ICRA: Incremental Cycle Reduction Algorithm for optimizing multi-constrained multicast routing

Transcription

1 ICRA: Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng Nauel Ben Al HANA Reseach Gup, ENI, Unvesty f Manuba, Tunsa nauel.benal@ens.nu.tn Mkls Mlna INA, Unvesty f Rennes 1, Fance mlna@sa.f Abdelfettah Belghth HANA Reseach Gup, ENI, Unvesty f Manuba, Tunsa abdelfattah.belghth@ens.nu.tn Abstact In futue Intenet, multmeda applcatns wll be stngly pesent. When a gup f uses s cncened by the same taffc flw, the multcast cmmuncatn can decease cnsdeably the netwk bandwdth utlzatn. The maj pat f ths knd f multcast cmmuncatn needs qualty f sevce (Q) specfcatn. Often, the Q s gven as a set f Q ctea and the cmputatn f feasble ptmal utes cespnds t a mult-cnstaned ptmzatn. Fng the multcast gaph espectng the defned Q equements and mnmzng netwk esuces s a NPcmplete ptmzatn task. Exhaustve seach algthms ae nt suppted n eal netwks. Geedy algthms was ppsed t fnd gd multcast sub-gaphs. In ths pape, we ppse a geedy algthm ICRA t mpve the mult-cnstaned multcast sub-gaph cmputed by the aleady ppsed algthm Mamca. We shw thugh dffeent examples that ICRA tackles all cases that Mamca fals t deal wth. Keywds: multcast utng, Qualty f evce, addtve cnstants, ptmsatn, geedy appach 1. INTRODUCTION Recently, the Intenet has shwn a temendus gwth. Emegent multcast applcatns lke aud/vde cnfeencng, vde n demand, IP-telephny, etc... usually have Qualty Of evce (Q) equements, whch nclude bandwdth, bunded delay, jtte and lss ate. eveal IP multcastng technques have been ppsed t suppt pnt-t-multpnt cmmuncatns by shang lnk esuces leag t a eductn n netwk esuce cnsumptn. All these technques ae based n IP multcast utng ptcls whch use shtest path tee algthm based n ne sngle metc, typcally delay hp cunt. Multcast applcatns tday needs t ptmze me than ne metc that s why mult-cnstaned Q utng s needed. Q utng s a utng scheme unde whch paths f flws ae attbuted by takng nt accunt flw equements and ae based n sme knwledge f esuce avalablty n the netwk. Multcast utng deplyed n the Intenet ams t use esuces effcently. In a pnt t multpnt sessn, p destnatns wll eceve the same nfmatn. eng p tmes ve each shtest path t each ndvdual multcast membe s neffcent. eng sngle packets thugh the shaed lnks and duplcatng them f t s necessay s me effcent. When we cnsde a sngle metc, multcast suce utng can be acheved by fwag packets ve the shtest path tee f example. When the veall cst f the tee must be mnmzed, the pblem must be tackled dffeently. Detemnng the mnmal cst multcast tee f a multcast gup cespnds t the Mnmum tene Tee pblem whch s shwn t be NP-cmplete [4]. An addtnal dmensn t the multcast utng pblem s t cnstuct tees sub-gaphs that wll satsfy multple Q equements. Rutng pblem (n the est f the pape, Q utng efes t mult-cnstaned Q Rutng) even n the uncast case s knwn t be NP-cmplete pblem and has been extensvely studed by the eseach cmmunty. [7] gves an vevew f the man ppsed Q utng algthms whch ty t fund a path between a suce and a destnatn nde that satsfes a set f cnstants. F the multcast case, a numbe f Q utng algthms based n sngle, dual and multple metcs have been ppsed. ngle metc Q multcast utng algthms have been ppsed f cst [1, 9], delay 1

2 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng [2, 5]. Dual metc based utng algthms have been ppsed f the fllwng cmbnatns: cst-delay [3] and delay-jtte [9]. F the mult-cnstant multcast utng pblem whch nvlves multple Q metcs, few algthms have been ppsed due t the cmplexty natue f ths pblem. Multcast Adaptve Multple Cnstants Rutng Algthm (Mamca) [6] attempts t fnd multple Q cnstant paths t the multcast membes n an effcent but nt always ptmal manne. The man dea f Mamca s t cmpute mult-cnstant shtest paths fm the suce nde t each destnatn usng a uncast Q utng algthm, amca [8]. The set f btaned paths s then ptmzed t btan a multcast sub-gaph that uses as few lnks fm the fst paths set as pssble. Mamca ppses a geedy heustc appach t slve ths secnd pblem. The qualty f the appxmatn sn t pved and the shtcmng f the ppsed geedy algthm can be mpved. Ths pape deals wth multple cnstant Q multcast utng pblem whch cnsttutes ne f the mst nteestng pblems f mult-bjectve ptmzatn n netwk feld. In ths pape, we wll fcus, essentally, n ptmzng the set f shtest paths t slve the multple cnstant Q multcast utng pblem. The set f paths can be btaned by amca any multple cnstant Q uncast utng algthm. Cnsdeng the dawbacks f Mamca s geedy algthm, we ppse ICRA, an mpvement vesn f the geedy algthm algthm t pvde a slutn whch can be clse t the ptmal slutn. Ths pape s ganzed as fllws. ectn 2 specfes the multple cnstant pblems f multcast Q utng and pvde an vevew f mst ppsed appaches t teat these pblems. We descbe als Mamca algthm ppsed t slve multple cnstant multcast utng pblem. Ths sectn emphaszes als Mamca s weak pnts. ectn 3 ppses a fmulatn f a new pblem amng t ptmze the multcast sub-gaph, the OM pblem. ectn 4 nvestgates hw the pblem f ptmzng multcast sub-gaph must be tackled f ncemental seach algthms ae adpted and t ppses ICRA, an ncemental algthm t slve the OM pblem. 2. MULTI-CONTRAINED MULTICAT ROUTING The pblem f Q utng, even n the uncast case, s knwn t be NP-cmplete. Befe pesentng the mechansms used t cnstuct mult-cnstaned Q multcast delvey stuctues, we need fst t specfy sme hyptheses used t slve these pblems and the ntatn used thughut ths pape. Then we pesent exstng appaches t acheve multcast Q utng Hyptheses Q utng appaches assume that the netwk-state nfmatn s tempaly statc and has been dstbuted thughut the netwk by any apppate taffc engneeng mechansm. The Q metcs ae categzed nt addtve (e.g., delay, jtte,..) and mn (esp. max) metcs (e.g. bandwdth). The cnstants n mn (esp. max) Q measues can be easly teated by punng all lnks (and dscnnected ndes) whch d nt satsfy the equested mn (esp. max) Q cnstants. In cntast, cnstants n addtve Q measues cause me dffcultes. Hence, wthut lss f genealty, all Q measues ae assumed t be addtve. A netwk tplgy s mdeled as an undected gaph G = (V, E), whee V s the set f vetces epesentng the netwk ndes and E s the set f edges epesentng the netwk lnks. Each lnk s chaactezed by m addtve Q metcs., we asscate t each lnk an m-dmensnal lnk weght vect f m nn-negatve Q weghts (w j, f = 1, 2,..., m and j s a lnk n E). The m Q cnstants (lmts) L f = 1, 2,..., m ae epesented by the cnstant vect L. A multcast gup g s cmpsed f a suce s and a set f membes D = {d 1, d 2,..., d p } whee p s the numbe f multcast membes Multcast Q utng pblem specfcatn F multcast Q utng, the cnstant vect L epesents the lmts allwed f path weghts fm the suce nde t evey membe f the multcast gup. The cnstant vect L s assumed t be the same f all multcast membes. In the fllwng, we pesent fstly the extng pblems specfed n the lteatue then we fmulate a new pblem dealng wth multcast Q utng Exstng multcast Q utng pblems Multcast Q utng pblem cnssts n fng a set f paths fm a nde suce s t p destnatn ndes d j (j = 1,..., p). In tadtnal multcast utng ths set f paths cespnds t a tee but n multcast Q utng t s nt cmpulsly a tee. In the geneal case, t cespnds t a sub-gaph M = (W, H), M G. M s egaded as a set f paths fm s t d j. Unde such hypthess, thee pblems wee fmulated n [6]. 2

3 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng blem I: Multple Cnstaned Multcast (MCM) Gven s and D, fnd M(W, H) such that, f each path P (s, d j ) fm s t d j D(j = 1,..., p): w (P ) L f = 1,..., m (1) Nte that, f f a cetan d j D n feasble path exsts, the pblem has n slutn. T fnd a slutn f the emanng sub-set f destnatns, d j shuld be emved fm D. blem II: Multple Paamete tene Tee (MPT) Gven s and D, fnd M(W, H), {s, D} W f whch l(m) s mnmum whee l s length fnctn. l(m) = max (w (M) ) (2) =1,...,m L whee w (M) = w (u, v) (=1,...,m) (u,v) H blem III: Multple Cnstaned Mnmum Weght Multcast (MCMWM) Gven s and D, fnd M(W, H) such that, f each path P (s, d j ) fm s t d j D(j = 1,..., p): w (P ) L f = 1,..., m and l(m) s mnmum. (3) We ntduce a new metc cl(m), the ctcal length f the multcast sub-gaph M. cl(m) s gven by: cl(m) = max d j D l(p (s, d j)) (4) In the est f papes, we wll fcus nly n the pblem MCM and eventually n the pblem MCMWM. Usng the metc cl(m) defned n (4), the MCM pblem can be fmulated as the detemnatn f a multcast sub-gaph M satsfyng (5) cl(m) 1. (5) Usng the cl metc, the MCMWM pblem cespnds t fnd M(V,H) satsfyng (6). cl(m) 1 and l(m) s mnmum. (6), t summaze, we can deduce that slvng the MCM pblem esults n satsfyng the Q equement f multcast membes. The MCMWM ptmzes the ttal weght f the multcast sub-gaph whle satsfyng Q equements Mult-cnstaned multcast utng ppsals F the multcast case, a numbe f Q utng algthms based n sngle, dual and multple metcs have been ppsed. Few wks, hweve, have dealt wth multple Q metcs. The Multcast Adaptve Multple Cnstants Rutng Algthm (Mamca) ppsed by [6] as the multcast extensn f amca, a uncast Q utng algthm [8] s the man ppsed algthme. The Mamca algthm cmputes the slutn f the multple cnstaned multcast utng pblem by fllwng tw steps. In the fst step, the set f shtest paths fm the suce nde s t all p multcast destnatn membes s cmputed usng a lghtly mdfed vesn f amca. The secnd step ams the eductn f the esultng sub-gaph such that the length functn s educed wthut vlatng the cnstants. The btaned sub-gaph s, t may cntan cycles. Mamca adpts a geedy appach t elmnate as many cycles as pssble. The esultng sub-gaph may nt be a tee and even f M cespnds t a tee, t s nt necessay a mnmal length tee. Ths s due t the geedy appach adpted by the Mamca eductn step. In de t decease the netwk essuces use, u ppstn cncens the mpvement f the multcast-gaph eductn. Auths n [10] analyze ths step and shwed that t suffes fm fu pblems. These pblems ae esumed as fllws: 3

4 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng The fst pblem s ased when cycles ae emved. In Mamca, nly ne pssblty s checked t elmnate a cycle. The secnd pblem s the eventual exstence f ntemedate destnatn ndes n a path. Geneally, nly a pat f the sub paths cmpsng a cycle can be emved nt t affect the eceptn f the ntemedate ndes. Multple pssbltes must be examned. The thd pblem s ased when sme lnks ae used by dffeent paths n the ppste dectn. The cten chce f cmmn ndes n the path eductn algthm shuld be specfed and ths cten nfluences the educed stuctue. The last pblem s the de adpted t emve cycles nfluences the numbe f cycles emved. That s due t the ncemental appach f Mamca used n emvng cycles. The fst pblem can be easly eslved by makng lttle mdfcatn t Mamca pcedue wthut changng Mamca pncples. But t tackle the thee last pblems, the appach must be evewed whlly and a glbal ptmzatn pcedue s needed. Befe ppsng algthms mpvng MAMCRA eductn pcedue, we must fst fmulate the pblem f ptmzng multcast sub-gaph. 3. OPTIMIZING MULTICAT UB-GRAPH (OM) PROBLEM Elmnatng edundances fm the mult-cnstant multcast utng stuctue whch s acheved dung the secnd step f Mamca can be fmulated as an ptmzatn pblem. In ths sectn, we ppse a fmulatn f the Optmal Multcast ub-gaph blem. Geneally, let us suppse that a set f feasble paths t the destnatns f a multcast gup s avalable (f example: the set f shtest paths cmputed by amca). Ou gal s the glbal ptmzatn f ths set: emve the me edundances wthut vlatng the Q cnstants. Optmzng Multcast ub-gaph (OM) blem: Let us suppse that, f a suce s V and a set f destnatns D = {d V, = 1,..., g}, a gaph G = (V, E ) s gven such as the unn f feasble paths fm the suce t each destnatn. The gal s t fnd M a sub-gaph f G ptmzng (mnmzng) an bjectve functn f(m). Ths bjectve functn can be ethe the length functn l(m) specfed n the MCMWM the numbe f hps h(m). T slve the OM pblem effcently, sme eductns ae pssble and we ppse t adpt sme defntns as fllws. A nde n G s a sgnfcant nde f t s the suce nde a destnatn nde a banchng nde havng a degee me than 2. A segment s a path cnnectng tw neghbng sgnfcant ndes f G. A segment s called atculatn segment, f ts deletn ncease the numbe f cnnected cmpnents n the gaph. F example, we can cnsde the example n (Fgue 1) as a gaph n whch nly the sgnfcant ndes ae epesented. (s, x 1 ) and (x 1, x 2 ) ae segments cntanng ntemedate ndes nt. (x 0, d 3 ), (d 5, d 2 ) and (x 3, d 5 ) ae atculatn segments n ths gaph. Reductn 1: A lnk belngs t the ptmal slutn f the OM pblem f and nly f the segment cntanng t s n the slutn. the set f paths that we am t ptmze can be cnsdeed as a set f segments. Thus, ptmzng multcast subgaph cnssts f detemnng the set f segments f G that shuld be emved. In ths way, a (pbably educed) gaph G cntanng the segments f G can be examned t slve the pblem. Anthe eductn f the seach space s pssble by detectng the segments whch belng t all slutns. These segments can neve be emved. F example, f we efe t the tplgy f Fgue 1, the segments (x 0, d 3 ), (d 3, d 2 ), (x 3, d 5 ) and (x 3,d 1 ) can nt be emved. 4

5 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng d 1 X 2 X 3 d 5 s X 1 d 4 X 0 d 3 d 2 Fgue 1: Decmpstn f a set f paths n segments Fm the pnt f vew f ts cmplexty, the OM pblem s equvalent t the gnal MCMWM pblem. fnd the ptmal sub-gaph f educed gaph G (nstead f ) spannng the multcast gup wth espect t the Q cnstants. In the wst case the sub-gaph G cespnds t the whle gaph G and f eductns ae nt pssble, G = G = G. In ths case OM cespnds exactly t the gnal MCMWM pblem whch s NP-cmplete [6]. T fnd the ptmal slutn f the OM pblem, exhaustve seach algthms can be magned (enumeatn f each cmbnatn f edundant segments f example). Incemental geedy algthms such as Mamca can be adpted. In the fllwng, we ppse ICRA, a geedy based algthm t slve OM pblem whch mpves Mamca but ptmalty f the cmputed slutns s nt guaanteed. 4. INCREMENTAL CYCLE REDUCTION lvng the OM pblem cnssts n fng a set f lnks fm the multcast subgaph G t be mtted whle fulfllng the membes equements. In ths sectn, cycle eductn wll be cnsdeed n an ncemental and geedy manne by adg the feasble paths f ne afte the the t the multcast utng stuctue M aleady examned Explng ssues ased by adg a new path t the utng stuctue Cntaly t Mamca algthm, n u ppstn the eductn s analyzed and executed egag all the exstng paths, wth whch the new path fms cycles., the eductn pcedue s ealzed wth a unque scan f the new path fm the fathest cmmn nde t the ne whch s clsest t the suce. Accg t the pesented eductns and t smplfy the analyss, the paths n M and the new path dented P n ae decmpsed n segments. Each segment s chaactezed by tw nfmatns: the weght w() and the numbe f lnks (the hp cunt) h(). The new added path P n fms a cycle n M, f t exsts a set f paths P n M havng cmmn ndes wth P n the then the suce nde. Let X = {x, = 1,...} be ths set f cmmn ndes between P n and the paths aleady n M (the then the suce nde). T smplfy the analyss f the elmnatn f a gven cycle P n (s, x )P (x, s), we ppse sme cntactns. All the segments f the new path fllwng the nde x can be cntacted n a sngle segment P +1 and the n weght vect and hp cunt metc f ths cntacted segment cespnd t the sum f weght vects and hp cunts f the cncened segments. The sub-path f ted at x can be cntacted n a sngle segment P +1 t analyze the pssble eductns. The weght epesent the weght f the ctcal path fm x t the fathest destnatn n and the hp cunt cespnds t the sum f hp cunts n the sub-path. Fgue 2 llustates the cntactn n a smple case. A new added path P n may have cmmn ndes wth many ld paths n M. T examne cycle elmnatn, we study hee thee cases. The fst case ccus when the examned cmmn nde x s n P n and n nly ne ld path. The secnde case s when x s n P n and n at least tw the ld paths. The last case s when 2 cmmn ndes n P n ae wthn an ld paths n nveted de. We wll study each case sepaately and we wll ppse an appach nvlvng all these cases. 5

6 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng d d Cntact sub-paths ted at x +1 x x +1 d d Fgue 2: Cntactn f cmpnents n the geneal eductn case Case 1: New Path P n has a cmmn nde wth nly 1 ld path P Ths case cespnds t the me fequent case. Wthut lss f genealty, let us suppse that the tw paths P n and P fm a cycle between the suce s and anthe cmmn nde x. Let n and n n be the last sgnfcant nde befe x, espectvely, n the ld path P and n the new ne P n. Elmnatng the cycle P n (s, x )P (s, x ) can be acheved by deletng ne f the tw segments P n (n n, x ) P (n, x ) whch pecedes x espectvely n P n and n P. The segment P n (n n, x ) can be elmnated, f mlaly, the segment P (n, x ) can be elmnated, f w((s, x )) + w( P +1) d n L w(p n (s, x )) + w( P +1) d L If the tw cndtns ae vefed smultaneusly and the tw segments ae canddates t the elmnatn, then the segment cespng t a hghe gan (f example, the segment wth hghe hp cunt) shuld be deleted. T llustate such a case, let s cnsde the example pesented n Fgue 3. If P shuld be euted thugh P n t elmnate cycle n x, nly the segment peceg x can be emved t ensue fwag data t ndes n 1, n 2 and n 3. We ecall that these ndes can nt be dscnnected as they ae whethe destnatn ndes banchng ndes whch fwad data t the destnatn ndes. In the the case, elmnatng the whle sub-path (s, x ) fm P n s allwed even f ndes such as n 4 and n 5 exst. In fact, f these ndes ae destnatn ndes, paths f these destnatns exst n G as emvng such paths fm the ntal gaph G ccus nly afte adg the new path P n. n2 n3 n1 x n4 n5 d Reutng ld path thugh new path Reutng new path thugh ld path n2 n3 n2 n3 n1 n1 x x n4 n5 n4 n5 d d Fgue 3: Cycle eductn when sub-paths cntans ntemedate destnatn(s) 6

7 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng Case 2: New Path P n has a cmmn nde wth a set f ld paths In a me geneal case, a cycle nvlves me than ne ld path. Ths case ccus when the cmmn nde s n the newly added path and n me than ne ld paths as t s depcted n Fgue 4. 1 x 2 n In ths case, cycle eductn can be acheved f : Fgue 4: Cntactn f cmpnents n the geneal eductn case eutng the newly added path thugh ne f the exstng paths s pssble due t a cmmn nde x : the sub-path P n (s, x ) s elmnated and ne path fm the ld paths can be used t fwad data t the segment +1 P n. eutng sme f the ld paths thugh the newly added path. In that case, sme f the segments peceg x and belngng t a set f ld paths can be mtted. In the best case, all the ld paths can be euted thugh P n. We shuld ntce that the aleady extng cycles fmed by the ld paths can nt be elmnated wthut the adg f the new path. In fact, when adg these paths all elmnatn pcedues have been tested and n elmnatn was pssble as the cycles pesst., t acheve cycle eductn, whle sub-paths a set f segments can be canddate t elmnatn. When me than ne canddate t elmnatn exsts, the canddate that ptmzes an bjectve functn g s deleted. Bascally, f g specfcatn, thee ae tw exteme pssbltes: mnmze the numbe f lnks used f data fwag maxmze the qualty at the destnatn ndes. In de t mnmze the numbe f used lnks, we ppse the elmnatn f the segment wth lage hp cunt. In the case f equalty, the Q ctea s dscussed and the slutn whch guaantees the smalle ctcal value f Q ctea s chsen. Ths decsn can be seen as a mult-cnstaned ptmzatn pblem n the plan hp dstance and ctcal length value. Othe selectn ctea can be als ppsed but ths s ut f scpe f u analyss. Case 3: New Path P n has 2 cmmn ndes wth at least ne ld path n nveted de Anthe ssue that must be nt neglected s the de f cnsdeed cmmn ndes n the ld and n the new path. In sme ae but pssble cases, the de f cmmn ndes n the ld paths n M can be dffeent fm the de n the new path. It can be examned by detectng the pstn f the nde x n the tw paths cmpaed t the peceg cmmn ndes. In fact, tw ndes may belng t tw paths but they ae used n nveted de. That s the case f ndes a and b n Fgue 5: a and b ae tw cmmn ndes f P 1 and P 2 but they ae used fm b t a n P 1 and fm a t b n P 2., when cnsdeng 2 cmmn ndes x and x 1, the de adpted hee s the de f these ndes n the new path P n. Elmnatn stategy must nvlve all the abve pesented cases. In the fllwng, we ppse a new algthm enhancng Mamca mechansms and cnsdeng all tplgy cases Impved Cycle Reductn Algthm ICRA algthm suppses that a feasble path s knwn fm the suce nde t each destnatn f the multcast gup. The set f these paths fms the gaph G whch can cntan cycles. ICRA algthms ams t elmnate edundances fm ths set f Q cnstaned paths. The set f paths, esult f the elmnatn pcedue, must cntan a feasble path f all destnatn ndes f whch a feasble path exsts n the ntal set. In the wlds, the 7

8 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng P2 P1 a b d1 d2 c Fgue 5: Decmpstn f a set f paths n segments gal s t decease the numbe f lnks n the esultng stuctue espectng the gven cnstants: fnd an appached slutn f the MCMWM pblem. As f Mamca algthm, the nputs f the algthm ae the fllwng. We suppse that the Q cnstants vect L s gven and the m dmensnal lnk value vects ae knwn f each lnk. Each path P k s a feasble path fm the suce s t a destnatn d k D f k = 1,..., d. The utput f u algthm s a set M cntanng paths uted at s. Each path cves a sub-set f destnatn ndes and the unn f the paths cves the ttalty f destnatns. The esult cntan nly feasble paths fm the suce nde t the cveed destnatns as t wll be e by the fllwng algthm. mlaly t Mamca, the algthm ICRA wks n a geedy manne and pceeds ncementally n tw phases: fst, selectng a path fm then attemptng t add t t the fnal utng stuctue whle emvng as much edundances as pssble. These phases ae detaled n the fllwng Path selectn T pcess the nput set f paths, we ppse the same pcedue f selectn as t s defned n Mamca. F each path P k, the end t end Q length l(p k ) and the destnatn ndes n the path ae knwn. The selectn f the next path t teat s based n the numbe f multcast membes pesent n the path. Paths wth lage numbe f multcast membes ae examned at fst. Ths mples that the selected path wll cve as many destnatns as pssble. In the case f equalty n ths metc, the algthm chses the path wth smallest Q length., at a fst tme the set f paths gws wth shtest paths and a sht kenel s ceated t facltate futue path addtns. If thee s an equalty n the tw mentned metcs, we ppse t select andmly ne f the equal paths. Even f the selectn nfluences the esult f the eductn, examnng all pssble des s expensve that s why we ppse the andm selectn Add a path t the set f multcast paths The pncpal steps f path addtn t the multcast sub-gaph ae as fllws: At fst, the cmmn ndes X = {x, = 1,...} between P n and the paths aleady n M ae detemned, but the suce nde s s excluded fm X. If thee s n ntesectn, then the path P n s added as a new path t the set M. In ths case, the the paths fm the suce t the destnatn ndes n P n can be deleted fm f they exst. If the path P n fms cycles wth sme paths n M, then examnng edundances s acheved n a geedy manne. We ppse a cycle eductn pcedue based n selectng fm a set f detemned canddate segments whch canddate t elmnate. Cmputng the set f segments canddate t elmnatn. In Mamca algthm, emvng cycles s pcessed by examnng cycles fmed between the new added path and lde paths ne at a tme. F exhaustve examnatn, lde paths ae selected ne afte the the usng the adg de. In ICRA algthm, the eductn s analyzed and executed egag all the exstng paths, wth whch the new path fms cycles. Cmmn ndes ae scanned fm the fathest cmmn nde t the ne whch s clsest t the suce accg t the de n the new path. The fst 8

9 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng step f the cycle eductn pcedue s t detemne the set f segments canddate t the elmnatn. Investgatn made n the pevus sectn leads us t state the fllwng eductn ules that must be used t detemne segments canddate t the elmnatn. Reductn Rule 1: A sub-path segment s p can be elmnated fm a path P k exstng between s and d k and t can be eplaced by a sub-path s p j f the eplacement esults a cntnual path fm s t d k and f w(p k ) w(s p ) + w(s p j ) d L Reductn Rule 2: When a sub-path fm suce nde f an ld path n M s canddate t elmnatn, nly the segment peceg the cuently examned cmmn nde can be emved. Ths last eductn ule s used t peseve ntemedate sgnfcant ndes fm deletn. In fact, when the ld path s subject t elmnatn, nly the segment peceg the cmmn ndes can be mtted nt t dscnnect ntemedate destnatn and sgnfcant ndes. electn f segments that shuld be elmnated. In the fllwng, each canddate t elmnatn s assmlated t a set f segments. If thee s me than ne canddate t elmnatn, the ne that ptmzes an bjectve functn g s chsen. In ICRA algthm, we ppse t defne the functn g as the smple sum f the hp numbes f the elmnated segments. Ths sum cespnds t the gan f the elmnatn and u bjectve s t maxmze t. Defntn : Gven a new path P n and an ld path P havng a set f cmmn ndes X = {x, = 1,...}, the elmnatn gan nduced by the emval f a set f segments P fm P n P when examnng the cmmn nde x s g = h(p ), f P s n P n; g P = g f P s n P and h(p ), f x and x 1 ae n the same de n P and P n ; g = h(p ) + h( 1 P ) f x and x 1 ae n nveted de n P and P n. whee j P s the segment the set f segments peceg x j. (7) It s mptant t ntce, that n (7) when j P cespnds t a set f segments (case f elmnatn f a set f segments fm P ), h( j P ) cespnds t the sum f the numbe f hps f each segments f the cnsdeed set f segments. T bette undestand the gan defntn, let us cnsde Fgue 6 whee Case A shws that elmnatng P n fm P n saves the use f h( P n ) hps. In the same way, elmnatng P fm P nduces a gan f h( P ) hps, f the cmmn ndes n P n and n P have the same de n the tw paths as t s n the Case B f the fgue. Let us ntce that the elmnatn extends n the whle sub-path n the new path, fm the suce t the last cmmn nde whee the tw eductn ules ae espected, whle the elmnatn cncens nly the last segment (f the cndtn s vefed f ths elmnatn) n the ld path. The easn s smple: the ntemedate destnatn ndes n the new path can be eached usng the wn exstng feasble paths n. These paths wll be examned late n the pcedue. Regag the ntemedate destnatn pesent n the ld path, thee s n me path n t each them., nly the last segment befe the examned cmmn nde canddates t be elmnated. The last case ccus when a segment P f the ld path P s canddate t elmnatn and when P and P n have at least tw cmmn ndes (f example: x and x 1 ) n nveted de as depcted n Case C f Fgue (6). Elmnatn f segments P mples the elmnatn f the last segment P 1 peceg the nde x 1 as taffc t nde d p wll be euted thugh P n. In ths case, chsen t elmnate the segments n the ld path saves h( P ) + h( P 1) hps. In geneal case, seveal cmmn ndes can be n nvese de n the tw paths. It s easy t shw that the elmnatn f the last segment n the ld path at a cmmn nde x mplcates the elmnatn f all the last segments whch ae befe the the cmmn ndes gng fm x t the last destnatn n the ld path. 9

10 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng x x-1 x d d Case A P0 g = h ( ) g = h ( ) P0 Case B -1 g = h ( ) x x-1 g = h ( )+ h( ) P0 P0 P0-1 d Case C Fgue 6: Cmputng gan asscated t segment elmnatn Fnally, let us cnsde the geneal case whee the examned cmmn nde x belngs t a set f ld path P O =, = 1,... Ths set can be pattned n 2 sets P O n and P O ut. P O n cntans ld paths havng cmmn ndes n the same de as n the new path. Unlke ths, P O ut cntans paths havng sme cmmn ndes n nveted de cmpaed t the de n P n. The gan asscated t the dffeent elmnatn pssbltes can be cmputed as fllws: If eutng P n thugh ne f the exstng paths n P O s feasble, then the gan cespnds t the ttal length f the sub-path f P n fm the suce t x and t s gven by g = h( P n ) If eutng a set f ld paths P R thugh P n whch s cnseved, the gan s g P R = g P + P (P R P O n) P (P R P O ut) g P Reductn ule 3: T decde whethe canddate t elmnate, cmpute the gan g asscated t each canddate then chse the elmnatn that maxmzes the gan Fmal descptn f ICRA algthm Usng eductn ules, defntns and the algthm gven pevusly, ICRA mechansms can be summazed as fllws. When adg a new path t the multcast gaph M, ICRA algthm detemnes the set X={x, x s, = 1,...} f cmmn nde that ae n the new path and n the ld paths. These cmmn ndes ae sted n ascendant de f the dstance fm the suce nde n the new path. They ae scanned nde by nde fm the fathest ne. Examnng a nde cnssts n detemnng the sub-paths and the segments that can be emved t elmnate edundances n ths nde. ub-paths ae canddate t elmnatn when eductn ule 1 s espected t fulfll destnatn equements. If eductn ule 2 s nt espected, the examned sub-path must be educed t the 10

11 ICRA:Incemental Cycle Reductn Algthm f ptmzng mult-cnstaned multcast utng last segment peceg the examned nde, as t s descbed abve, t avd the dscnnectn f ntemedate destnatn ndes. When the set f segments canddate t elmnatn s nt empty, ICRA chses the ptmal set f segments. Paths subject f segment emvals must be updated t ensue fwag multcast taffc t the cespng destnatn ndes. 5. CONCLUION In ths pape, we adpt an ncemental appach t slve mult-cnstaned multcast utng. It s ppsed t g beynd Mamca pblems that we pesented abve. Afte shwng hw Mamca fals n slvng sme pssble cases, we descbe hw ICRA tackles the ptmzatn f the multcast subgaph by takng nt accunt dffeent tplgy cases and by examnng dffeent elmnatn pssbltes. We shw n ths pape that ICRA vecmes the dffeent pblems that Mamca des nt handle. F the geneal case whch s handled by bth ICRA and Mamca, we plan t pefm smulatns t cmpae pefmances f ICRA and Mamca and ths ssue wll be nvestgated n u futue wk. And as ICRA s als a geedy methd, t s nteestng t see hw metaheustcs can be adpted t slve ths pblem and hw t behaves cmpaed t ncemental slutns such as Mamca and ICRA. Ths ssue wll be nvestgated n futhe wk. Bblgaphy [1]. Chen, K. Nahstedt, An Ovevew f Qualty-f-evce Rutng f the Next Geneatn Hgh-peed Netwks: blems and lutns, n IEEE Netwk, pecal Issue n Tansmssn and Dstbutn f dgtal Vde, pp ,(1998). [2] M. Falutss, A. Banejea, R. Pankaj, QMIC: Qualty f evce senstve Multcast Intenet ptcl, n ACM IGCOMM, pp ,(1998). [3] L. Henque, M. K. Csta,. Fdda, O.Cals, M. B. Duate, Hp-by-Hp Multcast Rutng tcl, In ACM IGCOMM, pp , (2001). [4] F. Hwang, D. Rchads, tene Tee blem, Netwks, vl. 22. n. 1, pp.55-89, (1992). [5] X. Ja, Y. Zhang, N. Pssnu, K. Makk, A dstbuted multcast utng ptcl f eal-tme multcast applcatns, n Cmpute Netwks 31, pp , (1999). [6] F. Kupes, P. V. Meghem, Mamca: A Cnstaned-Based Multcast Rutng Algthm, In Cmpute Cmmuncatns, Vl. 25, N. 8, pp ,(2002). [7] P. Van Meghem (ed.), F.A. Kupes, T. Kkmaz, M. Kunz, M. Cuad, E. Mnte, X. Masp-Bun, J. lé- Paeta,,. ánchez-lópez, Qualty f evce Rutng, Chapte 3 n Qualty f Futue Intenet evces, EU- COT 263 Fnal Rept, edted by mnv et al. n pnge LNC 2856, pp , (2003). [8] P. Van Meghem, H. De Neve, F.A. Kupes, Hp-by-Hp Qualty f evce Rutng, Cmpute Netwks, Vl. 37, N. 3-4, pp , (2001). [9] B. Wang, J. C. Hu, Multcast Rutng and ts Q Extensn: blems, Algthms and tcls, n IEEE Netwk, pp , [10] N. Ben Al, M.Mlna and A.Belghth, Mult-cnstaned Q Multcast Rutng Optmzatn, Techncal Rept 1882, Isa, febuay