. Then, we select only the paths that satsfy both of the QoS constrants n the selecte chromosome. We re-connect these selecte paths by our propose algorthm of crossover I (see Fgure 4). Fnally, the sconnecte estnaton noes wll be mounte to the sub-tree by our propose algorthm of ranom nvual creaton (see Fgure 3) Illegalty an Infeasblty The chromosomes generate ranomly n the ntal populaton an the offsprng prouce by the mutaton an crossover operators may be llegal or nfeasble. Illegalty refers to the phenomenon that a chromosome oes not represent a multcast tree; Infeasblty refers to the phenomenon that a chromosome, whch represents a multcast tree, oes not satsfy the problem constrants. Three strateges have been propose to eal wth these volatons: Rejectng strategy Penalzng strategy Reparng strategy The penalty methos are mostly use to hanle nfeasble chromosomes [36]. We have use ths strategy n our propose ftness functon. It s really ffcult to prove a reasonable penalzng factor for the llegal chromosomes n our stuy, because the llegalty can not be easly measure quanttatvely. The repar strategy oes nee surpass other stratege such as the rejectng or the penalzng stratege n ths case. We have use ths strategy n our propose mutaton I an crossover II algorthms. On the other han, we have propose another strategy to eal wth the llegalty problem. We wll refer to ths strategy as the avoance strategy. In ths paper, most of the propose algorthm such as the ntal populaton creaton algorthm, the crossover I algorthm, an the mutaton II algorthm, have been use ths strategy to avo creatng llegal nvuals.

9 4. Analyss of convergence Accorng the Theorem 2.7 n Ref. [34], the GA-base algorthms propose n ths paper coul fnally converge to the global optmal soluton. For a large-scale network, t s tme-consumng to obtan the optmal soluton to the banwth-elay-constrane least-cost multcast routng problem, whch s NPcomplete. Ths problem can be overcome by settng an approprate teraton tme of the genetc algorthm. In ths way, we can obtan a near-optmal soluton wthn a reasonable tme lmt. 5. Expermental Results In ths secton, we have use the smulaton experments to compare the performance of the propose GAbase algorthms wth the heurstc BSMA heurstc algorthm an some exstng GA-base algorthms. We have mplemente more than 2,000 lnes C++ program to smulate all of the propose algorthms. All smulaton experments are run on a Pentum III 800, 256 MB RAM, IBM PC. The experments are run repeately untl confence nterval of less than 5%, usng 95% confence level, are acheve for the smulaton results. A ranom graph generator base on the Salama [17] graph generator s use. The average egree of each noe n the ranom generate graphs s 4. The multcast group s ranomly selecte n the graph. The sze of multcast group s consere 5%, 15%, an 25% of the number of network noes. We have tune the propose GA-base algorthms an the followng parameter settngs are acheve: populaton sze pop-sze = 20, crossover probablty P c = 0.4 for crossover I, crossover probablty P c = 0.4 for crossover II, mutaton probablty P m = 0.01 for mutaton I, an mutaton probablty P m = 0.01 for mutaton II. The experments manly test the convergence ablty, the convergence spee, an the tree cost of the acheve solutons. Fgure 5, an 6 show the percentage tree cost of BSMA [3], Sun GA-base algorthm [28], an Wang GAbase heurstc algorthm [33] n comparson wth our propose GA-base heurstc algorthm for fferent network szes an fferent multcast group szes. These Fgures show that our propose GA-base heurstc algorthm can result n a smaller average tree cost than the mentone exstng algorthms. Fgure 7 shows a typcal example of the executon tme of our propose GA-base heurstc algorthm n comparson wth the mentone exstng algorthms. Ths Fgure shows that our propose GA-base heurstc algorthm can result n a smaller executon tme than the mentone exstng algorthms. Percentage excess cost ov er the propose GA-base algorthm BSMA Sun-GA Wang-GA Number of netw ork noes Fgure 5. Percentage excess cost over the propose GA-base algorthm versus number of network noe (Multcast group sze s 5% of the number of network noes)

10 Percentage excess cost ov er the propose GA-base algorthm BSMA Sun-GA Wang-GA Number of netw ork noes Fgure 6. Percentage excess cost over the propose GA-base algorthm versus number of network noe (Multcast group sze s 30% of the number of network noes) Executon tme (Secons) BSMA Sun-GA Wang-GA Propose-GA Number of netw orks noe Fgure 7. Executon tme of the propose algorthm n comparson wth other exstng algorthm 6. Conclusons In ths stuy, we have propose a GA-base heurstc algorthm to solve the banwth-elay-constrane least-cost multcast routng problem whch s known to be NP-complete. We have propose connectvty matrx of eges for representaton of the Stener trees. In our stuy, the followng new algorthms have been propose to ncrease the performance of the genetc algorthm: An algorthm for creaton of a ranom nvual: ranom nvual creaton Two heurstc algorthms for mutaton operator: mutaton I, II Two heurstc algorthms for crossover operator: crossover I, II We have use the penalzng strategy n the propose ftness functon to eal wth the nfeasble chromosomes an also the reparng strategy n the mutaton I an crossover II algorthms to eal wth the llegal chromosomes. On the other han, we have propose the avoance strategy to avo of creatng llegal chromosomes n the crossover I, mutaton II, an ranom nvual creaton algorthms. We have mplemente more than 3,000 lnes C++ program to smulate all of the propose algorthms. The smulaton results have shown that the propose GA-base algorthm has overcome all of the prevous algorthms n the lteratures. In ths stuy, we have focuse on the source routng an the future work shoul focus on mechansms to apply the propose algorthms to the herarchcal routng. References [1] S. L. Hakm, "Stener problem n graphs an ts mplcaton" Network Vol. 1, pp , [2] R. Karp, "Reucblty among combnatoral problem" n: R. E. Mller, J. W. Thatcher, "Complexty of computer computaton" Plenum Pres New York, pp , 1972.

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