3, 4 2, 4 4, 4 0, 4 1, 4 1, 3 4, 3 0, 3 2, 3 3, 3 1, 2 2, 2 4, 2 0, 2 3, 2 2, 1 0, 1 1, 1 3, 1 4, 1 2, 0 3, 0 4, 0 1, 0 0, 0

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1 00 Conrn on Inormtion Sins n Systms, Printon Univrsity, Mrh 0, 00 Cpity Provisionin n Filur Rovry or Stllit Constlltions Jun Sun n Eytn Moino 1 Lortory or Inormtion n Dision Systms Msshustts Institut o Thnoloy junsun, moino@mit.u Astrt This ppr onsirs th link pity rquirmnt or msh-torus ntwork unr uniorm ll-to-ll tri mol. Both primry pity n spr pity or rovrin rom link ilurs r xmin. In oth ss, w us novl mtho o uts on rph" to otin lowr ouns on pity rquirmnts n susquntly in lorithms or routin n ilur rovry tht mt ths ouns. Finlly, wquntiy th nits o pth s rstortion ovr tht o link s rstortion; spiilly, w in tht th spr pity rquirmnt or link s rstortion shm is nrly tims tht or pth s shm. I. Introution Th totl pity rquir y stllit ntwork to stisy th mn n prott it rom ilurs ontriuts siniintly to its ost. To imiz th utiliztion o suh ntwork, w xplor th minimum mount o spr pity n on h stllit link, so s to sustin th oriinl tri low urin th tim o link ilur. In nrl, or link ilur, rstortion shms n lssii s link s rstortion, or pth s rstortion. In th ormr s, t tri (i.. tri tht is suppos to o throuh th il link) is rrout ovr st o rplmnt pths throuh th spr pity o ntwork twn th two nos trmintin th il link. Pth rstortion rrouts th t tri ovr st o rplmnt pths twn thir sour n stintion nos [1,, 3, 5, 6]. Th ovious vnts o usin th link rstortion strty r simpliity n ility to rpily rovr rom ilur vnts. Howvr, s w will show ltr, th mount o spr pity n or th link s shm is siniintly rtr thn tht o pth s rstortion sin th lttr hs th rom to rrout th omplt sour-stintion usin th most iint kup pth. On th othr hn, th pth rstortion shm is lss lxil in hnlin ilurs [1,, 3]. W invstit th optiml spr pity plmnt prolm s on msh-torus topoloy whih is ssntil or th multistllit systms. An n n msh-torus is twoimnsionl (-D) n-ry hypru n irs rom inry hypru in tht h no hs onstnt numr o nihors (), rrlss o n. For th rminr o th ppr, w will rr to this topoloy simply s msh. In prtiulr, w r intrst in th snrio whr vry no in th ntwork is snin on unit o tri to vry othr no (lso known s omplt xhn or ll-to-ll ommunition) [7]. This typ o ommunition mol is onsir us th xt tri pttrn is otn unknown n n ll-to-ll mol is rquntly us s th sis or ntwork sin. Evn in 1 This work ws support y DARPA unr th xt Gnrtion Intrnt inititiv. th s o pritl tri pttrn, links o prtiulr stllit will xprin irnt tri mn s th stllit lis ovr irnt lotion on rth. Thus, h link o tht stllit must stisy th imum mn. Ain, llto-ll tri mol hlps pturin this t. Hn w lso ssum tht h stllit link hs n qul pity. Our rsults, whil motivt y stllit ntworks [9, 10, 11], r qully pplil to othr ntworks with msh topoloy suh smulti-prossor intronnt ntworks [, 13, 1] n optil WDM msh ntworks [, 3]. Furthrmor, whil our rsults r isuss in th ontxt o n n n msh or simpliity, thy n trivilly xtn to mor nrl n m topoloy, whih is typilly mor rprsnttiv in stllit onstlltions. Whn usin th pth rstortion shms, th rstortion n prorm t th lol lvl y rroutin ll th tri (oth thos t or unt y thlink ilur) in ntwork. Howvr, this lvl o rstortion rquirs romputin nw pth or h sour-stintion pir, thus it is imprtil i rstortion tim limit is impos or whn isruption o xistin lls is unptl. W n lso prorm pth rstortion t th lol lvl y rroutin only th tri whih is t y th link ilur. Oviously, th lol lvl roniurtion will rquir t lst s muh spr pity s th lol lvl roniurtion sin th ormr is sust o th lttr. vrthlss, s w show in stion IV, th lowr oun on th spr pity n, usin lol lvl roniurtion, n hiv y usin lol lvl roniurtion. To otin th nssry minimum spr pity, our pproh is to irst in th minimum pity, sy C 1, tht h link must hv in orr to support th ll-to-ll tri. W thn otin lowr oun, C, or th pity n on h link to stisy th ll-to-ll tri whn on o th links ils. Consquntly, th minimum spr pity n, C spr, shoul rtr thn th irn o C n C 1. Sin w o not rstrit th roniurtion (lol lvl or lol lvl) us to lult C ; C C 1 is lowr oun on C spr, oth t lol lvl n lol lvl. W willshow tht this lowr oun on C spr is hivl y usin pth s rstortion lorithm t lol lvl. Thus, th minimum spr pity n usin pth rstortion strty is C spr. Tl I summrizs pity rquirmnts unr link s n pth s rstortion. Communition on msh ntwork hs n stui in [, 11, 1]. In [], th uthors onsir prossors ommunitin ovr msh ntwork with th ojtiv o rostin inormtion. Th work in [11] prsnts routin lorithm nrtin minimum proption ly or stllit msh ntworks. In [1], th uthors propos nw lorithms or ll-to-ll prsonliz ommunition in msh-onnt multiprossors. Ths pprs mntion so r i not look into pity provisionin n spr pity rquirmnt o th msh ntwork.

2 o Link s Pth s rstortion rstortion rstortion 0, 1,, 3,, Totl Cpity ( o) Totl Cpity ( vn) Spr Cpity ( o) 0 Spr Cpity ( vn) ( 1) ( 1) ( 1) 0, 3 0, 1, 3 1,, 3, 3, 3 3,, 3, Tl 1: Cpity rquirmnts unr link s n pth s rstortion. 0, 1 1, 1, 1 3, 1, 1 0, 0 1, 0, 0 3, 0, 0 Pth s n link s rstortion shms hv n xtnsivly rsrh [1,, 3, 5]. In [1], th uthors stuy n ompr spr pity n y usin link s n pth s shms. Th work o [5] provis mtho or pity optimiztion o pth rstorl ntworks n quntiy th pity nits o pth ovr link rstortion. In [, 3], th uthors xmins irnt pprohs to rstor mshs WDM optil ntworks rom sinl link ilurs. In ll th ormntion pprs, th spr pity prolm is ormult s n intr linr prormmin prolm whih is solv y stnr mthos. Our ppr rsss th msh strutur or whih w n t los orm rsults or th spr pity. Th strutur o this ppr is s ollows: Stion II ivs nssry initions n sttmnt oth prolm. In stion III, lowr oun on C 1 is ivn lon with routin lorithm hivin this lowr oun. Th lowr oun C is prsnt lso. W thn show in stion IV tht th lowr oun on C spr, C C 1, n hiv y pth s rstortion lorithm. Stion V onlus this ppr. II. Prliminris W strt out with sription o th ntwork topoloy n tri mol, n ollow it with squn o orml initions n trminoloy tht will us in susqunt stions. Dinition 1. Th -imnsionl -msh is n unirt rph G =(V; E), with vrtx st V = ~ j ~ =( 1; ) n 1; Z ; whr Z nots th intrs moulo, n st E = (~; ~ ) j9j suh tht j ( j ± 1) mo n i = i or i 6= j; i; j 1; : Th ov inition is rom [7]. A -imnsionl -msh hs totl o nos. Eh nohstwo nihors in th vrtil n horizontl imnsion, or totl o our nihors. W ssoit h stllit with ix no, ( 1; ), in th msh. Unirt s o th msh r lso rrr to s links. Fi. 1 shows -imnsionl 5-msh. Th notion - imnsionl 1-msh is us to not th s whr is ritrrily lr, n it is th sm s n ininity ri. Dinition. A ut (S; V S) in rph G = (V; E) is prtition o th no st V into two nonmpty susts, st S n its omplmnt V S. Fiur 1: A -imnsionl 5-msh. Hr th nottion Cut-St(S; V S) =(~; ~ ) E j ~ S; ~ V S nots th st o s o th ut (i.. th st o s with on n no in on si o th ut n th othr on th othr si o th ut). Dinition 3. Th siz o Cut-St(S; V S) is in s C(S; V S) =j Cut-St(S; V S) j. For G =(V; E) n P(V ) not th powr st o th st V (i.. th st o ll susts o V ). Lt P n(v ) not th st o ll n-lmnts susts o V. Dinition. Lt G = (V; E) -imnsionl -msh, th untion " : Z +!Z + is in s " (n) = min C(S; V S): SPn(V ) Th untion " (n) rturns th minimum numr o s tht must rmov in orr to split th -imnsionl - msh into two prts, on with n nos n th othr with n nos. Similrly, "1(n) is in to th minimum numr o s tht must rmov in orr to split th 1-msh into two isjoint prts, on o whih ontinin n nos. To hiv th minimum spr pity, w onsir th shortst pth lorithm. Shortst pths on -imnsionl - msh r ssoit with th notion o yli istn whih w will in nxt []. Dinition 5. Givn thr intrs, i, j,, th yli istn twn i n j moulo is ivn y D (i; j) =min(i j) mo ); (j i) mo ): III. Cpity Rquirmnt without Link Filurs To otin th nssry pity, C 1, tht h link must hv in orr to support th ll-to-ll tri without link ilur, w irst provi lowr oun on C 1. An lorithm hivin th lowr oun will lso prsnt. For th proo o th lowr oun on C 1,w r wr o th xistn o simplr proo (usin Proposition 1 in []) thn th on w sri low. Howvr, th ut mtho w us hr will hlp us in th lowr oun, C, on th minimum pity n on h linkinthvnt o link ilur. Thror, w i to us th sm ut mtho onsistntly in provin th lowr oun on C 1 n th lowr oun C.

3 A. A Lowr Boun on th Primry Cpity To inlowr oun on C 1,w stt th ollowin lmms whih will prov to usul tools in th susqunt stions. Proos o ths lmms r omitt or rvity, n thy n oun in [16]. Lmm 1. Lt G =(V; E) n ininit msh. An ritrry st W n V suh tht "1(n) =C(W n; W n) must stisy th ollowin proprtis: 1. x W n; 9 y W n suh tht (x; y) E. In othr wors, nos in W n shoul onnt.. os in W n shoul lustr tothr to orm rtnulr shp (inluin squr) i possil. 3. "1(n) is n vn numr or ll n Z +.. "1(n) is monotonilly nonrsin untion o n. Lmm. Lt G =(V; E) n ininit msh, thn n "1(n + k) = "1(n )=n ρ n + or 1» k» n n + or n +1» k» n +1 or n; k Z + whr Z + nots th st o positiv intr. Th ov lmm ivs th minimum numr o s tht must rmov rom E in orr to split spii numr o nos rom th msh. Intuitivly, th st o n nos to rmov rom th msh must lustr tothr. Corollry 1. For "1(n) in in ov lmm, "1(n) p n or n Z +. Corollry. Lt G = (V; E) n ininit msh with n ritrry link ilur, thn n "1(n + k) = "1(n )=n 1 ρ n +1 or 1» k» n n +3 or n +1» k» n +1 or n; k Z + whr Z + nots th st o positiv intr. So r th untion "1(n) hs n th ous o our isussion. Sin th stllit ntwork tht w mol is - imnsionl -msh, it is ssntil to know " (n). In - imnsionl -msh, horizontl row o nos ( vrtil olumn o nos) orms horizontl (vrtil) rin. Whn n is vry smll ompr to, splittin st o n nos rom th -msh is similr to uttin th st o n nos rom 1-msh; mor prisly, "1(n) = " (n). Th rin strutur o th -imnsionl -msh os not t th minimum siz o ut whn n is rltivly smll. vrthlss, whn n is lr, tkin vnt o th rin strutur o th -imnsionl -msh will rsult in " (n) <"1(n). ow, lt's in th ollowin sts: A 1 1; ;::: ; ; A x j x +1;::: ; n (x mo ) 6= 0; A 3 x j x +1;::: ; n (x mo ) =0; O 1 1; ;::: ; 1 ; O x j x 1 +1;::: ; +1 n (x mo ) 6= 0; n O 3 x j x 1 +1;::: ; +1 n (x mo ) = 0: Lmm 3. Lt G =(V; E) -imnsionl -msh, or vn, < "1(n) or n A 1 " (n) = + or n A : or n A 3 or o, < "1(n) or n O 1 " (n) = + or n O : or n O 3 Thorm 1. On -imnsionl -msh, th minimum pity, C 1, tht h link must hv in orr to support llto-ll tri is t lst 3 or vn, n 3 Proo. Consir ix n twn 1 n 1. Th i is to us ut to sprt th ntwork (-msh) into two isjoint prts, with on prt ontinin n nos n th othr ontinin n nos. Bs on th ll-to-ll tri mol, w know th xt mount otri, C ross =n( n), tht must o throuh th ut. Thror, rom -low minut thorm [15] w know tht simply iviin C ross y th minimum siz o utst " (n)willivus lowr oun on C 1, n lt's ll this oun B n. It implis tht h linkinth ntwork must hv pity o t lst B n in orr to stisy th ll-to-ll tri mn. This prompts us to in B C 1 whih is th imum o B n ovr ll n 1;::: ; 1. W sy tht B C 1 is th st lowr oun or C 1 in th sns tht it is rtr or qul to ny othr lowr oun or C 1. For vn, lt» B C 1 ( = (1) = " (n)» ( ; "1(n)» ( ; na +» ( : () n1;::: ; 1 ρ na 1 na 3 Th s or o is th sm xpt tht A 1; A ; n A 3 in () r rpl y O 1; O ; n O 3. Solvin th imiztion prolm, w t < B C 1 = : n ; n o; o ; 3 (+1) 1 (+1) ; 3 o or vn or o

4 whr ( o) in th ov qution is th rsult o th irst trm o qution () or vn (o). Hr, xpliit vlution o n o is unnssry. Inst, y usin Corollry 1, n uppr oun on n o will suiint or us to solv th imiztion prolm. Sin "1(n) p n or n Z +, th ollowin qution hols: = na 1» nz +» (» "1(n) nz +» ( p n = 3 16 <» ( "1(n) o < 3 n shown similrly. Thus, w hv B C 1 = ( or vn or o Corollry 3. On -imnsionl -msh with n ritrry link il, th lowr oun, C, on th minimum pity tht h link must hv in orr to support ll-to-ll tri is or vn, n ( 1) B. Alorithm Ahivin th Lowr Boun on C 1 In this stion, w show tht th lowr oun on C 1 n hiv y usin simpl routin lorithm ll th Dimnsionl Routin Alorithm. As w hv mntion rlir, th routin lorithm will us th shortst pth twn sour n stintion nos. Blow is sription o th Dimnsionl Routin Alorithm: 1. From th sour no ~p =(p 1;p ), mov horizontlly in th irtion o shortst yli istn to th stintion no ~q =(q 1;q ); i thr is mor thn on wy to rout th tri, pik thonthtmovs in th (+) irtion (mo ), i.. (p 1;p )! ((p 1 +1)mo ; p )! ((p 1 +)mo ; p )!!(q 1;p ): Rout th tri or D (p 1;q 1) hops whr D (p 1;q 1) nots th shortst yli istn (hops) twn ~p n ~q in horizontl irtion.. Mov vrtilly in th irtion o shortst yli istn to th stintion no; i thr is mor thn on wy to rout th tri, pik th on tht movs in th (+) irtion (mo ). Rout th tri or D (p ;q ) hops whr D (p ;q ) nots th shortst yli istn (hops) twn ~p n ~q in vrtil irtion. Thorm. Lt G =(V; E) -imnsionl -msh, y usin th Dimnsionl Routin Alorithm ov, to stisy th ll-to-ll tri, th imum lo on h link is 3 or vn n 3 IV. Cpity Rquirmnt or Rovrin rom A Link Filur Unr th onition o n ritrry link ilur, w invstit th spr pity n to ully rstor th oriinl tri, usin th link s rstortion mtho n pth s rstortion mtho. A. Link Bs Rstortion Strty Consir tht n ritrry link, l ~u~v (onntin nos ~u n ~v), il in th -imnsionl -msh. W know rom th prvious stion tht thr r 3 ( 3 ) units o tri on l ~u~v hv to rrout or o (vn). Sin th link s rstortion strty is us hr, ths 3 units o tri in n out o no ~u hv to rrout throuh th rminin thr links onntin to no ~u (l ~u~v is lry rokn). W thn hv th ollowin thorm: Thorm 3. Usin link s rstortion strty in th vnt o link ilur, th minimum spr pity tht h link must hv in orr to support th ll-to-ll tri is 3 or o n 3 or vn. Proo in [16]. B. Pth Bs Rstortion Strty B.1 Lowr Boun on th Minimum Spr Cpity Thorm. On -imnsionl -msh with n ritrry il link, th minimum spr pity, C spr, tht h link must hv in orr to support ll-to-ll tri is t lst or ( 1) vn, n 3 ( 1) Proo. From Thorm, or rulr -imnsionl -msh, w know tht th pity tht h link must hv inorr to stisy ll-to-ll tri is 3 or vn, n 3 or o. In s o n ritrry link ilur, rom Corollry 3, t lst pity o ( ( 1) ) is n on h link to sustin th oriinl tri low or vn (o). W n to hv n xtr pity oc spr C C 1 on h link. Thus, w hv C spr ( 3 = 3 ( 1) ( 1) 3 = 3 ( 1) B. Alorithm Usin Minimum Spr Cpity or vn or o In this stion, w will show tht th minimum spr pity n on h link is 3 or ( 1) vn n 3 ( 1) In othr wors, th lowr oun in Thorm is tiht. W showthhivility y prsntin primry routin lorithm, n susquntly, pth-s rovry lorithm whih ully rstors th oriinl tri y usin th minimum spr pity ins o link ilur. W ous on th s o o or simpliity. To show th hivility or vn, irnt st o primry routin lorithm n rovry lorithm is n (not prsnt in this ppr). First, w sri th primry routin lorithm tht w ll Rottionl Symmtri Routin Alorithm, or RS Routin Alorithm, us to rout th ll-to-ll tri. W us th RS Routin Alorithm inst o th Dimnsionl Routin Alorithm s our primry routin lorithm us th ormr simpliy th onstrution n nlysis o th rstortion lorithm. Spiilly, with th Dimnsionl Routin Alorithm, th tri routs on horizontl n vrtil links r not symmtri; hn irnt rstortion lorithm woul rquir or vrtil n horizontl link ilur. In ontrst, th RS Routin Alorithm is symmtri n vrtil or

5 horizontl link ilur n trt usin th sm rovry lorithm. Th s o horizontl link ilur is th sm s th vrtil link ilur i w rott th topoloy y 90 i. RS routin lorithm Eh no ~ in -imnsionl -msh hs pir o intrs ( 1; ) ssoit with it. To rout on unit o tri rom th sour no ~p to th stintion no ~q, o th ollowin: 1. Chn oorint n omput th rltiv position o th stintion no with rspt to th sour no. Spiilly, shit th sour no to (0; 0) y pplyin th trnsormtion T ~p. Hr, th trnsormtion T ~p : Z Z!Z Z is in s ~ = T ~p (~q) = T ~p (q 1;q )=( 1; ), whr or i =1; >< i = >: q i p i; i 1» q i p i» 1 (q i p i) mo ; i ( 1)» q i p i < 1 ([ (q i p i)] mo ); i 1 <q i p i» 1 Hr, ( n) mo p is in s p n mo p i 0 < nmop<p. Thus, w willhv T ~p (~p) =(0; 0). Fi. illustrts this trnsormtion. 0, 0, 3 0, 1, 1, 3 1,,, 3, 3, 3, 3 3,,, 3, Dstintion o (q) Q = Q 3 = Q = n 0»» 1 (; ) j ; Z n 1 (; ) j ; Z n 1 (; ) j ; Z ; 0 <» 1 ;» <0; 1»» 0; 1»» 0; n 0 <» 1 ; 1»» 0:» <0; n 3. I ~ = T ~p (~q) (Q 1 [Q 3), rout th tri vrtilly in th irtion o shortst yli istn to th stintion no y D (p ;q )hops. Thn, rout th tri horizontlly in th irtion o shortst yli istn to th stintion no y D (p 1;q 1) hops. I ~ = T ~p (~q) (Q [Q ), rout th tri horizontlly in th irtion o shortst yli istn to th stintion no y D (p 1;q 1) hops. Thn, rout th tri vrtilly in th irtion o shortst yli istn to th stintion no y D (p ;q ) hops. ow, onsirin ll tri tht hs prtiulr no ~ s thir stintion, thir routin pths r rottionl symmtri y th ov lorithm. Tht is, rottin ll o th routin pths y nintr multipl o 90 i will rsult in hvin th sm oriinl routin oniurtion. RS routin lorithm lso hivs th lowr oun on C 1. Th proo is strihtorwr n thus omitt hr. A L A1 0, 1 1, 1, 1 3, 1, 1 0, 0 1, 0, 0 3, 0, 0 Sour o (p) Q T p -, -1, 0, 1,, Q1 A3 L A -,1-1, 1 0, 1 1, 1, 1 () -, , 0 1, 0, 0 -,-1 -,- -1,-1-1,- 0,-1 0,- 1,-1 1,-,-1,- Primry Routin Pth Q3 Dstintion o (q) Sour o (p) Q Fiur : Chn o oorint y usin trnsormtion T ~p.. Divi th nos o th -imnsionl -msh into our qurnts with th sour no s th oriin (shown in Fi. ). Spilly, lt Q 1 = (; ) j ; Z Rstortion Routin Pth () Fiur 3: Routin pth o th rstortion lorithm Our ol hr is to rovr th oriinl tri low y in n xtr mount o pity, whih is qul to th lowr oun

6 lult in Thorm, on h link. ow, w prsnt n xmpl to illustrt th ky is o th rovry lorithm. Without loss o nrlity, suppos tht link l ~ ~ il in th -imnsionl 7-msh shown in Fi. 3(). W n to in ll possil sour stintion pirs (S-D pirs) tht r t y th il link irst. From th RS routin lorithm, ths S- D pirs n trmin xtly. Spiilly, lt th sour no ~s n stintion no ~t. Th st o il tri F is in s F = F 1 [ F [ F 3 [ F [ F 5 [ F 6 whr F 1 = (~s;~t) j ~s A n ~t L ; D (s 1;t 1)» 1 n D (s ;t )» 1 ; F = (~s;~t) j ~s L n ~t A 3; D (s 1;t 1)» 1 n D (s ;t )» 1 ; F 3 = (~s;~t) j ~s A n ~t L ; D (s 1;t 1)» 1 n D (s ;t )» 1 ; F = (~s;~t) j ~s L n ~t A 1; D (s 1;t 1)» 1 n D (s ;t )» 1 ; F 5 = (~s;~t) j ~s L n ~t L ; D (s 1;t 1)» 1 n D (s ;t )» 1 ; n F 6 = (~s;~t) j ~s L n ~t L ; D (s 1;t 1)» 1 n D (s ;t )» 1 : In th -imnsionl 7-msh with link ilur, th sts A 1, A, A 3, A, L n L r shown in Fi. 3(). Mor nrlly, with il vrtil link onntin nos ~v =(v 1;v )n ~u =(v 1; (v +1)mo), tr tkin th trnsormtion T ~v, w n in ths sts s th ollowin: A 1 = (; ) j ; Z n 1»» 1 ; 1»» 1 ; A = (; ) j ; Z n 1»» 1; 1»» 1 ; A 3 = (; ) j ; Z n 1»» 1; [ 1 1]»» 0; A = (; ) j ; Z n 1» < 1 ; [ 1 1]»» 0; L = (; ) j ; Z n =0; 1»» 1 ; n L = (; ) j ; Z n =0; [ 1 1]»» 0: A simpl wy or rovrin il tri is to rvrs its routin orr. Tht is, i th primry routin shm is to rout th tri horizontlly in th irtion o shortst yli istn irst, th rovry lorithm will rout th tri vrtilly irst (shown in Fi. 3()). Thus, tri tht is suppos to o throuh th il link will irumvnt th il link. Consir now th vrtil links rossin lin in Fi. 3() n th t tri in th st F 1 [ F [ F 3 [ F. Rroutin (i.. rvrsin th routin orr) ll o th t tri in F 1 [ F [ F 3 [ F throuh th vrtil links rossin lin will n itionl units o tri on h o ths six vrtil links. Fi. () illustrts th rovrin pths o th tri (oriintin rom nos 0, 0,n 0 ) in th st F 1, whih r in rrout throuh th link l ~ 0 ~ 0. Rovrin pths or th tri in F, lthouh not shown hr, is just lip o Fi. () with rspt to th lin. Th totl mount o rrout tri in F 1 [ F on link l ~ 0 ~ 0, whih is, xs th lowr oun o spr pity, C C 1 = 3 = 7. ( 1) Howvr, utilizin th rin strutur o th msh topoloy, w n rrout hl o th t tri throuh links rossin lin i (illustrt in Fi. ()). This wy, whv totl o six units tri throuh th link l ~ 0 ~ 0 (thr rom F 1 n thr rom F ). For th tri in th st F 5 [F 6,w n rrout hl o thm (six units) throuh th link l ~~. Th rminin six units o tri n rout vnly throuh th six vrtil links rossin lin. Thus, w n rstor th oriinl tri low y usin only n itionl C C 1 mount o pity on h vrtil link. () Fiur : Rstortion pth or th -imnsionl 7-msh So r w hv only isuss th lo on vrtil link. ow, w will rss th qustion o whthr th itionl tri on h horizontl link will x C C 1. For xmpl, on th link l ~ 0 ~ in Fi. 3(), on my in tht th numr o rrout tri rom th st F 1 [ F, nin, xs C C 1 = 7 tr rvrsin th routin orr o th t tri. Howvr, s w rrout th t tri irumvntin th il link, w not only put n itionl nin units o tri (~s A ;~t = )onlinkl ~ ~ 0 ~ ut lso tk nin units o tri (~s L ;~t L 3) wy rom link l ~ 0 ~. Ovrll, w hv zro itionl rrout tri rom th st F 1 [ F o throuh link l ~ 0 ~. vrthlss, tri in th st F5 [ F6 os xtr units o tri on th link l ~ 0 ~. By rroutin hl o th tri in F 5 [ F 6 (six) throuh th link l ~~ (without usin ny horizontl link), w n thn istriut th rst o ()

7 th tri in F 5 [ F 6 (six) vnly, so s to stisy th spr pity onstrint. As w hv mntion rlir, only th tri in th st S 6 i=1 Fi r in rrout in our pth s rovry lorithm. Tri whih is unt y th il link rmins intt in th rovry lorithm. Lstly, w nnot inlu th ull tils o th pth s rstortion lorithm in this ppr u to sp limittion. For th sm rson, w stt th ollowin thorm, whih shows tht th lowr oun on th spr pit (C C 1)is in hivl, without proo. Thorm 5. On -imnsionl -msh, to rstor th oriinl ll-to-ll tri in th vnt o link ilur, w n spr pity o 3 on h link or ( 1) o n ( 1) or vn y usin th rstortion lorithm. V. Conlusion This ppr xmins th pity rquirmnts or msh ntworks with ll-to-ll tri. This stuy is prtiulrly usul or th purpos o sin n pityprovisionin in stllit ntworks. Anovl thniqu o uts on rph is us to otin tiht lowr oun on th pity rquirmnts. This ut thniqu provis n iint n simpl wy o otinin lowr ouns on spr pity rquirmnts or mor nrl ilur snrios suh s no ilurs or multipl link ilurs. Anothr ontriution o this work is in th iint rstortion lorithm tht mts th lowr oun on pity rquirmnt. Our rstortion lorithm is rltivly st in tht only thos tri strms t y th link ilur must rrout. Yt, our lorithm utilizs muh lss spr pity thn link s rstortion (tor o improvmnt). Furthrmor, in orr to hiv hih pity utiliztion, our lorithm mks us o pity tht is rlinquish y tri tht is rrout u to th link ilur (i.. stu rls [5]). Intrstin xtnsions inlu th onsirtion o no ilurs, or whih inin n iint rstortion lorithm is hllnin, s wll s onsirin th impt o multipl link ilurs. Finlly, or th pplition to stllit ntworks, it woul lso intrstin to xmin th impt o irnt ross-link rhitturs. [7] M.C. Azizolu n O. Eiolu Lowr ouns on ommunition los n optiml plmnts in torus ntworks", IEEE Trns. on Computrs, vol. 9, no. 3, pp , Mr [] B. Bos, R. Bro, Y. Kwon, n Y. Ashir, L istn n topoloil proprtis o k-ry n-us," IEEE Trns. on Computrs, vol., no., pp , Au [9] P. W. Lmm, S. M. Glnistr, n A. W. Millr, Iriium ronutil stllit ommunitions," IEEE Arosp n Eltronis Systms Mzin, vol. 1, no. 11, pp , ov [10] D. P. Pttrson, Tlsi: lol ron ntwork," 199 IEEE Arosp Conrn, vol., pp , 199 [11] E. Ekii, I. F. Akyiliz, n M. D. Bnr, A istriut routin lorithm or trm tri in LEO stllit ntworks," IEEE/ACM Trns. on tworkin, vol. 9, no., pp , Apr [] G. D. Stmoulis n J.. Tsitsiklis, Eiint routin shms or multipl rosts in hyprus," IEEE Trns. on Prlll n Distriut Systms, vol., no. 7, pp , Jul [13] E. Vrvrios, Eiint routin lorithms or ol-u ntworks," in Proins o th 1995 IEEE 1th Annul Intrntionl Phonix Conrn on Computrs n Communitions, pp , [1] Y. J. Suh n K. G. Shin, All-to-ll prsonliz ommunition in multiimnsionl torus n msh ntworks," IEEE Trns. on Prlll n Distriut Systms, vol., no. 1, Jn [15] D. P. Brtsks, twork Optimiztion: Continuous n Disrt Mols, Athn Sintii, 199. [16] J. Sun n E. Moino, Cpity Provisionin n Filur Rovry in Msh-Torus tworks with Applition to Stllit Constlltions," LIDS Rport: P-51, Msshustts Institut o Thnoloy, Sp Rrns [1] Y. Xion n L. Mson, Rstortion Strtis n Spr Cpity Rquirmnts in Sl-Hlin ATM tworks," in Proins o IFOCOM '97, vol. 1, pp , [] S. Rmmurthy n B. Mukhrj, Survivl WDM Msh tworks, Prt I Prottion," in Proins o IFOCOM '99, vol., pp , Mr [3] S. Rmmurthy n B. Mukhrj, Survivl WDM Msh tworks, Prt II Rstortion," in ICC '99 Proins, pp , [] E. Moino n A. Ephrmis, Eiint lorithms or prormin pkt rosts in msh ntwork," IEEE/ACM Trns. on tworkin, vol., no., pp 639-6, Au [5] R. R. Irshko, M. H. MGror, n W. D. Grovr, Optiml pity plmnt or pth rstortion in STM or ATM mshsurvivl ntworks," IEEE/ACM Trns. on tworkin, vol. 6, Jun [6] S.S. Lumtt n M. Mr, Towrs pr unrstnin o link rstortion lorithms or msh ntworks," in Proins o IFOCOM '01, vol. 1, pp , 001.