The Cost Optimal Solution of the Multi-Constrained Multicast Routing Problem

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1 Pulition Intrn l IRISA ISSN : PI 1957 Otor 2010 Th Cot Optiml Solution of th Multi-Contrin Multit Routing Prolm Mikló Molnár *, Ali Bll **, Smr Lhou *** miklo.molnr@lirmm.fr, li.ll@iri.fr, mr.lhou@iri.fr Atrt: In thi ppr, w tuy th ot optiml olution of th wll-known multi-ontrin multit routing prolm. Thi prolm onit in fining multit trutur tht pn our no n t of tintion no with rpt to t of ontrint. Thi optimiztion prolm i prtiulrly intrting for th multit ntwork ommunition tht rquir Qulity of Srvi (QoS) gurnt. Morovr, fining th multit trutur with rpt to th fin QoS rquirmnt whil minimizing ot funtion i n NP-iffiult optimiztion prolm. Aoring to th tt-of-th-rt, to olv th multi-ontrin multit routing prolm, mot of th propo lgorithm rh for multit tr. In thi ppr, w montrt tht th optiml onnt prtil pnning trutur tht olv th multi-ontrin multit routing prolm n iffrnt from prtil pnning tr. In, w how tht th minimum ot olution lwy orrpon to hirrhy, rntly propo impl gnrliztion of th tr onpt. For tht, w fin n nlyz th prtil minimum pnning hirrhi optiml olution for th multi-ontrin multit routing prolm. Morovr, w propo rnh n ut typ xt lgorithm tht i on om rlvnt proprti of th hirrhil olution. Ky-wor: hirrhy Multit, qulity of rvi, multi-ontrin Stinr prolm, hirrhy, prtil minimum pnning Routg multi-ontrint multit optiml Réumé : L routg multi-ontrint multit t un prolèm NP-iffiil. Dn ppir, nou étuion prolèm t nlyon l propriété l olution optiml. Contrirmnt à qui t onnu n l étt l rt, l rr n ont p toujour l trutur optiml pour l prolèm u routg multi-ontrint multit. Notr ontriution prinipl t éfinir l ntur l trutur optiml pour l prolèm trité. Nou prénton l hirrhi qui ont générlition rr t nou prouvon qu l olution optiml t toujour un hiérrhi. Nou nlyon églmnt l propriété l olution optiml t éfiion un nouvl lgorithm rnh n ut pour trouvr tt olution optiml. Mot lé : Multit, qulité rvi, prolèm Stinr multi-ontrint, hiérrhi, hiérrhi minimil ouvrmnt prtil Trvil pour l nouvll olltion pulition intrn l Iri vrion éltroniqu omptil v l RR Inri * Projt APR : LIRMM Monpllir ** Projt AtNt : équip ommun INSA t univrité Rnn 1 *** Projt AtNt : équip ommun INSA t univrité Rnn 1 IRISA Cmpu Buliu Rnn Cx Frn

2 2 M. Molnár & A. Bll & S. Lhou 1 Introution Qulity of Srvi QoS multit routing known multi-ontrin multit routing onit in ontruting multit trutur tht pn our no n t of tintion no. Thi multit trutur houl mt t of rquirmnt uh ly, jittr, nwith, lo rt n ot, for h tintion no. Th multi-ontrin multit routing prolm i known to NP-iffiult. In, th mot hllnging QoS multit routing thniqu im to upport pointto-multipoint ommunition y 1) tifying th QoS ontrint n 2) ling to rution in th ntwork rour onumption. Whn thr i on itiv ontrint to tify n no ot funtion to minimiz, n ffiint multit routing n hiv y forwring pkt ovr th hortt pth tr. If th omput tr ontin fil pth from th our to h tintion, th rquir QoS i ur, l thr i no olution to tify th ontrint for th tintion no. To fin ttr olution within th fil pth, ot funtion n introu. Th ot n n ritrry mtri n inpnnt from th QoS mtri. Th minimum ot olution n minimiz for intn th hop ount. Th ontrution of th minimum ot multit tr without QoS ontrint orrpon to th Stinr Tr Prolm. Thi prolm i known to NP-iffiult [10]. Morovr, whn QoS ontrint i prnt in th multit routing prolm n minimum ot tr i rquir, th orrponing Contrin Stinr Prolm i lo NP-iffiult hown in [11]. Noti tht mot of th propo work omput multit tr tht r prtil pnning tr. Howvr prtil pnning tr o not lwy orrpon to th optiml olution. In thi ppr, w onir th gnrl of th ontrin multit routing with multipl QoS ontrint. Morovr, w uppo tht n itiv ot funtion rflt th ntwork ug tht houl optimiz. Th two min ron to oupl optimiztion n multi-ontrin routing r: 1. Rl QoS rquirmnt r oftn on multipl ontrint n th multit trutur mut propo fil pth for h tintion. 2. Th intrt of th ntwork oprtor n thu impliitly of th ur, i to minimiz ntwork rour onumption. In [16], th uthor init tht gurnting QoS n optimizing rour utiliztion r two onfliting intrt n tr-off houl hiv. Evn if thi onflit xit in om, w r intrt in fining th t ot olution with rpt to t of QoS ontrint. Thu, w invtigt on th multi-ontrin ot optimiztion of multit trutur. Our ojtiv i to fin th minimum ot pnning trutur, whr th n-to-n pth, from th our to th tintion, tify th onir QoS ontrint. A it i montrt in [17], th multi-ontrin multit routing prolm lo known multi-ontrin prtil pnning prolm r NP-iffiult vn if thr i only on tintion. In [13], th uthor how tht fil prtil pnning trutur, tht olv th multi-ontrin pnning prolm, my iffrnt from prtil pnning tr. In thi ppr, our mot importnt rult i th xt hrtriztion of th optiml olution. W montrt tht n rlir propo impl gnrliztion of th tr onpt ll hirrhy lwy orrpon to th minimum ot olution n n u to ri ll fil olution. Thu, w invtigt on fining th prtil minimum pnning hirrhy th optiml olution of th multi-ontrin prtil pnning prolm. Th vntg to rpl th pnning tr onpt y th pnning hirrhy r onluiv: th hirrhi nl to fin th optiml olution. In, th optiml (minimum ot) multit trutur lwy orrpon to hirrhy. Th propo multi-ontrin prtil minimum pnning hirrhy prolm i NP-iffiult. Sin th hirrhi r iffrnt from tr, mot of th known lgorithm tht r on tr numrtion n no numrtion, to fin optiml tr, n not ppli to fin th ot optiml hirrhy. Thrfor, w propo rnh n ut xt lgorithm to olv thi prolm y oniring n pproprit olution trutur. To th t of our knowlg, our tuy i th firt to formult th optiml multit routing prolm uing th hirrhy onpt. To ppropritly prnt our ontriution, thi ppr i orgniz follow. Stion 2 pifi th multi-ontrin prtil pnning prolm for multit QoS routing n provi n ovrviw of th prviouly propo pproh to olv thm. Stion 3 prnt th hirrhy onpt gnrlizing pnning tr n giv om gnrl proprti of th hirrhi. In Stion 4, w prov tht th optiml olution of th multi-ontrin prtil pnning prolm i lwy hirrhy. Th hrn of th prolm i iu, n rlvnt proprti of th multi-ontrin miniml ot prtil pnning hirrhi r prnt. Th proprti r uful to ign n ffiint rnh-n-ut xt lgorithm ri in Stion 5. 2 Prolm Formultion n Rlt Work At th routing lvl, vrl multit multimi pplition rquir multi-ontrin multit trutur. In th litrtur, iffrnt ojtiv hv n trgt n vriou olution hv n propo. In thi tion, w prnt n ovrviw of th mot prtinnt formultion of th r prolm wll rif prnttion of th iffrnt propo

3 Multi-ontrin multit routing 3 rout omputtion lgorithm. To olv th r prolm, w rgu tht th t olution orrpon to minimum ot multit trutur with rpt to th QoS ontrint. Thrfor, w propo th xt formultion of th minimum ot multi-ontrin multit routing prolm. 2.1 Prolm Formultion Lt G = (V,E) th unirt grph rprnting th ntwork topology, whr V i th t of no n E i th t of g. Th our no n th multit tintion no t r not y V n D = { j V, j,j = 1,...,r} rptivly. Eh g E i oit with m QoS wight givn y wight vtor w () = [w 1 (),w 2 (),...,w m ()] T. Th n-to-n QoS rquirmnt i.., ontrint from th our to th tintion, r givn y n m-imnionl ontrint vtor L = [L 1,...,L m ] T. Th QoS mtri n roughly lifi into itiv uh ly, multiplitiv uh lo rt or ottlnk uh vill nwith. A it i xplin in [13], ottlnk mtri n ily lt with y pruning from th grph ll link tht o not tify th QoS ontrint, whil th multiplitiv mtri n trnform into itiv mtri y uing thir logrithm. Thrfor, n without lo of gnrlity, w only onir th itiv mtri. Th wight of pth p(, j ) orrponing to th mtri i i givn y w i (p(, j )) = p(, w j) i(). Thu, pth p(, j ) i fil if: w i (p(, j )) = w i () L i, for i = 1,...,m (1) p(, j) Unit QoS routing onit in fining fil pth p(, j ), twn our no n tintion no j. Oftn, ot funtion vluting th pth houl minimiz: min () (2) p(, j) Th multi-ontrin QoS routing with ot funtion, vn in th unit i known to NP-iffiult hown in [17]. Multit QoS routing im to fin multit trutur M(W,F), whr W i t of no tht n prnt vrl tim, n F t of g, whih n ontin vrl tim th m g. Th multit trutur M(W,F) mut ontin t lt on fil pth p(, j ) from th our no to h tintion j, j = 1...,r. Th optiml routing prolm lo im to minimiz th ot n thu voi ul runni. Th ot minimiztion i goo wy to tk into ount th onomil pt of ntwork ug. In, to olv th QoS multit routing prolm, mot of th xiting work r fouing on fining prtil pnning tr. Hrftr, w will how tht pnning tr nnot lwy olv th r prolm. Compr to th wll known Stinr prolm [10], whr th minimum ot prtil pnning tr i rquir, th ontrution of prtil pnning trutur tifying multipl QoS rquirmnt i vn mor omplx. To filitt th omprion of th multi-ontrin QoS pth, n to xpr th tiftion of th QoS rquirmnt, non-linr lr lngth funtion h n introu in [13]. Th lngth of pth p(, j ) i fin : l(p(, j )) = mx i=1,...,m (w i(p(, j )) L i ) (3) If ll of th ontrint r tifi, trivilly l(p(, j )) 1. Thu, fil pth n lo fin pth who lngth i l thn 1. To fin th multi-ontrin multit routing prolm, w propo th following formultion. Prolm 1. Th prolm of Multi-Contrin Minimum Cot Multit (MCMCM) l with fining th trutur G M with minimum ot (G M ), ontining t lt on pth p(, j ) from th our no to h tintion j D tht tifi th givn ontrint vtor L, n givn y th following Prto ominn: w(p(,j )) L n (G M ) i minimum. (4) Th ot funtion n th frquntly u hop-itn funtion or ny othr poitiv itiv ot xpr on th g. A w will hrftr, thi formultion i mor pproprit thn th propo on in th litrtur. In [13], thr formultion hv n propo. Prolm 2. Th Multipl Contrin Multit (MCM) prolm im to fin prtil pnning trutur G M = (W,F),W V n F E tht ontin pth p(, j ) from th our to h tintion j D uh tht: w i (p(, j )) L i,i = 1,...,m (5)

4 4 M. Molnár & A. Bll & S. Lhou (T) (T') (G') l(p1)= 0.7 l(p2)= 0.9 l(p"1)= 0.9 l(p''2)= 0.7 l(p'1)= 0.7 l(p'2)= 0.7 () () () or y uing th Prto ominn: Figur 1: Rlvn of th lngth funtion hoi w(p(,j )) L (6) A trivil olution n foun y omputing t of fil pth: on fil pth from th our to h tintion if uh pth xit. Prolm 3. In th Multipl Prmtr Stinr Tr (MPST) prolm, tr T M = (W,F),{,D} W tht minimiz lngth funtion l multit (T M ), i rquir. Thi lngth n n ritrry vtor norm. Typilly, th u non-linr lngth funtion in [13] i givn y: T l multit (T M ) = mx M w i () i=1,...,m (7) L i Noti tht thi non-linr lngth l multit of tr T M i not pproprit to hrtriz th qulity of th multiontrin multit rvi t th tintion til in th xmpl givn ftr Prolm 4. Prolm 4. Th Multipl Contrin Minimum Wight Multit (MCMWM) prolm omin th two prviou ojtiv. Th gol i to fin prtil pnning trutur G M = (W,F) tht ontin fil pth to h tintion j D uh tht th lngth l multit of G M i miniml: w(p(,j )) L n l multit (G M ) i minimum. (8) In th lt two formultion, whn th minimiztion of th totl non-linr lngth of th prtil pnning trutur i propo, w tt limittion in th u lngth funtion l multit, whih n onirly fft th qulity of th omput trutur. In, w noti tht thi non-linr lngth funtion u on th glol onirtion of th omput prtil pnning trutur o not onir th qulity of h of th n-to-n pth. To illutrt thi limittion w propo th xmpl prnt in Figur 1. In Figur 1, th link wight r init n th QoS ontrint r givn y: L= [10,10] T. For th multit group {, { 1, 2 }}, w noti tht thr r thr t of pth, to rh th two tintion. A tr T with lngth l multit (T) = 1, illutrt in Figur 1.(), nothr tr T with lngth l multit (T ) = 1, illutrt in Figur 1.(), n prtil pnning trutur G with lngth l multit (G ) = 1.2, illutrt in Figur 1.(). Th olution of Prolm 4 will th tr T, or th tr T. Howvr, if w onir th n-to-n qulity of omput pth, w noti tht th lngth l(p 1) in G i mllr thn l(p 1 ) in T, n l(p 2) in G n l(p 2 ) in T r qul. Thu, th trutur G i ttr oniring th rquirmnt of h of th tintion.

5 Multi-ontrin multit routing 5 W n onlu tht th non-linr lngth of th unit pth n th totl lngth of th gnrt trutur r not oviouly orrlt. In, from G to T (or T ), th totl lngth of th prtil pnning trutur r whil th lngth of th unit pth inr. Not tht th im of th routing i to fin trutur ontining t lt on fil pth from th our to h tintion, whil minimizing th ntwork rour ug xpr y n qut ot funtion. For intn, th ot i typilly itiv mtri, whih n pnnt or inpnnt from th m QoS mtri [5]. To hrtriz th rlvn (th tiftion of th QoS ontrint) of th multit trutur, th mot ritil vlu of th QoS mtri n u th imtr of th multit rout. In thi wy, w n lo fin th multi-ontrin multit routing prolm follow. Prolm 5. Th prolm of Multipl Contrin Minimum Dimtr Multit (MCMDM) im to fin trutur G M form y fil pth p(, j ) from th our to h tintion j D, whil minimizing th non-linr imtr givn y: D(G M) = mx (j=1,...,r) (l(p(, j )) (9) Th non-linr imtr D vlut th ritil qulity of th omput trutur G M on th n-to-n propriti of it pth. In, th non-linr imtr of G M xtly orrpon to th non-linr lngth of th mot ritil unit pth. If th mot ritil tintion i tifi, th othr tintion r lo tifi. All of th prnt pnning prolm with multipl ontrint r NP-iffiult. For th thr multit routing prolm prnt in [13], th proof n foun in th m ppr. Lmm 1. Th Multi-Contrin Minimum Cot Multit (MCMCM) n th Multipl Contrin Minimum Dimtr Multit (MCMDM) prolm r NP-iffiult. Proof. Sin th multi-ontrin minimum ot unit routing i NP-iffiult, th proof i trivil uppoing only on tintion. A onlu in [13], th olution of th MCM n MCMWM prolm i not lwy tr. Th olution orrpon to prtil pnning trutur, whih ontin fil pth to h tintion. It lo ontin yl tht n not limint. Th olution of th minimum ot ontrin prtil pnning prolm my lo ontin yl in th t of th fil pth. Hr w ri th following importnt qution: Wht kin of trutur orrpon to th optiml olution of th MCMWM n MCMCM prolm? Th hrtriztion of th olution llow to ign xt lgorithm n ffiint huriti to omput thi olution. 2.2 Rlt Work Th multi-ontrin multit routing prolm orrponing to th Multi-Contrin Prtil Spnning Prolm i known to NP-iffiult. To olv thi prolm, th iffrnt propo pproh n ivi in thr min l, oring to th ojtiv n th intrt of h l of prolm: (i) th firt l im to omput pnning tr tht minimiz givn ot funtion without hking th fiility of th olution, (ii) th on l omput minimum pnning tr, with rpt to t of QoS ontrint, (iii) n th thir l omput pnning hirrhy ( th finition in th nxt tion), with rpt to t of QoS mtri. Noti tht in th two firt l of prolm, th iffrnt propo olution im to omput pnning tr. Thr r two ron for ing multit trutur on multit tr (i) th t n trnmitt in prlll to vriou tintion long th tr link n (ii) th tr trutur voi runni. W lo noti tht, if th r multi-ontrin multit prolm h no ot funtion to minimiz, ny ffiint multi-ontrin unit lgorithm n u, uh SAMCRA propo in [21] n H MCOP propo in [12]. Thrfor, w minly fou th rlt work ovrviw on th thr ov it l of prolm. In th firt l of propoition, th olution im to minimiz th ot of th multit tr. Th i prolm i known th Stinr or Prtil Minimum Spnning Tr (PMST) Prolm. Thi NP-iffiult prolm h oliit lot of intrt, n lrg numr of xt [8] n huriti [20] lgorithm wr propo in th litrtur (f. [23] [10] for two goo ovrviw). Th mot known xt lgorithm r on th numrtion of th Stinr no 1 n Stinr tr, n r ll Stinr Tr Enumrtion Algorithm. Th firt xt lgorithm w propo in [8]. Among th propo huriti, w n it th Tkhhi n Mtuym lgorithm propo in [20]. Thi lgorithm u hortt pth to onnt itrtivly th tintion to th pnning tr. In th on l, th r prolm r known th Contrin Stinr Tr (CST) Prolm tht r lo NP-iffiult [11]. To olv thm, mny of th propo lgorithm onir on QoS ontrint in [2], whr th uthor propo rnh n oun lgorithm y uing th Lgrngn Rlxtion n huriti to gt lowr n uppr oun 1 A Stinr no i no tht h gr iggr thn 2 in th pnning tr.

6 6 M. Molnár & A. Bll & S. Lhou in th rnh n oun tr. Howvr, mot of th propo lgorithm for th CST prolm r huriti n gnrlly onir th n-to-n ly. In [11], th uthor propo huriti lgorithm tht ontrut low ot pnning tr, with rpt to oun ly on h multit tintion. Th propo lgorithm omput ly-ontrin lour 2 grph ovr th multit group. Thn, th lgorithm ontrut ontrin pnning tr of th lour grph uing th wll known Prim lgorithm [26]. Finlly, th lgorithm rpl th link in th pnning tr y th originlly omput pth, n rmov th gnrt loop. In [14], th uthor lo propo n lgorithm tht pproh th minimum ot pnning tr olution with rpt to th ly ontrint. For tht, th lgorithm ontrut two routing tr: hortt pth tr n n pproh Stinr tr. Thn, it intifi givn numr of tintion k, whr th iffrn twn th ly orv in th Stinr tr n th ly in th hortt pth tr for th tintion i lrg. For th tintion no, thir pth in th Stinr tr r rpl y thir orrponing pth in th hortt pth tr. Th uthor in [25] propo huriti lgorithm ll Th Boun Shortt Multit Algorithm (BSMA). Thi lgorithm omput lt-ly tr, tht pn th our no n th tintion no. Thn, it itrtivly rpl th link tht n rpl y othr link tht ru th totl ot of th tr, without violtion of th ly ontrint, until th totl ot of th tr n not furthr ru. Th BSMA lgorithm lwy fin ly ontrin tr, if on xit in it gin y th lt-ly pnning tr. Whn mor thn on QoS ontrint r onir, th prolm om mor omplx. In [22], th uthor u th Lgrngn Rlxtion Approh to ontrut pil tr ll LRATr. Thu, th LRATr lgorithm rlx th ontrint n ontrut nw prolm with on ojtiv funtion to minimiz (mx λ L(λ) = min (p(, j )) + r i=1 λ i(w i (p(, j )) L i ), for h tintion j D). λ i trmin how muh th violtion of th i th ontrint houl pnliz. Th lgorithm omput th minimum ot pth twn th our no n vry tintion no, thn omin th pth to otin n initil fil tr. Thi tr i upt whn th Lgrngn prmtr λ = [λ 1,...,λ m ] T i jut follow: λ k+1 = λ k +θ k ( w (p(, j )) L), with θ k = L(λk+1 λ k ) w i(p(, j)) L i 2. In [7], th uthor propo n lgorithm tht rh for fil olution to th prolm y fining, t firt, fil prtil tr tht pn th our n om of th tintion. Thn, it uil up th rmining tintion uing moifi vrion of th H MCOP lgorithm [12]. Thi lttr lgorithm omput th hortt pth twn two no y uing th omin non-linr lngth funtion lo prnt in Eqution 3. Howvr, prtil pnning tr my not lwy tify th rquir QoS ontrint, whil t of unit QoS pth n, n thi onirly ru th omplxity of th r prolm. Thrfor, tr-off twn ffiiny n omplxity i n. In th litrtur, fw propo lgorithm llow olution tht r iffrnt from pnning tr. Th tt of th rt prnt th Multit Aptiv Multipl Contrint Routing Algorithm (MAMCRA) [13] on of th mot prtinnt lgorithm tht ttmpt thi tr-off vn if thr r vrl QoS ontrint. In ft, th ov it lgorithm im to omput tr th only llow olution, n thi l to th CST prolm. MAMCRA i n lgorithm tht olv th multi-ontrin multit routing prolm y omputing pil routing trutur. For thi, MAMCRA pro in two tp: In th firt tp, th lgorithm omput t of optiml pth rgring fin lngth funtion (f. Eqution 3). Th omputtion of th hortt pth u lightly moifi vrion of SAMCRA [21], n xt multi-ontrin unit lgorithm. In th on tp, MAMCRA tri to limint th ul runni tht r prou in th firt tp. For thi, MAMCRA u gry lgorithm. Th gry lgorithm itrtivly ompr two pth tht hr t lt on no n lt th longut prfix 3 from th our no to th frt ommon no of on of thm, if th rult olution till fil. Furthrmor, w noti tht ll of th ov it prolm r mono-ojtiv, with ot or lngth th only ojtiv. Coniring th Multi-Ojtiv Multit Routing Prolm, thr r fw propo olution in[15][9], u of th ru ufuln n th high omplxity of thi prolm. Furthrmor, mot of th propo olution r on mt-huriti uh gnti lgorithm [6] n nt oloni [19]. 3 Hirrhi Spnning Strutur in Grph To intify th optiml olution of th multi-ontrin prtil pnning prolm, w propo rif ovrviw of th hirrhy onpt, whih w propo in [18]. 2 A lour grph on t of no i omplt grph in whih h link ot i qul to th hortt pth twn it no. 3

7 Multi-ontrin multit routing 7 f Figur 2: Exmpl of tr Uully, pnning tr r onir th minimum ot prtil pnning trutur. Thy orrpon to onnt u-grph without yl. In, if minimum ot trutur i rquir to olv prtil pnning prolm without ny ontrint, thi trutur lwy orrpon to pnning tr ll th Stinr tr. To olv th optiml multi-ontrin multit routing prolm, om prviou work propo minimum lngth prtil pnning tr. For intn, th olution of th MSTP prolm prnt in [13] orrpon to prtil pnning tr minimizing non-linr lngth funtion. A minimum tr olution lwy xit ut it o not lwy tify th n-ton ontrint it i tt in [13]. Trivilly, th minimum lngth tr o not lwy ontin fil pth to h tintion. Furthrmor, rl QoS multit rqut o not olutly n tr, ut thy minly rh for fil pth from th our to ll tintion, whil minimizing th ot of th olution. Thu, if thr i no tr tifying th rqut, th minimum ot olution n iffrnt trutur ontining fil pth for h tintion. For intn, th olution n t of fil pth it i th rult of th firt tp of MAMCRA. Among th olution offring fil pth for ll tintion, thr i lwy n optiml olution with th mllt lngth (or ot), it w formult in MCMWM n MCMCM prolm. Hrftr, w montrt tht thi optiml olution orrpon to gnrliztion of tr. Tr r onnt u-grph without yl. In th following, impl nottion i u for root tr: th hilrn of no r numrt twn prnthi ftr th prnt no, it i illutrt y th following xprion orrponing to Figur 2: T = ((,(,),f)) (10) In root tr, h onrn no i prnt only on n h t mot on prnt no, xpt th root whih h no prnt. Th hirrhy onpt w firt introu in [18], whr tr-lik trutur ontining no rptition wr fin. Th following finition n onir th xtnion of th tr onpt to fin uful hirrhil trutur rlt to grph. In thi ppr, w only onir th root hirrhi. In ft, root hirrhi r mor pproprit to ri th multi-ontrin multit routing prolm. Dfinition 1 (Root hirrhy). A non-mpty root hirrhy in grph i onnt trutur, whr h no ourrn h t mot on prnt no. A root hirrhy n givn y th hirrhil tr-lik numrtion of no ourrn, illutrt in Figur 3. H = ((((f)),((,)))) (11) A hirrhy i not nrily xmpt from rptition: no n g of th rlt grph my prnt vrl tim in hirrhy. In our xmpl, th no n r prnt twi. It i importnt to mphiz tht no n hv mny ourrn in hirrhy ll no ourrn, n thi i lo ppli for g n r. Sin th iffrnt ourrn of th m lmnt my ply iffrnt rol in th hirrhy, th itintion n th intifition of th ourrn i utntil. In, no ourrn n n intrmit no in hirrhy whil nothr ourrn of th m no n lf, lik th no in Figur 3 (). If no ourrn h t lt two hilrn thn it i ll rnhing no ourrn, lik th no in Figur 3 (). Noti tht hirrhy i not u-grph, ut it gnrt u-grph in th i grph. Thi u-grph i ll th img of th hirrhy in th i grph. Th img of th mntion hirrhy in th i grph i hown in Figur 3 (). Hirrhi n irt or unirt. Thy lo n pn t of no or ll no. Howvr, to ri th hr nlyz ingl our QoS multit routing prolm, irt prtil pnning root hirrhi n ppli. Th hirrhi r irt from th our to th tintion. W will ltr tht th u of irt root hirrhi llow n urt ription of th olution for th multi-ontrin prtil minimum pnning prolm. Th numrtion of th mot importnt proprti of th hirrhi filitt thir ontrution to olv th ontrin pnning prolm.

8 8 M. Molnár & A. Bll & S. Lhou f () f () Figur 3: Exmpl of n unirt root hirrhy in n unirt grph 3.1 Bi Proprti of Hirrhi Rll tht tr r pil hirrhi whr h no h t mot on ourrn. Thu, tr i hirrhy. Noti tht, om proprti of tr r tru for hirrhi ut not ll, whr ll proprti hrtrizing hirrhi r tru for tr. Som importnt proprti of hirrhi n formult follow. Sin h no ourrn i iffrnt in hirrhy, th ylomti 4 numr of hirrhy i qul to 0. Howvr, th ylomti numr of th img of hirrhy n iffrnt from 0 in th gnrt u-grph n ontin yl. In hirrhy, thr i on n only on pth twn two no ourrn. A hirrhy my ontin mor thn on no ourrn of no n vntully mor thn on g/r ourrn of n g/r. Morovr, whn th m no pir i rpt in hirrhy, th r limit y th no i lo rpt. If grph lmnt i prnt vrl tim, it ourrn houl itinguih. In th following, w will itinguih th ourrn of no x y iffrnt xponnt x 1,x 2,... whn n. 3.2 Hirrhi U y MAMCRA Th mot ffiint huriti lgorithm to olv th multi-ontrin multit routing prolm i MAMCRA [13]. Thi lgorithm tht i rifly prnt in Stion 2 omput t of optiml pth with vntul runni, thn u gry lgorithm to limint om of th runni. At firt, w propo rif nlyi of th trutur u y MAMCRA. Noti tht MAMCRA o not gurnt th optiml olution. W will tht th t of optiml pth rturn in th firt tp of MAMCRA o not nrily ontin th pth longing to th optiml olution nithr of th MCMCM nor of th MCMWM prolm. Aftr th rif nlyi of th pth propo y MAMCRA, om importnt proprti of th optiml minimum ot or minimum lngth olution r prnt in th nxt tion. In th following, w will prov tht th t of pth tht r omput in th firt tp of MAMCRA wll th finl olution of MAMCRA r hirrhi. Lmm 2. A t of pth from th m our no to iffrnt tintion orrpon to hirrhy. Proof. Th lmm i trivil, in th t of pth i onnt u to th ommon our no n h no ourrn h t mot on prnt in thi t. Thu th firt tp of MAMCRA ontrut hirrhy. Lmm 3. Th multit routing trutur otin ftr th on tp of MAMCRA i hirrhy. Proof. Th firt tp of MAMCRA omput hirrhy. Th on tp limint om runni uing th following oprtion. Lt p 1 (,x 1, 1 ) n p 2 (,x 2,2) two pth hring ommon no x. Unr om onition xplin in [13], th prt from to x of on of th pth i omitt: for xmpl p 1 (,x 1 ) i lt n th ontntion p 2 (,x 2 )+ p 1 (x 2, 1 ) i u for th tintion 1, if thi nw pth till fil. To prov our lmm, it i uffiint to montrt tht th runny limintion lgorithm o not hng th trutur: th otin trutur i lo hirrhy. Sin p 2 (,x 2 ), p 1 (x 2, 1 ) n p 2 (x 2,2) r pth, n thy hr only th no ourrn x 2, th hilrn of x 1 will hng thir prnt no to x 2, n thy onquntly till hving on prnt no. Thrfor, MAMCRA rturn hirrhy. 4 Th ylomti numr of grph i fin th numr of inpnnt yl in th grph.

9 Multi-ontrin multit routing 9 p 1 p 2 p 3 p p 2 p 4 1 p p 4 () () p 1 p 4 1 () 2 2 () Figur 4: Th hirrhi omput y MAMCRA Figur 4 how th volution of th multit trutur omput y MAMCRA, whr four tintion,, n houl pnn. Figur 4 () prnt th four pth omput y SAMCRA in th i grph, whil Figur 4 () prnt th pth hirrhy. Thi hirrhy nl to itinguih th iffrnt ourrn of th no tht r hr y mor thn on pth. A pth p 1 (, 1 ) n p 2 (, 2 ) hv th m qun of no, on of thm n omitt. Thrfor, om n intrmit tintion no in p 1. Similrly, th tintion n onir n intrmit no in p 4. Th implifition r til in [13]. W n onir tht th four tintion r pnn y two fil hortt pth rturn y th firt tp of MAMCRA illutrt in Figur 4 (). Th tintion n r intrmit no in th pth p 1 n p 4 rptivly. Morovr, nothr ourrn of th no i rly no in th pth p 1. Aoring to Lmm 2, thi t of pth orrpon to hirrhy. Lt u uppo, tht th ontntion of p 1 (, 1 ) n p 4 ( 2,) i fil pth. In thi, th gry lgorithm in th on tp of MAMCRA rpl th pth p 4 y thi ontntion it i init in Figur 4 (). Th rulting trutur i hirrhy. It i importnt to mphiz tht thr i no oviou orrltion twn th hortt pth n th optiml olution. Morovr, vn if w u n xt lgorithm to limint th runni, th rult hirrhy my iffrnt from th optiml olution. Lmm 4. Th minimum lngth olution of th MCMWM prolm n th minimum ot olution of th MCMCM Prolm o not nrily long to th t of hortt pth omput y th firt tp of MAMCRA uing th non-linr lngth. Proof. Th proof i on n xmpl. Figur 5 illutrt tht th hortt pth, oniring th non-linr lngth, r not nrily inlu in th optiml olution. In th givn grph, th ot n th link wight w r init. Lt u uppo tht i th our no, n thr r two tintion: n. Th QoS ontrint r givn y L = [7,7] T. In Figur 5 (), w how tht th hortt pth (, ),(, )), uing th ot n th non-linr lngth funtion rptivly, o not ontin nithr th minimum ot olution nor th olution with minimum non-linr lngth. Th optiml olution orrpon to th tr (((,))). Figur 5 () illutrt th ft tht th tr with miniml non-linr lngth i not th optiml olution for th n-to-n ontrin multit routing prolm. In thi xmpl, th optiml olution of th MCMWM prolm i th tr (((, ))). Thi tr h grtr non-linr lngth t th lv thn th tr ((,)). Morovr, th olution of oth prolm n iffrnt. In, th minimum ot olution o not nrily orrpon to th minimum lngth olution of MCMWM.

10 10 M. Molnár & A. Bll & S. Lhou w=[5,5] =5 =3 w=[3,3] =5 w=[5,5] =4 =3 w=[3,3] =4 w=[5,5] w=[5,5] =3 =3 w=[3,3] w=[3,3] =3 =3 w=[1,3] w=[3,1] ) ) Figur 5: Th t of hortt pth n th t of minimum ot pth o not ontin th optiml olution 4 Th Minimum Prtil Spnning Hirrhi to Solv th Multi-Contrin Multit Routing It i known from [13] tht om fil multit trutur unr multipl n-to-n ontrint o not orrpon to pnning tr. It i th with th olution rturn y MAMCRA. Howvr th xt trutur of th olution w not yt nlyz. Th introution of th hirrhi llow n urt finition of th multit trutur. Morovr, our nlyi im to trmin th optiml olution of th multi-ontrin multit routing. W r prtiulrly intrt in riing th optiml olution of Prolm 1 (MCMCM). W minly invtigt on thi minimum ot olution, whr th pnning trutur olving Prolm 4 (MCMWM) hv th m proprti. 4.1 Th Minimum Cot Solution Lt M th optiml (of minimum lngth or minimum ot) multit trutur, with rpt to th givn QoS ontrint. Thorm 1. Th optiml multit trutur M with rpt to multipl ontrint on poitiv itiv mtri i lwy irt prtil pnning hirrhy. Proof. Th optiml multit trutur mut ontin t mot on irt fil pth from th our to h tintion. Conquntly, thi trutur i irt n onnt. Th trutur o not oligtorily pn th ntir no t, it i prtil pnning trutur. It i uffiint to prov tht th minimum ot prtil pnning trutur nnot ontin ny uprfluou yl. In othr trm, h no ourrn h t mot on prnt no n on r twn thi prnt n th no ourrn in th omput trutur. Lt u uppo tht no ourrn v 1 h two prnt no (or two inoming r from th m prnt no) in th minimum ot trutur M pnning {} D. Thn on of th pror r of v 1 n ropp without lo of th onntivity n th rmin trutur ovr {} D. Thrfor, M nnot th multit trutur with minimum ot. In th following, w ll thi olution th multi-ontrin minimum prtil irt pnning hirrhy (rvit y MC-MPDSH). Uing th hirrhy onpt, th MCMCM prolm n r-formult follow. Th optiml (minimum ot) multi-ontrin multit routing tht onit in fining minimum ot trutur hving t mot on irt pth p(, j ) from th our to h tintion j D, with rpt to th givn ontrint w(p(,j )) L i quivlnt to fin th multi-ontrin minimum prtil irt pnning hirrhy with rpt to th givn ontrint. Similrly, th MCMWM prolm n lo formult uing th hirrhy onpt. 4.2 Proprti of th MC-MPDSH In th following, w numrt om proprti of th MC-MPDSH. Thi proprti nl to i) ttr intify th optiml olution ii) montrt om limittion of known omputtionl lgorithm of pnning trutur n iii) ign mor ffiint xt n huriti lgorithm. Thu, w invtigt on th proprti of th minimum ot olution. Th following two proprti r lwy tru in th optiml multit routing trutur. Proprty 1. Th lv in th optiml pnning hirrhy M r tintion. Proof. Lt u uppo tht t M i lf no ut it i not tintion. In thi, t n it prnt r n ropp u ll of th tintion r pnn with th rmin onnt trutur. Thrfor, M n not th minimum pnning trutur.

11 Multi-ontrin multit routing 11 1 (1,3) (2,3) (4,1) (2,7) (5,2) (2,2) (5,3) 2 (3,2) (3,1) (2,5) (3,1) 3 Figur 6: An g n u vrl tim in n MC-MPDSH v 2 1 Figur 7: A no i prnt twi in pth Noti tht tintion my lo orrpon to n ritrry intrmit no ourrn. Proprty 2. In th optiml hirrhy M th irt pth from th our to n ritrry no ourrn v 1 longing to M i fil pth. Proof. M ontin fil pth for h tintion. For v 1 D, th proprty i trivil. Lt u uppo tht v 1 / D (n thu, ftr Proprty 1, v 1 i n intrnl no of pth towr tintion lf). Lt u lo uppo tht th pth (,v 1 ) i not fil. Conquntly, th pth from th our to th tintion whih r xtn from v 1 r not fil u th mtri r poitiv n itiv. Thi i in ontrition with th ft tht M ontin fil pth to ll tintion. W how tht n g/r my long vrl tim to th olution. In our prolm, th olution i uppo to irt hirrhy in n unirt grph. Figur 6 giv n xmpl tht n g of th grph n u y hirrhy vrl tim n in oth irtion. In thi xmpl, th g (,) i u thr tim in th optiml pnning hirrhy. Thi hirrhy i th only on proviing fil pth from th our to th tintion 1, 2 n 3 whn th QoS ontrint r givn y th vtor L = [13,13] T. Lt u noti tht thr r uppr oun on th numr of ug of n g n no in th optiml hirrhy. In th following, our ojtiv i to fin th uppr oun to ign ffiint hirrhy omputtion lgorithm. Aoring to Proprty 1, lf of th MC-MPDSH i lwy tintion. Thrfor, th MC-MPDSH h t mot D lv. Th following thr proprti r not trivil in MC-MPDSH, ut thy onirly hlp th ontrution of th optiml olution. Proprty 3. In irt pth from th our to n ritrry tintion j in th MC-MPDSH, no i prnt t mot on. Proof. Lt u uppo tht no v i prnt twi in irt pth of th MC-MPDSH it i illutrt in Figur 7. If thr i tintion 1 twn th two ourrn of v, th lt gmnt ( 1,v 2 ) from 1 to th on ourrn of v n limint, n th otin hirrhy h lowr ot n ttr QoS vlu t th tintion no. If thr r vrl tintion in th yl, trivilly, th lt gmnt from th lt intrmit tintion no ourrn to v 2 n limint. If thr i no tintion, thn th ntir yl n lt. Thu, th optiml olution n not ontin two ourrn of no in th m irt pth from th our. Thi proprty i lo tru rgring th our no itlf: no irt pth in n MC-MPDSH mk it poil to rturn to th our no. Conquntly, th our i prnt only on in n MC-MPDSH. Lt u noti tht no iffrnt from th our my prnt vrl tim in th optiml hirrhy ut h ourrn of th no long to iffrnt pth. Proprty 4. Th numr of th ourrn of no in th MC-MPDSH i uppr oun y D.

12 12 M. Molnár & A. Bll & S. Lhou () () 3 6 Figur 8: A mximum numr of rnhing no ourrn of no in th hirrhy Proof. Aoring to Proprty 3, no i prnt t mot on in ny pth of th MC-MPDSH. Lt u uppo tht th no v i prnt in ll of th irt pth of th MC-MPDSH. In th wort, thr r D irt pth uh tht ny pth (, j ) o not inlu ny othr tintion k. Sin v n long to ll of thm, th uppr oun i trivily qul to D. If om ourrn of no r rnhing no in th optiml hirrhy, thn th numr of it ourrn n mllr thn D. Proprty 5. If tintion i lf no in th MC-MPDSH, thn it h only on ourrn (th lf ourrn) in th optiml olution. Proof. Lt u uppo tht th no 1 i prnt twi in th MC-MPDSH: firt no ourrn 1 1 i lf in th pth p 1 (, 1 1) n 2 1 i nothr ourrn in n othr pth p 2 (, 2 ). Sin th MC-MPDSH ontin only fil pth, th pth p 2 (, 2 ) n it prfix p 2 (, 2 1) r oth fil. Thu, th pth p 1 (, 1 1) i ul for 1, n t lt th lt r of thi pth n lt without ffting th fiility of th olution. If thr r intrmit tintion in p 1 (, 1 1), thn only th lt gmnt from th lt tintion to 1 1 n lt. Thrfor, th two ourrn of 1 r not poil in th optiml olution. Proprty 6. In n MC-MPDSH, no n hv t mot D 2 rnhing no ourrn. Proof. Th mximl numr of rnhing no ourrn of givn no v i prou whn ll of th rnhing no ourrn orrpon to v n h ourrn i ommon no of t mot two pth towr two tintion lv. Not tht thr r t mot D lv in th hirrhy. If D i vn, th numr of rnhing no ourrn of v i D El if D i o, th lt ourrn of v i not rnhing no, in it ontin only on uor. Thrfor, th uppr oun of th rnhing no ourrn of v i. D 2 2. Figur 8 illutrt th wort introu in Proprty 6. Th lt two proprti n gnrliz in Proprty 7. Proprty 7. Lt v i th i th ourrn of th no v in u-hirrhy H in n MC-MPDSH n + H (vi ) it out-gr. Lt l H th numr of lv in H. For th ourrn of th m no v, th following inqulity lwy hol: + H (vi ) l H (12) v i H Proof. If v h lf ourrn in H, thn oring to Proprty 5, thi ourrn i th only on of th no v in H, n v houl tintion. In thi, Proprty 7 i trivil. If th iffrnt ourrn of v r not lv, thn h no ourrn h t lt on hil. Mor prily, th ourrn v i h + H (vi ) u-hirrhi. In h u-hirrhy thr i t mot on lf no. Aoring to Proprty 3, h pth to lf ontin t mot on ourrn of v. Not tht th mtri in th grph r poitiv n itiv. Suppoing n optiml olution for givn QoS rqut, wht n tt out th u-optimlity of th u hirrhi of th optiml hirrhy? Th following proprty nl to tlih n importnt rltion twn th QoS ontrint n giv u-optimlity proprty.

13 Multi-ontrin multit routing 13 Proprty 8. Lt M n MC-MPDSH root t n tifying th ontrint L in th lv. Lt M v u-hirrhy of M root t v. Lt w(p(,v)) th wight vtor of th pth p(,v). Th u-hirrhy M v i n MC-MPDSH from v to th tintion ourrn tht it ontin with rpt to th ontrint L v = L w(p(,v)). Proof. A tintion no h only on tintion ourrn in th MC-MPDSH (othr ourrn of th no n u intrmit rly no ourrn towr othr tintion ut th no houl riv multit mg tintion only on).thrfor, w rfr to tintion y tintion ourrn. At firt, w prov tht w(p(v, j )) L v = L w(p(,v)) for ll tintion j M v. Lt u uppo tht for th tintion j M v th wight vtor w(p(v, j )) o not omint L v. In thi, w(p(, j )) o not omint L n thu M n not fil. Sonly, w prov tht M v i th minimum ot hirrhy in th t of th fil hirrhi pnning th tintion in M v. Lt u uppo tht M v i not minimum ot olution ut only fil olution for th prtil prolm. In thi, thr i n MC-MPDSH M v root t v th ot of whih i l thn th ot of M v. By rpling M v y M v in th hirrhy M, w otin fil pnning hirrhy with l ot, whih i in ontrition with th ft tht M i n MC-MPDSH. To our knowlg, our tuy i th firt on to nlyz th miniml olution of th multi-ontrin prtil pnning prolm orrponing to th QoS multit routing prolm. To olv thi prolm, th ign of xt or huriti lgorithm i n importnt hllng for th QoS multit routing. 5 An Ext Algorithm to omput th Multi-Contrin Minimum Prtil Spnning Hirrhi A it w prnt rlir, th multi-ontrin multit routing prolm i NP-iffiult. Unfortuntly, th thniqu whih r known to olv th Stinr prolm uh th Stinr Tr Enumrtion Algorithm n th Topology Enumrtion Algorithm n not ppli hr, u of th multipl ourrn of th grph lmnt in th optiml olution. Prily, th olution w try to omput i not u-grph ut hirrhy. Th rnh n ut lgorithm i goo nit to xtly olv th r prolm. Furthrmor, th u of th ut oprtion orrponing to th optiml hirrhy proprti r intrting to lrt th omputtion. In th following, w prnt our firt propol to omput th optiml MC-MPSH. W rgu tht vrl xt lgorithm n foun n furthr rrh i n to fin mor ffiint olution. 5.1 A Brnh n Cut Algorithm Th u of th rnh n ut lgorithm llow th numrtion of th irt prtil pnning hirrhi tifying th limittion mntion in Stion 4.2. Th hr propo lgorithm lt hirrhi in n inring orr of thir totl ot. Th firt hirrhy in th our of th numrtion, whih ovr ll tintion n tifi th n-to-n ontrint, orrpon to th optiml olution. Eh no of our virtul rh tr orrpon to irt hirrhy root t th our no. Th lgorithm gin with hirrhy only ontining th our no. Th ot n th n-to-n wight on th pth from th our to th lv r omput. Trivilly, hirrhy i not fil if it ontin lf whih o not tify th givn QoS ontrint. At h tp of th rnh n ut lgorithm, th hirrhi orrponing to th lf no of th rh tr r xmin n th hirrhy with th lowt ot i lt. If thr i no lf hirrhy in th rh tr tifying th n-to-n ontrint, th prolm h no olution in th givn grph with th givn ontrint. Lt K th t of lv of th rh tr. Th mt-o of th lgorithm i givn y Algorithm 1. Th ruil qution of thi lgorithm i how th vli uor of th lt hirrhy H n omput. Th uor r th hirrhi nlrg only y jnt r from th lv of H tht tify th QoS ontrint. Thi kin of r r ll fring r, xprion orrow from [24]. Th fring r of hirrhy r illutrt y Figur 9. Lmm 5. Algorithm 1 numrt ll vli pnning hirrhi in inring orr. Thi numrtion o not lo ny vli hirrhy. Proof. Noti tht for givn vli hirrhy H, Algorithm 1 fring link to th lv of H n ontrut ll vli hirrhi y omining th link. In, for k fring link th lgorithm ontrut t mot k i=1 ( k! i!(k i)! ) nw vli hirrhi. Lt u uppo tht th lgorithm omit to ontrut vli H tht ontin n r t n intrmit no of H n H i vli. In th rh tr of Algorithm 1 thr i no v in hirrhy H for whih th r i fring r. Sin H i vli, th uor of H ontining i vli n th lgorithm h lry numrt thi hirrhy.

14 14 M. Molnár & A. Bll & S. Lhou Algorithm 1 Brnh n ut lgorithm to fin MC-MPDSH Rquir: th wight grph G = (V,E), th wight vtor w n th ot () for h g E, th our no, th tintion t D n th ontrint vtor L Enur: H th MC-MPDSH if it xit H 0 {initiliz th hirrhy with th our} K {(H 0 )} { t of th lv in th rh tr} whil K o lt H th lf in K with miniml ot (H) if D H thn rturn H {it i th optiml olution} STOP n if S uor hirrhi of(h) for ll H S o K K {(H )} { H to K} n for K K \ (H) {lt H from K} n whil STOP without olution H i m f j g k h n o p l g h f 1 k H fring link Figur 9: Th fring link of hirrhy Morovr, if hirrhy H i not vli, u th n-to-n ontrint n/or th proprti of th optiml hirrhy r not tifi, ing t of fring link n not improv th hirrhy. Thu, th uor of non vli hirrhy r non vli. Proprti 3, 4 n 6 n irtly ppli to hoo th vli ontinution of th lt hirrhy. An r ling to no, whih o not orrpon to th proprti i not uful for ontinuing th ontrution of th olution. For intn, lt u uppo tht no v i lry prnt on th irt pth p(,n), whih onnt th our to th trt no n of th r (n,v). Aoring to Proprty 3, th optiml olution n not ontin th pth otin y ontnting p(,n) n p(n,v): th r (n,v) i not poil jnt r to ontinu th hirrhy H ontrution towr otining th optiml olution. Morovr, th numr of lv wll th numr of th ourrn of n ritrry grph no in th optiml olution r limit y D hown in Proprty 4, n no n rnhing no t mot D 2 tim hown in Proprty 6). Only th fring r orrponing to th oun r uful. To numrt ll of th vli uor of H in th rh tr, th ut of th poil fring r houl xhutivly numrt. Morovr, thi numrtion nur tht hirrhy i viit only on y th rnh n ut lgorithm. Th omputtion of th t of uor of givn hirrhy H i ri y Algorithm 2. i f 2 j n k Thorm 2. Algorithm 1 rturn th optiml olution of th multi-ontrin minimum prtil pnning hirrhy prolm if uh olution xit. Proof. Th lgorithm numrt ll vli pnning hirrhi from th our no. Thu, th optiml olution i lwy viit for othr vli olution if uh olution xit hown in Lmm 5. Lt u uppo tht nothr hirrhy with lowr ot i rturn for th optiml olution. It i in ontrition with th ft tht th optiml olution i th minimum ot olution of ll fil olution.

15 Multi-ontrin multit routing 15 Algorithm 2 Computtion of uor hirrhi Rquir: th wight grph G = (V,E), th hirrhy H orrponing to th urrnt no in th rh tr Enur: S th t of uor hirrhi of H with rpt to th n-to-n QoS rquirmnt n Proprti 3, 4, 6 S A {t of poil jnt r} Stp 1: Gnrtion of ll poil fring r from H for ll lf n of H o A n jnt r of (n) {jnt r of n} V n prnt of (n) {t of prnt no of n in H} for ll r A n o {omputtion of poil jnt r of n} t oppoit no of () no t numr of ourrn(t,h) w(t) w(n) + w() {umult wight t t} if ( w(t) L)&(t / V n )&(no t < D ) thn A A {} n if n for n for Stp 2: Comintion of th fring r omput in Stp 1 for ll omintion C(A) of th r in A o H H C(A) { th r of C(A) to H} if (numr of lv(h ) D ) & (rnhing no ourrn(oppoit no()) D 2, C(A)) thn S S {H } { H nw uor to S} n if n for

16 16 M. Molnár & A. Bll & S. Lhou 6 Conluion n Prptiv Th min rult of our invtigtion on th multi-ontrin multit routing prolm i th hrtriztion of th optiml olution trutur. Thi trutur lwy orrpon to hirrhy. Th ontrution of th minimum ot n th minimum non-linr lngth hirrhi r NP-iffiult optimiztion prolm. Th lgorithm omputing huriti olution, lik MAMCRA, mnipult hirrhi uh t of pth root t th our n mor or l runny-fr pnning trutur. In our ppr, w propo n initil tuy of th proprti of th hirrhy-typ olution. Evn if thr i no imultion rult in thi ppr, w rgu tht it i utntil to firt intify th optiml olution trutur of th r prolm. An itionl rult of our nlyi i tht th optiml olution o not long nithr to th t of minimum ot pth nor to th t of hortt pth omput y uing th frquntly u non-linr lngth. Sin hirrhi my ontin vrl ourrn of no n n g, mot of th numrtion lgorithm r not pproprit to omput th optiml olution. Thrfor, w propo n ffiint rnh n ut lgorithm to fin th optiml hirrhy. Following thi firt tuy on multi-ontrin minimum prtil pnning hirrhi, importnt futur work on xt n huriti lgorithm of thi prolm i n to fin ffiint olution. Rfrn [1] H. Lin n Z. Yu-Lin n R. Yong-Hong, Two multi-ontrin multit QoS routing lgorithm, ighth ACIS Intrntionl Confrn on Softwr Enginring, Artifiil Intllign, Ntworking, n Prlll/Ditriut Computing, [2] V. Aggrwl n Y. P. Anj n K. P. K. Nir,Miniml pnning tr ujt to i ontrint, Journl of Computr & Oprtion Rrh, volum 9(4), pp , [3] F. Bur n A. Vrm, ARIES: A Rrrngl Inxpniv Eg-B On-Lin Stinr Algorithm, INFOCOM, pp , [4] Z. Wng n J. Crowroft, Qulity-of-Srvi Routing for Supporting Multimi Applition, IEEE Journl of Slt Ar in Communition, volum 14(7), pp , [5] N. Bn Ali n M. Molnár n A. Blghith, Multi-ontrin QoS Multit Routing Optimiztion, IRISA, Rrh Rport, numr 1882, [6] J. Crihigno n B. Brn, A Multit Routing Algorithm uing Multiojtiv Optimiztion, ICT, Springr-Vrlg, pp , [7] G. Fng, A multi-ontrin multit QoS routing lgorithm, Journl of Comptur Communition, volum 29, pp , [8] S. L. Hkimi, Stinr prolm in grph n it implition, Journl Ntwork, volum 1, pp , [9] S. P. Hong n S. J. Chung n B. H. Prk, A fully polynomil iritri pproximtion hm for th ontrin pnning tr prolm, Oprtion Rrh Lttr, volum 32(3), , [10] F. K. Hwng n D. S. Rihr, Stinr Tr Prolm, Journl of Ntwork, volum 22, pp , [11] P. V. Kompll n J. C. Pqul n G. C. Polyzo, Multit routing for multimi ommunition, IEEE/ACM Trntion on Ntwork, volum 1(3), pp , [12] T. Korkmz n M. Krunz, Multi-ontrin optiml pth ltion, INFOCOM, pp , [13] F. A. Kuipr n P. Vn Mighm, MAMCRA: ontrin- multit routing lgorithm, Journl of Computr Communition, volum 25(8), pp , [14] S. Kumr n P. Rolvov n D. Thlr n C. Alttinoglu n D. Etrin n M. Hnly, Th MASC/BGMP Arhittur for Intr-Domin Multit Routing, SIGCOMM, pp , [15] M. S. Lvin n R. I. Nuirikhmtov, Multiritri Stinr Tr Prolm for Communition Ntwork, Informtion Thnology in Enginring Sytm, volum 9(3), pp , [16] X. Mip-Bruin n M. Ynnuzzi n J. Domingo-Pul n A. Font n M. Curo n E. Montiro n F. Kuipr n P. Vn Mighm n S. Avllon n G. Vntr n P. Arn-Gutirrz n M. Hollik n R. Stinmtz n L. Innon n K. Slmtin, Rrh Chllng in QoS Routing, Computr Communition, volum 29(1), 2006.

17 Multi-ontrin multit routing 17 [17] P. Vn Mighm n F. A. Kuipr, Conpt of xt QoS routing lgorithm, IEEE/ACM Trntion on Ntwork, volum 12(5), pp , [18] M. Molnár, Optimition ommunition multit ou ontrint, Univrity Rnn 1, 2008, [19] D. Pinto n B. Brán n R. Frgt, Multi-Ojtiv Multit Routing B on Ant Colony Optimiztion, Proing of th 2005 Confrn on Artifiil Intllign Rrh n Dvlopmnt, pp , [20] H. Tkhhi n A. Mtuym, An pproximt olution for th Stinr prolm in grph, Journl of Mthmti Jponi, volum24(6), pp , [21] P. Vn Mighm n H. D Nv n A. F. Kuipr, Hop-y-hop Qulity of Srvi Routing, Computr Ntwork, volum 37(3-4), pp , [22] D. Wng n F. Ergun n Z. Xu,Unit n Multit QoS Routing with Multipl Contrint, Qulity of Srvi in Multirvi IP Ntwork, volum 3375, pp , [23] P. Wintr, Stinr prolm in ntwork: urvy, Journl of Ntwork, volum 17, pp ,1987. [24] X. Zhng n J. Wi n C. Qio, Contrin multit routing in WDM ntwork with pr light plitting, INFOCOM, volum 3, pp , [25] Z. Qing n P. Mhr n J. J. Gri-Lun-Av, A Sour-B Algorithm for Dly-Contrin Minimum-Cot Multiting, INFOCOM, pp , [26] R. C. Prim, Shortt onntion ntwork n om gnrlition, Bll Sytm Thnil Journl, volum 36, pp , 1957.

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