Link-Disjoint Paths for Reliable QoS Routing

Transcription

1 Link-Disjoint Pths or Rlil QoS Routing Yuhun Guo, Frnno Kuiprs n Pit Vn Mighm # Shool o Eltril n Inormtion Enginring, Northrn Jiotong Univrsity, Bijing, 000, P.R. Chin Fulty o Inormtion Thnology n Systms, Dlt Univrsity o Thnology, P.O. Box 0, 600 GA Dlt, Th Nthrlns Astrt: Th prolm o ining link/no-isjoint pths twn pir o nos in ntwork hs riv muh ttntion in th pst. This prolm is irly wll unrstoo whn th links in ntwork r only spii y singl link wight. Howvr, in th ontxt o Qulity o Srvi routing, links r spii y multipl link wights n rstrit y multipl onstrints. Unortuntly, th prolm o ining link/no isjoint pths in multipl imnsions s irnt onptul prolms. This ppr prsnts irst stp to unrstning ths onptul prolms in link-isjoint Qulity o Srvi routing n proposs huristi link-isjoint QoS lgorithm tht irumvnts ths prolms. Introution Th prolm o ining isjoint pths in ntwork hs n givn muh ttntion in th litrtur u to its thortil s wll s prtil signiin to mny pplitions, suh s lyout sign o intgrt iruits, survivl sign o tlommunition ntworks n rstorl/rlil routing. Pths twn givn pir o sour n stintion nos in ntwork r ll link isjoint i thy hv no ommon (i.. ovrlpping) links, n no isjoint i, sis th sour n stintion nos, thy hv no ommon nos. With th vlopmnt o optil ntworks n th ploymnt o MPLS or GMPLS ntworks, th isjoint pths prolm is riving rnw intrst s st rstortion tr ntwork ilur is ruil in suh kin o ntworks. In roust ommunition ntworks, onntion usully onsists o two link- or no-isjoint pths: on tiv pth, n on kup pth. A srvi low will rirt to th kup pth i th tiv pth ils. Lo lning, nothr importnt spt or ommunition ntworks to voi ntwork ongstion n optimiz ntwork throughput, lso rquirs isjoint pths to istriut lows. Roustnss n lo lning r, mong othrs, oth spts o Qulity o Srvi (QoS) routing. In this ppr w will ous on ining QoS-wr link-isjoint pths. In gnrl link-isjoint pths lgorithm n xtn to no-isjoint lgorithm with th onpt o no splitting, i.. rpling on no with two nos tht r link togthr y link with zro wights [6]. Throughout this ppr, w us th ollowing nottion. A ntwork is not y irt grph G(V,E), whr V is th st o nos n E is th st o links. A irt link rom no u to no v is rprsnt s u v, u, v V. Eh link is hrtriz y link wight vtor w r onsisting o M link mtris w m (u v), or m =,, M. W ssum tht only nonngtiv link mtris r ssign to h link. Howvr, in th pross o omputing isjoint pths, ngtiv link # Corrsponing uthor Th rsrh or this ppr ws onut whn Y. Guo ws visiting sintist t Dlt Univrsity o Thnology, sponsor y NUFFIC (Nthrlns orgniztion or ooprtion in highr ution) n CSC (Chins sholrship ounil).

2 wights my ssign to links. QoS mtris n ) itiv,.g. ly, jittr, in whih s th pth-wight vtor onsists o summing th link-wight vtors o th links ining th pth, ) multiplitiv,.g., on minus th pkt loss proility, whih n onsir s itiv tr tking th logrithm n ) min-mx,.g. nwith, n poliy lgs, in whih s th minimum (or mximum) link wight ins th wight o pth. Min/mx links tht o not oy th onstrints n prun rom th topology, whih is ll topology iltring. Aitiv mtris us mor iiultis n thror without loss o gnrlity, w ssum ll mtris to itiv [9]. In th ontxt o QoS routing or multi-onstrin routing, pth is ll sil whn its wight vtor os not violt th onstrints spii y th vtor L r. Sin w minly ous on ining link-isjoint pths, pth P, twn sour s n stintion t is onsir to st o links tht ompos this pth. With slight us o nottion, w hoos P to not th pth s wll s its link st. I pth P is link-isjoint with P, thr is no ommon link lmnt in th link st rprsnting h pth n P I =, ls P I. P P Dinition o pth lngth: Givn grph G(V,E) with M mtris pr link, th non-linr lngth o pth P rom sour no s to stintion no t is in s []: wm ( P) l( P) = mx ( ) () m=, K, M L whr w m (P) = (u v) P w m (u v). Th normliztion in () y th onstrints L r srtins tht i l(p)>, thn on o th onstrints hs n violt. For M =, th non-linr lngth o pth s in in () rus to linr on, n th link wight vtor w r rus to slr w(u v). Whn no onstrint is rquir, s in th LPP prolm stt low, th linr lngth o pth is omput s (u v) P w(u v), i.. L =. For simpliity o rprsnttion, th ov nottion o pth lngth l(p) is still us. I pth P is link-isjoint with P, i.. P I =, w hv l P U P ) = l( P ) + l( ) or M =. But or M > P m ( P, w hv l P U P ) l( P ) + l( ). Our trgt in this ppr is to in st o two link-isjoint pths tht oth ( P oy multipl onstnts. W in th totl lngth o two pths s or M. l P ) + l( ) () ( P Link-isjoint Pth Pir (LPP) Prolm. Givn grph G(V,E) with mtri pr link (M = ), or sourstintion pir (s,t), in st o two pths P n P, suh tht P I P =, n th totl lngth l(p ) + l(p ) is minimiz. Th LPP prolm n solv in polynomil tim [][][6]. Multipl Constrin Pth (MCP) Prolm. Givn grph G(V,E) with M > mtris pr link n onstrint vtor L r, or sour-stintion pir (s, t), in pth tht oys th onstrint vtor L r,

3 whr w ( P), or m =,, M, m L m w ( P) = w ( u v), or m =,, M. m m u v E Th MCP prolm is NP-omplt [8][0]. Multipl Constrin Link-isjoint Pth Pir (MCLPP) Prolm. Givn grph G(V,E) with M > mtris pr link n onstrint vtor L r, or sour-stintion pir (s, t), in pir o link-isjoint pths P n P, suh tht P I =, n oth pths oy th onstrint vtor L r. P Thorm. MCLPP is NP-omplt. S 0 i i+ S- i i Figur. Th ssignmnt o link wights to th links in th hin topology twn nos i n i+ Proo: Givn hin topology with n+ nos n n links, h with two-omponnt wight vtor s pit in Figur n st o numrs i A, 0 i S, or i=,...,n, whr S = hosn s ollows: L = ns-(s/), n L =(S/). n i i=. Th onstrints r To solv th MCLPP prolm, w n to in two pths P n P rom no to no n+ tht oy th onstrints. Sin, or ll link wight vtors, th sum o th omponnts quls S, w hv tht w (P)+w (P)=nS n w (P )+w (P )=ns. Aoringly, solution stisying th onstrints is only oun i w (P n P )=ns-(s/) n w (P n P )=(S/). Th prolm hs now om n instn o th wll-known NP-omplt prtition prolm [8] n n only solv y ining th st A A, or whih i =(S/). A sil pth P xists i A' th st A xists. A sil pth P onsists o th lowr link i i A n th uppr link i i A. Th pth P thn ollows th rmining links. i In this ppr w ous on solving th MCLPP prolm. Rlt work on ining isjoint pths in on imnsion twn sour n stintion will rviw in Stion n simpl link-isjoint lgorithm LBA will xplin in Stion. In Stion n xtnsion o LBA to multipl imnsions is isuss n shown to iiult. Thror, huristi lgorithm DIMCRA or solving th MCLPP prolm is propos in Stion. W onlu this rtil in Stion 6.

4 Rlt work. Link-isjoint Pths Routing in On Dimnsion An intuitiv mtho to trmin two shortst link-isjoint pths twn pir o sour n stintion nos onsists o two stps. Th irst stp rtrivs th shortst pth twn givn pir o nos in grph. Th son stp is to rmov ll th links o tht pth rom th grph, n to in th shortst pth in th prun grph. W will rr to this mtho s th Rmov-Fin (RF) mtho. Although th RF mtho is irt n simpl, it hs t lst two isvntgs u to th rmovl o links longing to th irst shortst pth: () provi tht two link-isjoint pths xist, thr is no gurnt tht thy will oun s illustrt in Stion. n () th son link-isjoint shortst pth my hv signiintly lrgr lngth. To surmount th isvntgs o th RF mtho, othr mthos hv n vis to in pir o shortst link-isjoint pths with miniml totl lngth [][][][9][][][6][]. In [], Suurll proposs n lgorithm, rrr to s Suurll s lgorithm, to in K no-isjoint pths with miniml totl lngth using th pth ugmnttion mtho. Th pth ugmnttion mtho is originlly us to inrs th siz o mthing with n ugmnting pth [6] n to in mximum low or minimum ost low in ntwork [7][]. Th prolm to in link/no isjoint pths n viw s spil s o th minimum ost low prolm s monstrt in [][][6]. Th si i o Suurll s lgorithm is to onstrut solution st o two isjoint pths s on th shortst pth n shortst ugmnting pth. K isjoint pths n otin y ugmnting th K- optiml isjoint pths with this lgorithm. In 98, Suurll n Trjn [6] improv Suurll s lgorithm suh tht pirs o link-isjoint pths rom on sour no to n stintion nos oul iintly otin in singl Dijkstr-lik omputtion. This lgorithm is rrr to s th S-T lgorithm. To in n pirs o isjoint pths, th S-T lgorithm rquirs O( E log( + E / n) n) tim n Suurll s lgorithm O ( n log n), whr n is th numr o stintion nos n E is th numr o links. Kr t l. [] n Koilm n Lkshmn [][] inorport th S-T lgorithm into thir lgorithms to in pir o link-isjoint pths srving s tiv n kup pths or routing nwith gurnt onntions. Ling [7] xtn th S-T lgorithm to in two link-isjoint pths twn pir o nos with optimiztion in oth ntwork lo n routing ost. In 99, Bhnri [] propos n lgorithm to in pir o spn-isjoint pths twn two nos in optil-ir ntworks. Th isjoint pths lgorithm us y Bhnri is moii vrsion o Suurll s lgorithm [] tht rquirs spil link wight trnsormtion to ilitt th us o Dijkstr s. Bhnri m simpliition to Suurll s lgorithm y irtly stting ll th link wights on th irst shortst pth ngtiv. Shikh [] m n xtnsion to Bhnri s lgorithm [] to solv th spn-isjoint pths prolm in mor omplit strutur optil ntworks. It is prov in [6][] tht th LPP prolm will NP-omplt i it is rquir tht th mximl lngth o th two isjoint pths, i.. mx(l(p ), l(p )), is minimiz. In ition, Ho n Mouth [0] propos nothr optiml ojt untion α l(p ) + l(p ), whr P n P r th tiv pth n th kup pth, rsptivly. Th prmtr α n st lrg or shr prottion shm (:N or M:N) n oul s smll s unity or it prottion shm (:). Whn α =, it rus to th ojt untion us in [][][6].

5 Huristi lgorithms s on mtrix lultion [8] or rursiv mtrix-lultion [0] to solv th K- shortst link-isjoint pths prolm with oun hopount hv n propos s wll. Thr r lso som lgorithms or ining K-st pths, i.. K isjoint or mximlly isjoint pths with minimum totl lngth twn pir o nos, in trllis grph [][]. An optiml lgorithm or ining K-st pths without hopount limittion twn pir o nos is givn y L n Wu in [], whr thy trnsr th K-st pths prolm into mximum ntwork low n minimum ost ntwork low lgorithm vi som moiitions to th originl grph. Distriut lgorithms or th link/no-isjoint pths lgorithms n oun in [][9][].. Disjoint Pths Routing in Multipl Dimnsions To th st o our knowlg thr is no litrtur on th MCLPP prolm. Rntly som pprs on isjoint pths in QoS routing hv mrg. Howvr, thy only onsir nwith n/or ly s thir QoS mtris [][][][9][8]. Th mximlly isjoint shortst n wist pths (MADSWIP) lgorithm rom Tt-Plotkin, t l. [7], involvs moii vrsion o th S-T lgorithm to in pir o isjoint pths. MADSWIP n prou pir o wist or shortst mximlly link-isjoint pths rom sour no to ll othr nos. Morovr it tris to in two pths simultnously to stisy th mximlly link-isjointnss to h othr in QoS routing ontxt. Howvr th link mtris us in thir lgorithm r nwith n ly, whr only th lttr mtri is itiv. Pth Augmnttion or Solving LPP In this stion w will prsnt simplii vrint o Bhnri s Algorithm [], rrr to s LBA (Link-isjoint vrsion o Bhnri s Algorithm), whih n prou n optiml solution or th LPP prolm. Th si stps o LBA r givn in Stion.. Th unmntl onpts o this lgorithm r isuss in Stion.. Th optimlity is prov in Stion. n in Stion., LBA is shown to loop-r.. Th stps o LBA Bhnri s lgorithm [] ws sign to in pir o spn-isjoint pths in n optil ntwork. W moiy Bhnri s lgorithm into link-isjoint pth pir lgorithm LBA y omitting th no-splitting oprtion tht nsurs th no-isjointnss n th grph trnsormtions tht nsur spn-isjointnss. Bor xplining th oprtion o LBA w irst introu som nottions tht will us urthr. I w rvrs th irtion n th sign o th link wights o h link on th pth P twn s n t, i.. w(v u) = w(u v), (u v) P, thn w will hv pth irt rom t to s, not y P, whih onsists o th rvrs P links. W in l( P ) = l(p ). A st, whih onsists o th P links whos rvrs links ppr on ~ P n vi vrs, is not s P I P = {( u v) n ( v u) ( u v) P n ( v u) }. In ll P th igurs, ol lins rprsnt links on th shortst pth(s) in grph or its orrsponing moii grph, sh lins rprsnt rvrs links whih o not xist in th originl grph n ol sh lins rprsnt suh rvrs links tht ppr on th shortst pth. Th stps o th LBA lgorithm r s ollows: With th inition o lngth in (), w hv l( P ) = l(p ) only or M =.

6 Givn irt grph G(V, E), or sour-stintion pir (s, t), Stp. Fin th shortst pth P rom no s to no t; Stp. Rpl P with P, moii grph G(V,E ) is rt; Stp. Fin shortst pth P rom no s to no t in th moii grph G(V,E ); i P os not xist, thn stop; Stp. Tk th union o P n P, rmov rom th union th link st whih onsists o th P links whos rvrs links ppr in P, n vi vrs, thn group th rmining links into two pths n, ~ i.. U = P U P ) \ ( P I ). ( P 6 6 () Stp () Stp 6 6 () Stp () Stp Figur. Exmpl o th oprtion o LBA W will xplin th stps o LBA with n xmpl in Figur. Suppos tht w r rquir to in st o two shortst isjoint pths twn n. In Stp, th shortst pth rom to is oun s P =, with minimum lngth. In Stp, moii grph G(V,E ) is rt y rvrsing th irtion n th sign o th wight o h link on P. For instn, th link with wight is rpl y th link with wight. ~ In Stp, th shortst pth in th moii grph P = hs lngth 6. In Stp, P I P ={, } is rmov rom th union P U P. Th solution st o isjoint pths { P, } ={, } is otin. Th totl lngth o this pth st quls + = 0, whih is xtly th miniml totl lngth o two link-isjoint pths in this grph. I thr xist mor thn on shortst pth in th originl grph or in th moii grph, ithr on o thm n hosn. Choosing irnt shortst pths my l to irnt solution sts. Howvr, ths solution sts will hv th sm minimum totl lngth. 6

7 6 6 () Stp () Stp Figur. Exmpl o th oprtion o RF () Stp () Stp Figur. Exmpl o th oprtion o RF For omprison, in Figur, w pply th RF mtho on th sm topology with th sm rquirmnts. In stp th shortst pth is rtriv. In stp, moii grph is rt y rmoving ll th links on Th shortst pth in th moii grph is with lngth. Thus th st {, } hs totl lngth + =, whih is longr thn 0 s oun with LBA. This xmpl illustrts tht th RF mtho nnot gurnt to in th optiml solution. Mor importnt, in th grph shown in Figur (), lthough thr xist two linkisjoint pths twn n, RF nnot in th son pth in stp s shown in Figur (). LBA, on th othr hn, still rturns th optiml st in this s.. LBA is Bs on th Shortst Pth In this sustion, w will lriy why th optiml solution st o LBA, s wll s othr pth ugmnttion lgorithms [,8,9], is s on th shortst pth. Although th thory prsnt hr is s on (or n riv rom) th thory o min-ost low [7][], it is instrutiv to giv n outlin. W will irst show tht th optiml st or th LPP prolm is s on th shortst pth. Sonly, w will show tht th optiml st o two link-isjoint pths hs th smllst irn in lngth rom th shortst pth mong ll th possil sts o link-isjoint pths. Finlly, w will show tht th logil irn st (in low) n viw s pth. Givn igrph G(V,E) n pir o sour-stintion nos (s, t), th rltion twn st o two linkisjoint pths {P, P } n th shortst pth P longs to on o th ollowing typs:. P itsl is P or P, i.. P = P or P = P ;. P ovrlps with oth pths P n P, i.. P I P, P P n P I P, P P ; 7

8 . P only ovrlps with on pth in th st {P, P }, ut not with th othr on, i.. P I P, P P n P I = (or P I P, P P n P I P = ); P. P is link-isjoint with oth pths in {P, P }, i.. P I ( P ) U P =. Lmm. Givn irt grph G(V, E) n sour-stintion pir (s, t), i th optiml st P, } o LPP { P xists, P U must ontin ithr th irst shortst pth P itsl or som P links on h o its two pths. Proo: I P U is o typ (), thn h pth in { P, P } is link-isjoint with P. As P is th shortst pth, oth P, } n P, } hv totl lngth shortr thn P, }. Hn th optiml st P, } nnot o { P { P { P { P typ () n P U must ontin som or ll P links to th optiml st. I P U is o typ (), only on pth in P U P ontins som P links, without loss o gnrlity, suppos ontins som P links, n th othr pth is link-isjoint with P, thn { P, P } is st whih is shortr thn P, }, Hn th optiml st P, } nnot o typ (). { P { P Thror, i th optiml st { P, } xists, P U must ithr o typ () or (). Proprty. Th optiml st P, } hs th smllst irn in lngth { P Y = l( ) + l( ) l( P ) 0 () rom th shortst pth P, mong ll th possil sts o link-isjoint pth pirs. In th st U U ( P ), th P links ontin in th st P U will orm loops with th P links. For xmpl, i P link u v is ontin in th st P U, thn it will rt loop with th link v u on P twn th nos u n v. Th lngth o this loop is zro us w(v u) = w(u v). Lt us not O l ~ = U ) I ( ), whih mns tht th st O l onsists o h P link in th union o P U P n its ( P orrsponing P link. W in th logil irn st twn P U n P s ( P U P = U U ) \O l. In t, l(o l ) = 0 us th st O l onsists o loops with zro lngth, h ) ( P onsisting o pir o opposit P n P links. With l( P )= l(p ), w hv l(( U ) P ) = l(( U ) U ( P )) l( Ol ) = l( ) + l( ) + l( P ) = l( ) + l( ) l( P ), Th logil irn st P P lso n omput s P P ={(u v) (u v) P \(P P )} U { (v u) (u v) P \(P P )}, whih mns tht i link u v o P os not ppr on P, thn this link longs to th irn st P P, n i link u v o P os not ppr on P, thn its irtion rvrs link v u longs to th irn st P P, with link wight w(v u) = w(u v). In st thory, th irn oprtion is in s P P = P \ (P P ), n th symmtri irn oprtion is in s P P = (P U P )\ (P P ). Th onpt o logil irn st in this ppr rsmls th symmtri irn st ut it is not th sm. 8

9 whih is xtly Y in (). Lmm shows tht th logil irn st orms th shortst pth in th moii grph whr P is rpl with P. Lmm. Givn irt grph G(V,E) n pir (s, t) n lt P th shortst pth in this grph. W in G(V,E ) s th grph G(V,E) or whih th pth P is rpl with P. Th logil irn st P U P twn th optiml st o two link-isjoint pths P, } n th shortst pth P orms th shortst pth P { P rom no s to no t in G(V,E ). Proo: W will irst prov tht P = U P is omplt pth rom s to t in G(V,E ), thn w will prov tht P is th shortst pth in G(V,E ). Prt A. From Lmm, th optiml st o two link-isjoint pths P U must ontin ithr th irst shortst pth P itsl or som P links on h o its two pths. I ( U ) P, without loss o gnrlity, suppos P = P, thn O l = P U ( P ). With th inition o logil irn st, w hv P = (( U ) U ( P )) \ Ol = ( P U U ( P )) \ ( P U ( P )) =. Hn P must omplt pth rom s to t. I P U ontins som P links on h o its two pths, s P is th pth rom t to s in G(V,E ), n nithr nor ontins ny P links, thn th union U U ( P ) ontins two yls: on yl onsists o n P, th othr onsists o n P. Whn th st O l is rmov rom th union st, th rmining links ompos th logil irn st P. Hn P must omplt pth rom s to t. Prt B. Assum tht th shortst pth in G(V,E ) is P P, thn w must hv l P ) < l( ). As ( P l( P ) = l( ) + l( ) l( P ) w hv lp ( ) + lp ( ) < lp ( ) + lp ( ), whih ontrits th ssumption tht { P, } is th optiml st.. LBA Is Loop-r Mny routing lgorithms ssum non-ngtiv link wights to voi loop o ngtiv lngth ppring on pth. Howvr, ngtiv link wights introu to grph in LBA will not us loops in th routing pross.... s v v i... v i+ v n t u i () Th shortst pth P (s,t)... s v v i... v i+ v n t u i () A loop ontining som P link Figur. A loop ontins som ngtiv link 9

10 Thorm : Givn igrph G(V,E) n sour-stintion pir (s, t) n lt P th shortst pth in this grph. Th moii grph G(V,E ) is in s th grph G(V,E) or whih P is rpl with P. A loop ontining som ngtiv link(s) in G(V,E ) will not hv ngtiv lngth. Proo: Assum sv... vi vi+... vnt is th shortst pth P rom no s to no t in G(V,E), s shown in Figur (). Th orrsponing pth P in G(V,E ) (Figur ()) hs link (v i+ v i ) whih pprs on loop P l = u i v i+ v i u i. Suppos th loop P l hs ngtiv lngth l(p l ) = w(u i v i+ ) + w(v i+ v i ) + w(v i u i ) < 0. Bus w(v i+ v i ) = w(v i v i+ ), w must hv w(v i u i ) + w(u i v i+ ) < w(v i v i+ ). Hn th su-pth sv... viuivi+ is shortr thn th su-pth sv... v v i i+. This ontrits th ssumption tht sv... v i v i+... v nt is th shortst pth.. Optimlity o th solution prou with LBA Thorm. Givn irt grph G(V,E) n sour-stintion pir (s,t), th lgorithm LBA rturns th optiml st or th LPP prolm. Proo: Lt P th shortst pth in th originl grph G(V,E) oun in stp o LBA n P th shortst pth in th moii grph G(V,E ), oun in stp o LBA. P, } is th solution st gnrt y LBA. Th proo { P onsists o thr prts. Prt A. (Proo o Link-isjointnss) By onstrution o th solution st, w must hv I P =. Prt B. (Proo o Miniml Totl Lngth) Suppos th optiml st o link-isjoint pths is P, } inst o { P P, }. Aoring to Lmm, th logil irn st o P, } with P is th shortst pth in th moii { P { P grph G(V,E ). This ontrits tht P is th shortst pth in moii grph G(V,E ). Prt C. (Proo o Loop-rnss) On Thorm, LBA is loop-r. Thus th solution st rturn y LBA must th optiml st. Extning LBA to Multipl Dimnsions Th xtnsion o LBA to multipl imnsions using SAMCRA [9] is ll MLBA (Multipl-onstrin LBA). A ri sription o SAMCRA, whih srvs s th multipl-onstrin shortst pth routing lgorithm in MLBA, is givn in Stion.. Th si stps o MLBA (Multipl-onstrin LBA) r prsnt in Stion.. Th prolms ppring in multipl imnsions r rss in Stion... Bri Introution o SAMCRA SAMCRA [9] is n xt multipl-onstrin routing lgorithm s on thr onpts: () non-linr pth lngth, () k-shortst pth routing, n () non-ominn. Th non-linr lngth untion in in () is nssry or xtnss n implis tht su-pth o shortst pth is not nssrily shortst itsl. It is thror nssry to kp trk o multipl su-pths t h intrmit no on th pth twn pir o nos. A (su)-pth P is omint y (su)-pth P i wm ( P ) wm ( P ), or m =,,M, with n inqulity 0

11 sign or t lst on m. This oprtion rus th srh sp n rmovs loops rom rout whn nonngtiv link wights r us.. Oprtions o MLBA Th si stps o MLBA r th sm s thos or LBA xpt tht th shortst pth routing lgorithm is rpl with SAMCRA in MLBA. W will illustrt th oprtion o MLBA with th xmpl topology shown in Figur 6(). For th sk o simpliity, w hv ssign h link two-imnsionl wight vtor, ut it is lso possil to us n M-imnsionl wight vtor (M>). Th omplxity o solving th MCLPP prolm will inrs with M, ut s shown in [9], th omplxity my rs (n vn om polynomil) i M tns to ininity. To solv th MCLPP prolm, w r rquir to in two link-isjoint pths rom sour no A to stintion no B tht oth oy th onstrints vtor L r = (0, 0). Among th solutions to MCLPP w prr th on with th minimum totl lngth. Th shortst multipl-onstrin pth rom no A to no B is th pth. Its pth wight vtor is (, ). Th optiml st o two shortst link-isjoint pths (oring to ()) in this topology is {, }, with pth vtors (, 6) n (, ) rsptivly n minimum totl lngth =,,,,,,,6 6,,6 6, () Stp () Stp,,,,,,,,6 6,,6 6, () Stp () Stp Figur 6. Exmpl o th oprtion o MLBA 0.. Now lt us run MLBA on this topology. In Stp, th shortst pth P = is oun. In Stp, th originl grph is moii y rpling ll th P links with P links. In this s, h omponnt o link wight vtor o P link is st ngtiv. For instn, th link with wight vtor (, ) is rpl with th link with orrsponing wight vtor (, ). In Stp, th shortst pth in th moii grph oun with

12 SAMCRA is P =, with pth wight vtor () + (, ) + () = (6, 6). In Stp, th st O l onsisting o pir o opposit P n P links ( ) n ( ) r rmov rom th union o P n P. Thn th optiml solution st {, } is rturn.. Prolms u to th Non-linr Lngth in Multipl Dimnsions.. Loops us y Ngtiv Link Wights For M =, SAMCRA ts just lik Dijkstr s lgorithm, thror MLBA rus to LBA n ngtiv link wights long P will not us loop in th routing pross o MLBA. For M >, Thorm still hols n loop ontining som P link(s) still hs non-ngtiv lngth. Howvr, som o th omponnts o th loop wight vtor my ngtiv, using MLBA to pss this loop init numr o tims. W will xplin this looping through Figur 7, whr h link posssss two link mtris. Suppos tht th shortst pth P is s, pit with ol lins in Figur 7(). Th link wights vtor (x, x ) o link must hosn to nsur tht th pth s is longr thn s, i.. w ( s ) + w ( ) + x + w ( ) w ( s ) + w ( ) + w ( ) mx > mx. w ( s ) + w ( ) + x + w ( ) w ( s ) + w ( ) + w ( ) Numrilly, x w ( ) w ( ) mx > mx = = () x w ( ) w ( ) Atr Stp o MLBA is xut, thr pprs loop P l = shown with oul lins in Figur 7(), ontining th link with ngtiv link wights (, ).,,, x, x s,8,,,,6,,,,, x, x s,,8,,6,,, 6,,7 g 6,,7 g () Th shortst pth is s. () Th loop ontins ngtiv link. Figur 7. Non-ominn my il to rmov loop in th s o M>. I qution () hols n h omponnt o vtor (x, x ) is grtr thn, thn th su-pth s will omint y th irt link s with wight vtor (,8) n will rmov y th non-ominn hk in SAMCRA. Howvr, i qution () hols ut on omponnt o (x, x ) is not grtr thn, sy x <, x >, thn

13 w ( Pl ) = w ( ) + w ( ) + w ( ) = ( ) + + x < 0, w ( Pl ) = w ( ) + w ( ) + w ( ) = ( ) + + x > 0 whr (w (P l ), w (P l )) is th pth vtor o th loop P l. In this s, th su-pth s is not omint y th link s, lthough l(p l ) > 0. Hn, loops n our in MLBA tht ontinu until on o th onstrints is violt. Unortuntly, hking ll pths to ssur tht thy r loop-r is omputtionlly too xpnsiv. As mntion in Stion, in Suurll s lgorithm [] n th S-T lgorithm [6], trnsormtion o link wights w ( u v) = w( u v) + ( u) ( v) is ppli to h link, whr (u) is th istn rom sour no s to no u on th shortst pth tr. This trnsormtion is m to gurnt tht th links on th shortst pth tr hv zro link wights n thos links not on th tr hv link wights grtr thn zro in th moii grph. Howvr, n rtit o non-linr lngth is tht sustions o shortst pths r not nssrily shortst pths [][9]. Consquntly, or M >, Suurll s trnsormtion nnot nsur non-ngtiv link wights n loops my mrg... Totl Lngth o th Solutions Prou with MLBA W ssum or th momnt tht th onstrints r lrg nough suh tht ll pths r sil. I M =, it is prov in Stion. tht th solution st P, } prou with MLBA hs th minimum totl lngth. With th { P totl lngth in in (), Lmm in stion still hols or M >. Th optiml solution st o two linkisjoint multipl-onstrin pths with minimum totl lngth ithr ontins th irst shortst pth P itsl or som P links on h o its two pths. Also, th optiml st P, } still oys Proprty. Unortuntly, th { P logil irn st ( P U ) P is not nssrily th shortst pth P in th moii grph, sin l(( P U ) P ) = l( ) + l( ) l( P ) os not nssrily hol or M >. Hn, Lmm my not hol or M > n th solution st onstrut s on P n P is not nssrily th optiml st with minimum totl lngth. Morovr, th solution st my lso violt th onstrints or sil solution my not oun. DIMCRA In th prvious stion, w hv shown tht it is not trivil to xtn LBA to multipl imnsions. Du to th prolms xisting in MLBA, w propos huristi lgorithm DIMCRA (link-disjoint Multipl Constrints Routing Algorithm) or th MCLPP prolm.. Oprtions o DIMCRA DIMCRA (G, s, t): Givn irt grph G(V, E), onstrint vtor L r n sour-stintion pir (s, t), Stp. Fin th shortst pth P oying L r with SAMCRA; i P os not xist, thn stop; Stp. Rvrs th irtion o ll th links on th shortst pth P, n st th sign o thir link wights zro, w m ( v u) = 0, ( u v) P n m =,, M. A moii grph G is rt;

14 Stp. Fin th shortst pth P onstrin y L r in th moii grph G with SAMCRA; i P os not xist, thn stop; Stp. Mk th union o P n P, rmov rom th union th P links whos rvrs links ppr on P, n vi vrs, thn group th rmining links into st o two pths P, }, i.. ~ U = P U P ) \ ( P I ). ( P { P Stp. Chk th lngth o h pth in th st P, }. I th pth ( i ) violts th onstrints, thn { P upt th moii grph G y rmoving th link st ( I P ) rom it, n go to Stp. Othrwis stop n rturn th urrnt solution st P, }. { P i i \ i Compr to MLBA, DIMCRA uss irnt trnsormtion to rt th moii grph. In Stp o DIMCRA, th shortst pth links r still rvrs in irtion ut th orrsponing irtion rvrs links r ssign with zro link wight vtors inst o ngtiv ons. Thror th loop prolm us y ngtiv link wights tht minly stroys th iiny o MLBA is ypss. In MLBA, P is rquir to oy th r r onstrints, whih my us som sil sts to ignor y MLBA. In t, whn w( P ) > L, i P ontins no rvrs P link(s), thn P, } is tully {P, P } n nnot sil st. But i P ontins som { P rvrs P link(s), it is possil tht { P, } is sil st, or instn, l(p ) = 0.6, l ( P ) = 0.8, l ( P ) = 0.9, r r r r r r n l(p ) =.. Howvr, i w( P ) > L, thn w must hv w P + P ) = w( P + P P ) > w( P ) L, ( r whr P r nots th st o P links whos rvrs links ppr on P, n P r must propr sust o P. Thror, in Stp o DIMCRA, th onstrint hk on pth P in SAMCRA is prorm with L r s th onstrints vtor, othrwis sil solution st my not oun. W hv lso n xtr stp, Stp r r o DIMCRA, to hk tht th onstrints r oy. I only with w( P + P ) L DIMCRA os not lwys r r r r nsur oth pths within onstrints, i.. w( P ) L n w( P ) L. Hn Stp o DIMCRA hks oth pths in th solution st rturn t Stp. I h o thm oys th onstrints, DIMCRA will rturn th solution st n stop. On th othr hn, i ithr o thm os not oy th onstrints, DIMCRA is rirt to Stp to ontinu th srh or sil st. In Stp, i no P xists, DIMCRA will stop with no solution. W will illustrt th oprtion o DIMCRA with th ollowing xmpls. Exmpl : Consir th xmpl grph in Figur 8(). W r rquir to in st o two link-isjoint pths twn n, h within th onstrints L r = () n prrly with th minimum totl lngth. In Stp, th shortst pth P = is oun. In Stp, h P link is rvrs n is ssign with zro link wights. In Stp, th shortst pth in th moii grph G is oun s P =, with pth vtor () + () + () = (7,8), s shown with ol lins in Figur 8(). In Stp, only or th P link, its rvrs link pprs on P n vi vrs. Thus ths two links r rmov rom th union o P n P, n th rmining

15 ,,,,6 6,,6 6, () Stp () Stp,,,,6 6,,6 6, () Stp () Stp Figur 8. Exmpl o th oprtion o DIMCRA links r group into st o two pths P, } ={, }, shown with ol lins. In Stp, th onstrints { P hk is xut on oth pths. As h o thm oys th onstrints, DIMCRA stops. In this s, th optiml solution st o {, } is rturn y DIMCRA. Th solution st tht woul hv n rturn y RF, is not optiml. Exmpl : Consir th grph in Figur 9(), whih is th sm s in th prvious xmpl xpt tht th,,,,6,,6, () Stp () Stp,,,,6,,6, () Stp () Stp Figur 9. Exmpl o th oprtion o DIMCRA

16 link is ssign irnt vtor (,). Th onstrints rmin th sm. In this xmpl th optiml st o () Stp () Stp,,,,6, () Stp Figur 0. Exmpl o th oprtion o DIMCRA two link-isjoint multipl-onstrin pths is still th st {, } with pth vtors (,) n (,6) rsptivly, n th minimum totl lngth /0 + 6/0 = 0.. In Stp, th shortst pth in th moii grph is oun s P = with pth vtor (7,7), shown in Figur 9(). In Stp, s or h P link, its rvrs link os not ppr on P, or vi vrs, th solution st P, } is onstrut s {, }, xtly P n P { P thmslvs. Th totl lngth o {, } is /0 + 7/0 = 0.6. In this xmpl, DIMCRA il to rturn th optiml st, ut DIMCRA s solution st is los to th optiml on n oth pths oy th onstrints. RF woul hv rturn th sm solution. Exmpl. W gin onsir Exmpl xpt with irnt onstrints (6,6). Running DIMCRA, w otin th sm rsults s in Exmpl (or Stp to Stp ). But in Stp, whn th onstrints hk is m on h pth in th solution st { P, } ={, }, th longr pth = with pth vtor (7,7) os not oy th onstrints. This mns tht th urrntly uilt solution st is not sil. Th links tht only ppr on =, i.. link n, r rmov rom th moii grph shown in Figur 9(). Th upt moii grph is shown in Figur 0() n DIMCRA is rirt to Stp. In Stp, shortst pth in th upt moii grph is oun s P =, pit in Figur 0(). In Stp, th solution st is {, }, s shown in Figur 0(). At lst, in Stp, h pth in th urrnt solution st oys th onstrints. Th optiml st {, } is rturn n DIMCRA stops. RF woul hv il to rturn solution. In Stp th onstrints r st to L r. With ths moii onstrints, i th shortst pth P in th moii grph violts th onstrints L r ut oys L r, it n rturn y SAMCRA in Stp, hn sil st 6

17 ,,,, 6,, 6,, () Stp () Stp,,,, 6,, 6,, () Stp () Stp () Stp Figur. Exmpl o th oprtion o DIMCRA with onstrints (0,0) rlt to suh kin o P will not ignor, s illustrt in Exmpl. Morovr, i pth P os not xist in th upt moii grph, DIMCRA will stop. Thus DIMCRA will not oun k n orth twn Stp n. With th onstrints hk on h pth in th solution st, Stp gurnts tht DIMCRA rturns sil st o two link-isjoint multipl-onstrin pths, s illustrt in th ov xmpls. Howvr it my our, s illustrt in Figur, tht DIMCRA nnot rturn sil st vn i thr xists on. Th RF mtho lso ils to rturn th sil st in this s.. Proprtis o DIMCRA As prov in Stion, th wy to onstrut solution st y rvrsing th shortst pth P, ining shortst pth P in th moii grph n onstruting th solution st s on ths two shortst pths P n P gurnts th isjointnss o th two pths in th solution st. Stting th irtion-rvrs P links with zro link wights gurnts th loop-rnss o DIMCRA. For, i no ngtiv link wights r us in grph, loop n voi y th non-ominn hk in SAMCRA. Compring with th oprtion o stting irtion-rvrs P links ngtiv, th oprtion o stting suh rvrs P links with zro link wights still nourgs th hoi o suh rvrs P links on pth ut with lss intnsity. Unortuntly DIMCRA os not lwys in th st o sil link-isjoint pths. Hn, it my possil to urthr optimiz DIMCRA, suh tht it n gurnt to lwys in st o sil link-isjoint pths, i thy xist. Howvr, DIMCRA in its urrnt stt is ttr thn th RF mtho (s ws init in th xmpls). Both mthos rturn th sm solution whn P, } ={P, P } n P I =. In ll othr ss DIMCRA { P P 7

18 ithr rturns mor optiml solution thn RF or RF os not in solution whr DIMCRA os. Sin, to our knowlg, no othr lgorithms or solving MCLPP xist, th prormn o DIMCRA is iiult to ssss. 6 Conlusions Th Link-isjoint pth prolm ours in ntwork sign whr spts s survivility, lo lning n ntwork rsour utiliztion r striv or. This prolm hs rly n invstigt in th QoS routing ontxt whr pth is hrtriz y multipl mtris. A simpl lgorithm or solving th LPP prolm or is prsnt in this ppr. Th prolms surrouning th xtnsion o this simpl lgorithm to multipl imnsions r isuss. A huristi lgorithm DIMCRA is propos to in link-isjoint multipl-onstrin pths twn pir o sour n stintion nos. I DIMCRA rturns link-isjoint pir o pths thy lwys oys th onstrints. Howvr, DIMCRA s solution is not nssrily optiml in trms o minimizing th totl lngth o th rturn pths or gurnting to lwys in th sil st. Its prormn howvr is ttr thn th simpl Rmov-Fin mtho. Som opn issus rmin, nmly: mking DIMCRA xt whilst still iint, llowing mximlly isjoint pths or rigs n simulting th prormn. Rrn: [] R. Bhnri, Optiml Divrs Routing in Tlommunition Fir Ntworks, Pro. IEEE INFOCOM 9, Toronto, Ontrio, Cn, Vol., pp.98-08, Jun 99. [] Y. Bjrno, Y. Britrt, A. Or, R. Rstogi, n A. Sprintson, Algorithms or Computing QoS Pths with Rstortion, Pro. o IEEE INFOCOM 0, April 00. [] D.A. Cstnon, Eiint lgorithms or ining th K st pths through trllis, IEEE Trns. on Arosp n Eltroni Systms, Vol. 6, No., pp. 0-0, Mrh 990. [] C. Chng, S.P.R. Kumr n J.J. Gri-Lun-Avs, A istriut lgorithm or ining K isjoint pths o miniml totl lngth, Pro. 8th Annul Allrton Conrn on Communition, Control, n Computing, Urn, Illinois, Otor 990. [] H. D Nv n P. Vn Mighm, TAMCRA: tunl ury multipl onstrints routing lgorithm, Computr Communitions, vol., No. 7, pp , Mrh 000. [6] R. Distl, Grph Thory, Grut Txts in Mthmtis, Springr-Vrlg Nw York, 997. [7] L.R. For n D.R. Fulkrson, Flows in Ntworks, Printon Univrsity Prss, Printon, Nw Jrsy, 96. [8] M. R. Gry n D. S. Johnson, Computrs n Intrtility, A Gui to th Thory o NP-Compltnss, Frmn, Sn Frniso, 979. [9] K.P. Gummi, M.J. Prp n C.S.R. Murthy, An Eiint Primry-Sgmnt Bkup Shm or Dpnl Rl-Tim Communition in Multihop Ntworks, ACM/IEEE Trnstions on Ntworking, vol., no., pp. 8-9, Frury 00. [0] P-H Ho n H.T. Mouth, Issus on ivrs routing or WDM msh ntworks with survivility, Pro. Tnth Intrntionl Conrn on Computr Communitions n Ntworks, pp. 6-66, 997. [] G.F. Itlin, R. Rstogi n B. Ynr, Rstortion Algorithms or Virtul Privt Ntworks in th Hos Mol, Pro. o IEEE INFOCOM 0, 00. [] K. Kr, M. Koilm n T. V. Lkshmn, Routing Rstorl Bnwith Gurnt Conntions using Mximum -Rout Flows, Pro. IEEE INFOCOM 0, 00. [] M. Koilm n T. V. Lkshmn, Dynmi Routing o Bnwith Gurnt Tunnls with Rstortion, Pro. IEEE INFOCOM 00, 000. [] M. Koilm n T.V. Lkshmn, Rstorl Dynmi Qulity o Srvi Routing, IEEE Communitions Mgzin, pp. 7-8, Jun 00. [] S.W. L n C. S. Wu, A k-st pths lgorithm or highly rlil ommunition ntworks, IEICE Trns. on Commun., Vol. E8-B, No., pp.86-80, April 999. [6] C-L Li, S.T. MCormik, D. Simhi-Lvi, Th omplxity o ining two isjoint pths with min-mx ojtiv untion, Disrt Appli Mthmtis, Vol. 6, No., pp. 0-, Jnury 990. [7] W. Ling, Roust routing in wi-r WDM ntworks, Pro. o th Int'l Prlll n Distriut Prossing Symp., Sn Frniso, April 00. 8

19 [8] C-C Lo n B-W Chung, A Novl Approh o Bkup Pth Rsrvtion or Survivl High-Sp Ntworks, IEEE Communitions Mgzin, Mrh 00. [9] R.G. Ogir, V. Rutnurg n N. Shhm, Distriut lgorithms or omputing shortst pirs o isjoint pths, IEEE Trns. on Inormtion Thory, Vol. 9, No., pp. -, Mrh 99. [0] E. Oki n N. Ymnk, A rursiv mtrix lultion mtho or isjoint pth srh with hop link numr onstrints, IEICE Trns. Commun., Vol. E78-B, No., pp , My 99. [] C.H. Ppimitriou n K. Stiglitz, Comintoril Optimiztion Algorithms n Complxity, Prnti-Hll, In., Englwoo Clis, Nw Jrsy, 98. [] A. Sn, B.H. Shn, S. Bnyophyy n J.M. Cpon, Survivility o lightwv ntworks - pth lngths in WDM prottion shm, Journl o High Sp Ntworks, vol. 0, no., pp. 0-, 00. [] S.Z. Shikh, Spn-isjoint pths or physil ivrsity in ntworks, Pro. o IEEE Symposium on Computrs n Communitions, pp. 7-, 99. [] D. Sihu, R. Nir n S. Allh, Fining isjoint pths in ntworks, ACM SIGCOMM Computr Communition Rviw, Pro. o th onrn on Communitions rhittur & protools, Vol., No., August 99. [] J.W. Suurll, Disjoint Pths in Ntwork, Ntworks, Vol., pp. -, 97. [6] J.W. Suurll n R.E. Trjn, A Quik Mtho or Fining Shortst Pirs o Disjoint Pths, Ntworks, Vol., pp. -, 98. [7] N. Tt-Plotkin, B. Bllur n R. Ogir, Qulity-o-Srvi Using Mximlly Disjoint Pths, Pro. o IWQoS (Intrntionl Workshop on Qulity-o-Srvi), Jun 999. [8] Y. Tnk, F. Ru-Xu n M. Akiym, Dsign Mtho o Highly Rlil Communition Ntwork y th Us o Mtrix Clultion, IEICE Trns., Vol. J70-B, No., pp. -6, 987. [9] P. Vn Mighm, H. D Nv n F.A. Kuiprs, "Hop-y-hop Qulity o Srvi Routing", Computr Ntworks, Vol. 7. No -, pp. 07-, 00. [0] Z. Wng n J. Crowrot, QoS Routing or supporting Multimi Applitions, IEEE J. Slt Ars in Communitions, Vol., No.7, pp. 8-, Sptmr 996. [] J.K. Wol, A.M. Vitri n G.S. Dixon, Fining th Bst st o K pths through trllis with pplition to multitrgt trking, IEEE Trnstions on Arosp n Eltroni Systms, Vol., No., pp , Mrh