Centralized Multi-Node Repair in Distributed Storage

Transcription

1 Cntalizd ulti-nod Rpai in Distibutd Stoag awn Zogui, and Zhiying Wang Cnt fo Pvasiv Communications and Computing (CPCC) Univsity of Califonia, Ivin, USA Abstact In distibutd stoag systms, multipl stoag nod failus a fqunt and fficintly coving thm is cucial fo high systm pfomanc. In this wok, w consid th poblm of paiing multipl failus in a cntalizd way, which can b dsiabl in many data stoag configuations. W fist stablish th tadoff btwn th pai bandwidth and th stoag siz fo functional pai. Using a gaph-thotic appoach, th optimal tadoff is idntifid as th solution to an intg optimization poblm, fo which w div a closd-fom xpssion. Whn th numb of asus satisfis k, k bing th minimum numb of nods ndd to constuct th nti data, th tadoff ducs to a singl point, fo which w povid an xplicit cod constuction. Expssions of th xtm points, namly th minimum stoag multi-nod pai (SR) and minimum bandwidth multi-nod pai (BR) points, a also divd. Futhmo, w pov that functional BR point is not achivabl fo lina xact pai cods. Finally, fo k and d, wh d is th numb of hlp nods duing pai, w show that th functional pai tadoff is not achivabl und xact pai, xcpt fo mayb a small potion na th SR point, which paallls th sults fo singl asu pai by Shah t al. Indx Tms gnating cods, distibutd stoag, multinod pai. I. INTRODUCTION Ensuing data liability is of paamount impotanc in modn stoag systms. Rliability is typically achivd though th intoduction of dundancy. Taditionally, simpl plication of data has bn adoptd in many systms. Fo instanc, Googl fil systms optd fo a tipl plication policy [1]. Howv, fo th sam dundancy facto, plication systms fall shot on poviding th highst lvl of liability. On th oth hand, asu cods a optimal in tms of th dundancy-liability tadoff. In asu cods, a fil of siz is dividd into k pics, ach of siz k. Th k fagmnts a thn ncodd into n fagmnts using an (n, k) maximum distanc spaabl (DS) cod and thn stod at n diffnt nods. Using such a schm, th data is guaantd to b covd fom any n k nod asus, poviding th highst lvl of data liability fo th givn dundancy. Howv, taditional asu cods suff fom high pai bandwidth. In cas of a singl nod asu, thy qui to download th nti data of siz to pai a singl nod stoing a fagmnt of siz k. This xpansion facto mad asu cods not pactical in som applications using distibutd stoag systms. In th last dcad, th pai poblm has gaind incasing intst and motivatd th sach fo a nw class of asu cods with btt pai capabilitis. Th sminal wok in [2] poposd a nw class of asu cods, calld gnating cods, that optimally solv th pai bandwidth poblm. Intstingly, th authos in [2] povd that on can significantly duc th amount of bandwidth quid fo pai and th bandwidth dcass as ach nod stos mo infomation. Fomally, suppos any k out of n nods a sufficint to cov th nti fil of siz. Assuming that d nods, tmd hlps, a paticipating in th pai pocss, dnoting th stoag capacity of ach nod by α and th amount of infomation downloadd fom ach hlp by β, thn, an optimal (, n, k, d, α, β) gnating cod satisfis k 1 = min{α, (d i)β}. Th quation dscibs th fundamntal tadoff btwn th stoag capacity α and th bandwidth β. Two xtm points can b obtaind fom th tadoff. inimum stoag gnating (SR) cods cospond to th bst stoag fficincy with α = k, whil minimum bandwidth gnating (BR) cods achiv th lowst possibl bandwidth at th xpns of xta stoag p nod. Following th sminal wok in [2], th has bn a fluy of intst in dsigning pactical gnating cods achiving th optimal tadoff, focusing mainly on th xtm SR and BR points [3] [11]. Th authos in [12] psntd a poduct-matix famwok that allows dsign of BR cods fo any valu of d and dsign of SR cods fo d 2k 2. Th poduct-matix constuction njoys simpl ncoding and dcoding and nsus optimal pai of all nods. Th afomntiond fncs, as most of th studis on gnating cods in th litatu, focus on th singl asu pai poblm. Howv, in many pactical scnaios, such as in lag scal stoag systms, multipl failus a mo fqunt than a singl asu. oov, many systms [13] apply a lazy pai statgy, which sks to limit th pai cost of asu cods: instad of immdiatly paiing vy singl failu, on waits until asus occu, thn, th pai is don by downloading th quivalnt of th total infomation in th systm to gnat th asd nods. In this wok, w consid th pai poblm of multipl asus in a cntalizd mann. Th famwok quis th contnt of any k out of n nods in th systm to b sufficint to constuct th nti data. Upon failu of nods in th systm, th pai is caid out by contacting any d nods (hlps) out of th n availabl nods, and downloading β amount of infomation fom ach of th d hlps. Ou objctiv is to chaactiz th functional pai tadoff btwn th stoag p nod α and th pai bandwidth β und th cntalizd multipl failu pai famwok.

2 Th cntalizd pai famwok is intsting in many pactical situations. Indd, th a situations in which, du to achitctual constaints, it is mo dsiabl to gnat th lost nods at a cntal sv bfo dispatching th gnatd contnt to th placmnt nods [13]. Fo instanc, on can think of a ack-basd nod placmnt achitctu [14] in which failus fquntly occu to nods cosponding to a paticula ack. In this scnaio, a cntalizd pai of th nti ack is favoabl to paiing th ack on p-nod basis. Futhmo, [14] showd that a cntalizd pai famwok can hav intsting applications to communication fficint sct shaing. Finally, cntalizd pai can b usd in a boadcast ntwok, wh th pai infomation is tansmittd to all placmnt nods (.g. [15]). Fo th abov asons, w bliv that chaactizing th pai-bandwidth tadoff und th cntalizd pai famwok is impotant fom both, an infomation-thotic and also a pactical pspctiv. A. Rlatd wok Coopativ gnating cods (also known as coodinatd gnating cods) hav bn studid to addss th pai of multipl asus [16], [17]. In this famwok, ach placmnt nod downloads infomation fom d hlps in th fist stag. Thn, th placmnt nods xchang infomation btwn thmslvs bfo gnating th lost nods. Th pai is caid out in a distibutd way. Coopativ gnating cods achiving th xtm points on th coopativ tadoff hav bn dvlopd: minimum stoag coopativ gnating (SCR) cods [18] and minimum bandwidth coopativ gnation (BCR) cods [19]. Th poblm of cntalizd pai has bn considd in [20], in which th authos stictd thmslvs to DS cods, cosponding to th point of minimum quid stoag p nod. [20] showd th xistnc of DS cods with optimal pai bandwidth in th asymptotic gim wh th stoag p nod (as wll as th nti infomation) tnds to infinity. In [21], th authos povd that Zigzag cods, which a DS cod dsignd initially fo paiing optimally singl asus [11], can also b usd to optimally pai multipl asus in a cntalizd mann. In [14], th authos indpndntly povd that multipl failus can b paid in Zigzag cods with optimal bandwidth. oov, [14] dfins th minimum bandwidth multi-nod pai cods as cods satisfying th popty of having th downloadd infomation dβ matching th ntopy of nods. Basd on that, th authos divd low bound on β fo systms having a ctain ntopy accumulation popty and thn showd achivability of th minimum bandwidth using BCR cods. Howv, th optimal stoag p nod siz α is not known und ths cods. In [22], th authos povidd an xplicit DS cod constuction that povid optimal pai fo all n k and k d n k. Th authos in [15] studid th poblm of boadcast pai fo wilss distibutd stoag which is quivalnt to th modl w study in this pap. B. Contibutions of th pap In this pap, w fist stablish th tadoff btwn th pai bandwidth and th stoag siz fo functional pai wh th paid nods a not ncssaily th sam as th faild nods. W obtain th tadoff using infomation flow gaphs. Whn th numb of asus satisfis k, k bing th minimum numb of nods ndd to constuct th nti data, th tadoff ducs to a singl point, fo which w povid an xplicit cod constuction. Futhmo, w pov that functional minimum bandwidth multi-nod pai point is not achivabl fo lina xact pai cods, whil lina cods achiv such point fo singl asu [12]. Finally, w show that th functional pai tadoff is not achivabl und xact pai, xcpt fo mayb a small potion na th minimum stoag multi-nod pai point, which paallls th sults fo singl asu pai [23], fo k, d. Th maind of th pap is oganizd as follows. A dsciption of th systm modl is povidd in Sction II. Th analysis of th functional tadoff is dtaild in Sction III. Sction IV dscibs ou cod constuction in cas k. W pov th non-achivability of BR cods und lina xact pai in Sction V. Th non-achivabilty of th intio points und xact pai is invstigatd in Sction VI. Sction VII daws conclusions. Finally, som of th poofs a lgatd to Sction VIII. II. SYSTE ODEL Th notations k and k a usd to dnot whth k is a multipl of, o not, spctivly. u = [u 1,..., u m ] dnots a vcto of lngth m. Fo a st A, A dnots th siz of A. W wit [i] = {1,..., i} fo any intg i 1. Th cntalizd mutli-nod pai poblm is chaactizd by paamts (, n, k, d,, α, β). W consid a distibutd stoag systm with n nods stoing amount of infomation. Th data lmnts a distibutd acoss th n stoag nods such that ach nod can sto up to α amount of infomation. Th systm should satisfy th following two poptis: Rconstuction popty: a data collcto (DC) conncting to any k n nods should b abl to constuct th nti data. Rgnation popty: upon failu of nods, a cntal nod is assumd to contact d k hlps and download β amount of infomation fom ach of thm. Nw placmnt nods join th systm and th contnt of ach is dtmind by th cntal nod. Th pai bandwidth is givn by β. Th total bandwidth is dnotd γ = dβ. W consid functional pai and xact pai. In th fom cas, th placmnt nods a not quid to b xact copis of th faild nods. Ou objctiv is to chaactiz th tadoff btwn th stoag p nod α and th pai bandwidth β und th cntalizd multipl failu pai famwok. On th optimal tadoff, th minimum bandwidth mutli-nod pai (BR) point has th minimum possibl β, and th minimum stoag mutli-nod pai (SR) point has th minimum possibl α. III. FUNCTIONAL STORAGE-BANDWIDTH TRADEOFF A. Infomation flow gaphs Simila to [2], th pfomanc of a stoag systm can b chaactizd by th concpt of infomation flow gaphs

3 (IFGs). Ou constuctd IFG dpicts th amount of infomation tansfd, pocssd and stod duing pai. An IFG has diffnt kinds of nods. It contains a singl souc nod s that psnts th souc of th data objct. Each stoag nod i of th IFG is psntd by two distinct nods: an input stoag nod x i in and an output stoag nod xi out. Each nod x i out is connctd to its input nod x i in with an dg of capacity α, flcting th stoag constaint of ach individual nod. Th infomation flow gaph is fomd with n initial nods, ach with stoag siz α connctd to th souc nod with dgs of capacity. Th IFG volvs with tim. Upon failu of nods, nw nods join simultanously th systm. Each of th placmnt nods x j is similaly psntd by an input nod x j in and an output nod xj out, linkd with an dg of capacity α. To modl th cntalizd pai natu of th systm, w add a vitual nod x i vitual that links th d hlps to th nw stoag nods. Likwis, th vitual nod consists of an input nod x i vitual,in and an output nod xi vitual,out. Th input nod x i vitual,in is connctd to th d hlps with dgs ach of capacity β. Th output nod x i vitual,out is connctd to th input nod x i vitual,in with an dg of capacity α, flcting to th ovall siz of th data to b stod in th nw placmnt nods. Th output nod x i vitual,out is thn connctd to th input nods x j in of th placmnt nods, with dgs of capacity. Each IFG psnts on paticula histoy of th failu pattns. Th nsmbl of IFGs is dnotd by G(n, k, d,, α, β). Fo convninc, w dop th paamts whnv it is cla fom th contxt. Givn an IFG G G, th a ( n k) diffnt data collctos conncting to k nods in G. Th st of all data collcto nods in a gaph G is dnotd by DC(G). Fo an IFG G G and a data collcto t DC(G), th minimum min-cut valu spaating th souc nod s and th data collcto t is dnotd by mincut G (s, t). B. Ntwok coding analysis Th ky ida bhind psnting th pai poblm by an IFG lis in th obsvation that th pai poblm can b cast as a multicast ntwok coding poblm [2]. Clbatd sults fom ntwok coding [24], [25] a thn invokd to stablish th fundamntal limits of th pai poblm. Dtmining th functional tadoff of th cntalizd pai poblm follows along th sam ida as th singl asu tadoff and th coopativ gnating cods [2], [17]. Accoding to th max-flow bound of ntwok coding [24], fo a data collcto to b abl to constuct th data, th minimum cut (min-cut) spaating th souc to th data collcto should b lag o qual to th data objct siz. Considing all possibl data collctos and all possibl failu pattns, th following condition is ncssay and sufficint fo th xistnc of gnating cods 1 satisfying th liability constaint: min min mincut G(s, t). (1) G G t DC(G) 1 Stictly spaking, this is only valid whn th numb of failus/pais is boundd. A igoous poof is quid to dop th bounddnss assumption as [17], [26] Analyzing th minimum cut of all IFGs sult in th following thom. Thom 1. Fo fixd systm paamts (, n, k, d,, α, β), gnating cods satisfying th cntalizd multi-nod pai condition xist if and only if ( g ) i 1 min u P wh min(u iα, (d u j)β) min f(u), (2) u P f(u) = i 1 min(u i α, (d u j )β), (3) i 1 P = {u : 1 u i such that g u i = k, g k}. (4) W not that (2) was also indpndntly dvlopd in [14]. Poof: Consid a covy scnaio u P in which a data collcto DC conncts to a subst of k nods {x i out : i I}, wh I is th subst of contactd nods. Th siz of th suppot of u cosponds to th numb of pai goups of siz taking pat in th constuction pocss, whil u i cosponds to th numb of nods contactd fom pai goup i. As all incoming dgs of DC hav infinit capacity, w only xamin cuts (Ū, U) with S U and {xi out : i I} Ū. Evy dictd acyclic gaph has a topological soting, which is an oding of its vtics such that th xistnc of an dg x y implis x < y. W call that nods within th sam pai goup a paid simultanously. Sinc nods a sotd, nods considd at th i th stp cannot dpnd on nods considd at j th stp with j > i. Consid th i-th goup, consid th cas { x i in U} = m and th maining nods a such that x i in Ū. if x i in U, thn th contibution of ach nod is α. Th ovall contibution of ths nods is mα. ls: x i in Ū, thn if xi vitual,out U, th contibution of this nod is. Thus, w only consid th cas x i vitual,out Ū. Thn, w discuss two cass if x i vitual,in U, th contibution to th cut is α. ls, sinc th i-th goup is th topologically i-th pai goup, at most i 1 u j dg com fom output nods in Ū. Thus, th contibution is (d i 1 u j )β. Thus, th contibution of this nod is min(α, (d i 1 u j )β). As fo ths nods, x i vitual,out Ū, w do not nd to account fo oth simila nods. Thus, if m = u i, th contibution of th i-th pai goup is u i α. If m < u i, th contibution is mα + min(α, (d i 1 u j )β), which can b ducd to min(α, (d i 1 u j )β) if m = 0. Thus, to low th cut capacity, ith m = u i in cas (d i 1 u j )β > u i α o m = 0 othwis. Thus, th total

4 contibution of th i-th pai goup is i 1 min(u i α, (d u j )β). Finally, summing all contibutions fom diffnt pai goups and considing th wost cas fo u P implis that min min mincutg(s, t) = ( g ) i 1 min(u iα, (d u j)β), G G t DC(G) min u P with P dfind as in (4). Thfo, th xistnc of gnating cods is guaantd by [24] as long as min min mincut G(s, t). G G t DC(G) In th squl, w will us th notation k = a +, such that a = k and = k mod. C. Solving th minimum cut poblm In this sction, w div th stuctu of th optimal configuation u in (2) fo any st of paamts (α, β). Fo instanc, w show that fo (i 1) < k i, th numb of optimal pai goups g (th suppot of u) is qual to i. Th sult is fomalizd in th following poposition. Thom 2. Fo an (, n, k, d,, α, β) stoag systm, th scnaio u cosponding to th minimum cut ov all infomation flow gaphs (cf. (2)) is chaactizd as follows: k, if k, [,..., ], ls if k = a, }{{} u a tims = [,,..., ], ls if k = a + and α d+a a }{{} β, a tims [,...,, ], othwis, }{{} a tims wh 0 < <. In poving th sult of Thom 2, w fist chaactiz th optimal solution in cas k. Insight and intuition gaind fom th fist cas a usd to div and motivat th gnal solution. W fist stat th following lmma, which psnts a ky stp towads poving ou sult. Lmma 3. Lt α, β, u 1, u 2, d,, l b non-ngativ als such that u 1 + u 2 = s, thn th following inquality holds f([u 1,,...,, u }{{} 2 ]) min(f([s,,..., ]), f([,...,, s])), }{{}}{{} l tims l tims l tims wh f(u) is dfind as in (3). Poof: To pov th sult, w cast it as an optimization poblm as follows minimiz u=[u 1,u 2] subjct to 0 u 1 s, l 1 min(u 1 α, dβ) + min(α, (d i u 1 )β) + min(u 2 α, (d (l + 1) u 1 )β) 0 u 2, u 1 + u 2 = s. (5) Substituting u 2 by s u 1 in (5), using th idntity min(a, b) = a+b a b 2 and aft liminating constant tms, (5) bcoms quivalnt to minimiz u 1 l 1 u 1lβ u 1α dβ α dβ + iβ + u 1β sα u 1(α β) (d l)β subjct to 0 u 1 s. (6) Th objctiv function in (6), as function of u 1, is concav on th intval [0, s]. Th concavity is du to th convxity of x x. Thfo, th minimum is achivd at on of th xtm valus. Equivalntly, u 1 = s o u 1 = 0. 1) Cas k : In this scnaio, conncting to k nods fom th sam pai goup yilds th wost cas scnaio fom an infomation flow pspctiv. Givn a paticula pai scnaio chaactizd by a vcto u, fo any two adjacnt pai goups (i.., two adjacnt ntis in u) with n 1 and n 2 nods spctivly, w hav u 1 + u 2. On can goup ths two goups into a singl pai goup to achiv a low cut valu. Indd, fom th cut xpssion in (2), th contibution of th initial st [u 1, u 2 ] to th cut is min(u 1 α, lβ) + min(u 2 α, (l u 1 )β), fo som l. Aft gouping th goups into a singl pai goup, th contibution of th nwly fomd pai goup is min((u 1 + u 2 )α, lβ), which is low than th initial contibution by vitu of Lmma 3, thus achiving a low cut. This mans that stating fom an IFG, w constuct a nw IFG that has on lss pai goup and low min-cut valu. This pocss can b patd until w nd up with a singl pai goup consisting of k nods, which cosponds to th minimum cut ov all gaphs in this cas. Thfo, th tadoff in (2) is simply chaactizd by min(kα, dβ). oov, α SR = α BR = k and β SR = β BR = d. Equivalntly, th functional stoag bandwidth tadoff ducs to a singl point givn by (α SR, β SR ) = (α BR, β BR ) = ( k, d ). 2) Cas < k: otivatd by th pvious cas, th intuition is that, givn a scnaio u, on should fom a nw scnaio which xhibits as many goups of siz as possibl. Subsquntly, on constucts a scnaio u such that all its ntis, xcpt mayb on nty qual to, a qual to. Lmma 3 addsss th cas u 1 + u 2. Gnalizing it to th cas wh u 1 + u 2 2 follows th sam appoach. Coollay 4. Assum that u 1 +u 2 = +s. Thn, th following inquality holds f([u 1,,...,, u 2]) min(f([s,,..., ]), f([,...,, s])), (7) }{{}}{{}}{{} l tims l+1 tims l+1 tims wh f(u) is dfind as in (3). Poof: Fist, w notic that u 1 = + s u 2 s as u 2. Thn, th poof follows along simila lins as that of Lmma 3 by placing th constaint in (6) by s u 1. Fo a fixd β, as a function of α, w dnot th min-cut cosponding to u = [,...,,,,..., ] by C }{{}}{{} j (α), j = jtims a j tims

5 0,..., a. As will b shown lat in th poof of Thom 2, a caful analysis of th bhavio of th a + 1 diffnt configuations C j (α) is ndd to dtmin th ovall optimal scnaio lading th lowst minimum cut. W stat th sult in th following lmma, whos poof is lgatd to Appndix VIII-A. Lmma 5. Th xists a point α c (a) [ d β, d β] such that, fo any 0 j a, { C 0 (α), if α α c (a), C j (α) (8) C a (α), if α α c (a), with d + a a α c (a) = β. (9) Poof of Thom 2: Now that w hav th ncssay machiny, w pocd as follows: givn any configuation u, w kp combining and/o changing pai goups by mans of succssiv applications of Lmma 3 and Coollay 4 until w can no long duc th minimum cut. Th algoithm convgs bcaus at ach stp, ith th numb of pai goups in u is ducd by on, o th numb of pai goups of full siz is incasd by on. As th numb of pai goups is low boundd by a, and as th numb of pai goups of full siz is upp boundd by a, th algoithm must convg aft a finit numb of stps. It can b sn thn that th abov duction pocdu has a finit numb of outcoms, givn by u = [,..., ] if k = a, }{{} a tims u = [,...,,,,..., ] whn k = a +, }{{}}{{} jtims a j tims with 0 < < and j {0,..., a}. Thfo, if k, thn th optimal scnaio cosponds to considing xactly a pai goups. On th oth hand, if k, thn, it is optimal to consid xactly a + 1 pai goups. Howv, th optimal position of th pai goup with nods nds to b dtmind. Thn, using th Lmma 5, th sult in Thom 2 follows. Exampl 1. Lt u = [1, 3, 2, 3, 2] with = 3. Thn, on can stat by ducing th fist th pai goups [1, 3, 2]. This lads to u = [3, 3, 3, 2]. Anoth appoach would b to consid th st [2, 3, 2]. Rducing this st lads to ith u = [1, 3, 3, 3, 1] o u = [1, 3, 1, 3, 3]. Rducing futh u = [1, 3, 3, 3, 1] lads to u = [2, 3, 3, 3] o u = [3, 3, 3, 2]. Rducing u = [1, 3, 1, 3, 3] lads to u = [3, 2, 3, 3] o u = [2, 3, 3, 3]. It mains to compa th cuts givn by u = [3, 3, 3, 2], u = [3, 3, 2, 3], u = [3, 2, 3, 3] and u = [2, 3, 3, 3]. Following Thom 2, ith u = [2, 3, 3, 3] o u = [3, 3, 3, 2] givs th lowst min-cut. D. Explicit xpssion of th tadoff Having chaactizd th optimal scnaio gnating th minimum cut in th last sction, w a now ady to stat th admissibl stoag-pai bandwidth gion fo th cntalizd multi-nod pai poblm. Thom 6. Fo an (, n, k, d,, α, β) stoag systm, th xists a thshold function α (, n, k, d, γ, ) such that fo any α α (, n, k, d,, β), gnating cods xist. Fo any α < α (, n, k, d,, β), it is impossibl to constuct cods achiving th tagt paamts. Th thshold function α (, n, k, d, γ, ) is dfind as follows: if k, thn: α = k, γ [, + ), ls if k = a, thn: {, k γg 0 (i) γ [f0(a 1), + ), α = i, γ [f 0(i 1), f 0(i)], i = a 1,... 1, (10) ls: k = a + with 1 1, thn: γ [f(a 1), + ), k α γg (i) =, γ [f +i (i 1), f (i)], i = a 1,... 1, γg (0) d, γ [ (a+1)d ( a+1 2 ), f(0)], wh (11) 2d f (i) = k (k ) + 2kd 2 (i 2 + i) 2i, (12) (a i)( d a i) g (i) =. 2d (13) Poof: S Appndix VIII-B. Rmak 1. In cas divids k, th following quality holds fo all points on th tadoff = min(α, (d i)β) min(α, ( d i)β). = Thfo, th tadoff btwn α and β is th sam as th singl asu tadoff of a systm with ducd paamts givn by, k = a and d. Th xpssion of th tadoff in this cas can b covd fom [2] with th appopiat paamts. W now hav th xpssions of th two xtmal points on th optimal tadoff. W focus on th cas < k, as othwis th optimal tadoff ducs to a singl point. Th SR point is th sam ispctiv of th lation btwn k and, and it is givn by α SR = k, γsr = k d d k +. (14) Intstingly, th BR point dpnds on whth divids k o not. If k = a, w obtain γ BR = 2d k 2 + k + 2kd = d da ( a 2 ), (15) α BR = γ BR. (16) Th amount of infomation downloadd fo pai is qual to th amount of infomation stod at th placmnt nods. This popty of th BR point is simila to th minimum bandwidth point in th singl asu cas [2] and also th minimum bandwidth coopativ pai point [17]. If k = a +, w obtain 2 γ BR = (k + )(2d k + ) = d d(a + 1) ( ) a+1, (17) d + a a α BR = γ BR. (18) d 2

6 Compaing th amount of infomation stod to th total bandwidth, w hav α BR γ BR = 1 + ( )(d a) d > 1. (19) This situation is novl fo multipl asus as th nods nd to sto mo than th ovall downloadd infomation. This is an xta cost in od to achiv th low valu of th pai bandwidth. IV. CONSTRUCTION WHEN k In cas k, th optimal paamts satisfy α = k, β = d and γ =. W not that th ovall pai bandwidth and th constuction bandwidth a th sam. Thfo, on can achiv α and γ, by dividing th data into k packts and ncoding thm using (n, k) DS cod (fo xampl, a Rd- Solomon cod). Th pai can b don by downloading th full contnt of any k out of d hlps whil not contacting d k nods. Such pai is asymmtic in natu. W dscib on appoach fo achiving th pai with qual contibution fom d hlps. 1) Divid th oiginal fil into kd symbols (i.., packts) (that is = kd) and ncod thm using an (nd, kd) DS cod. 2) Sto th ncodd packts at n nods, such that ach nod is stoing α = d ncodd packts. 3) Fo constuction, fom any k nods, w obtain kd diffnt symbols. By vitu of th DS popty, w can constuct th data. 4) Fo pai, ach hlp nod tansmits any β = d = k symbols. Th placmnt nods civ dk diffnt codd symbols, which a sufficint to constuct th whol data and thus gnat th missing symbols. Rmak 2. Th abov pocdu woks fo a spcific pdtmind d. Howv, it can b gnalizd to suppot any valu of d satisfying k d n. Fo instanc, lt δ = lcm(k, k + 1, k + 2,..., n ) (lcm dnots th last common multipl). Th fil of siz is thn dividd into kδ packts and ncodd into nδ with an DS cod. Each nod thn stos α = k = kδ k = δ codd symbols. Fo pai with a spcific d, ach nod tansmits any β = d = k δ d codd symbols fo his nod. Similaly, it can b sn that constuction and xact pai is always fasibl fo any k d n. Not that th constaint of th fild siz aiss fom th nd fo an (nδ, kδ) DS cod. Th fild siz nds to b no lss than nδ,.g. Rd Solomon cods. V. NON-EXISTENCE OF EXACT BR REGENERATING CODES Exact gnating cods a of intst in pactic. Exact gnating cods achiving th SR point hav bn constuctd [14], [21], [22], [27]. In this sction, w xplo th xistnc of lina xact BR gnating cods. Unlik th singl asu pai poblm [12] and th coopativ pai poblm [19], w pov that lina xact gnating cods do not xist. Following [12], [19], w pocd by invstigating subspac poptis lina xact BR cods should satisfy. Thn, w pov that th divd poptis ovconstain th systm. A. Subspac viwpoint Lina xact gnating cods fo th BR point can b analyzd fom a viwpoint basd on subspacs. A lina stoag cod is a cod in which vy stod symbol is a lina combination of th souc symbols. Lt f dnot an - dimnsional vcto containing th souc symbols. Thn, any symbol x can b psntd by a vcto h satisfying x = f t h such that h F, F bing th undlying finit fild. Th vctos h dfin th cod. A nod stoing α symbols can b considd as stoing α vctos of th cod. Nod i stos h (i) 1... h(i) α. It is asy to s that lina opations pfomd on th stod symbols a quivalnt to th sam opations pfomd on th ths vctos: γ i f t h i = f t ( γ i h i ). Thus, ach nod is said to sto a subspac of dimnsion at most α. W wit W A to dnot th subspac stod by all nods in th st A. Fo gnation, ach nod passs β symbols. Equivalntly, ach nod passs a subspac of dimnsion at most β. W dnot th subspac passd by nod j to pai st R of nods by Sj R. Th subspac passd by a st of nods A to pai a st R of nods is dnotd by SA R. Th symbol j A j dnots th dict sum of subspacs A j. Notation. Fo a gnal xact gnating cod, which can b nolina, w us by abus of notation W A, SA R to psnt th andom vaiabls of th stod infomation in nods A, and of th tansmittd infomation fom hlps A to faild nods R. Poptis that hold using ntopic quantitis fo a gnal cod do hold whn considing lina cods. Fo instanc, consid two sts A and B. Thn, w not th following H(W A ) dim(w A ), (20) H(W A W B ) dim(w A ) dim(w A W B ), (21) I(W A, W B ) dim(w A W B ), (22) wh th symbol mans tanslats to. Whn sults hold fo gnal cods, w only pov fo th ntopy poptis, and th poof fo th subspac poptis of lina cods is omittd. All sults on ntopic quantitis a fo gnal cods, and all sults on subspacs a fo lina cods. In this sction, w focus on symmtic cods. Namly, th sults do not dpnd on th indics of th nods. Not that on can always constuct a symmtic cod fom a non-symmtic cod [28]. W now stat by poving som poptis xact gnating cods should satisfy. Th following popty [21, Lmma 4] is valid fo all optimal xact gnating cods, not ncssaily BR cods. Lmma 7. Lt B [n] b a subst of nods of siz, thn fo an abitay st of nods A, such that A d, B A =, H(W B W A ) min( B α, (d A )β). (23) Poof: If nods B a asd, consid th cas of having nods A and nods C as hlp nods, C = d A. Thn,

7 th xact pai condition quis 0 =H(W B S B A, S B C ) = H(W B S B A ) I(W B, S B C S B A ) H(W B S B A ) H(S B C ) H(W B S B A ) (d A )β H(W B W A ) (d A )β. oov, w hav H(W B W A ) H(W B ) B α and th sults follows. Th poof was also givn in [21]. In th nxt subsction, w focus on th cas wh k. B. Cas k Points on th tadoff satisfy = min(α, (d j)β), d k + β α d β. j=0 Lmma 8. (Entopy of data stod): Fo an abitay st L of stoag nods of siz, and a disjoint st A such that A = m < k fo som intg m Fo lina cods, H(W L ) = α, (24) H(W L W A ) = min(α, (d m)β). (25) dim(w L ) = α, (26) dim(w L ) dim(w L W A ) = min(α, (d m)β). (27) Poof: By constuction quimnt, w wit = H(W [k] ) (28) = H(W [] ) + H(W {j+1,...,(j+1)} W [j] ) (29) α + min(α, (d j)β) (30) min(α, (d j)β) (31) j=0 =, (32) wh (30) uss Lmma 7 and (31) follows as α dβ fo all points on th tadoff. Thus, all inqualitis must b satisfid with quality. Rmak 3. Lmma 8 stats that th contnts of any goup of nods a indpndnt. In paticula, fo ach nod i, w hav H(W i ) = α. Coollay 9. At th BR point, fo any st L of siz and disjoint st A of siz A = m < k, w hav dim(w L W A ) = mβ. Poof: dim(w L ) dim(w L W A ) = min(α, (d m)β) = (d m)β. Using th fact that dim(w L ) = α = dβ, w obtain th sult. Lmma 10. Fo any st E of siz, th BR point satisfis W E = j S E j, dim(s E j ) = β. Th subspacs Sj E and SE j a linaly indpndnt. oov, ach subspac has to b in th span of W E : Sj E W E. Poof: Fo xact pai, w nd W E j dβ = dim(w E ) dim( j S E j ) dβ = α. S E j. Thus, Thus, vy inquality has to b satisfid with quality. Lmma 11. At th BR point, fo any st E of nods and any oth disjoint st Q of siz Q k, w hav S E Q = W E W Q, dim(s E Q) = Q β. (33) Poof: Consid Q nods such that Q k hlping in th pai of a st E of nods. Lt J contains Q such that J = k. Fom Coollay 9, w hav dim(w E W J ) = (k )β. On th oth hand, fom Lmma 10, w hav SJ E W E. oov, by dfinition, SJ E W J. Thus, SJ E W E W J. As th dimnsions match, it follows that SJ E = W E W J. Not that SA E W E W A holds fo any subst A of siz A d. Now, w wit S E J = W E W J = W E (W Q + W Q c) W E W Q + W E W Q c S E Q + S E Q c = SE J. This implis that all inclusion inqualitis hav to b satisfid with quality and th sult follows. Th nxt lmma plays an impotant ol in stablishing th non-xistnc of xact BR cods. It only holds tu whn 2, which confoms with th xistnc of singl asu BR cods. Lmma 12. Consid th BR point. Whn 2, fo any st of + 2 k nods, labld 1 though + 2, it holds that dim(w +2 (W [+1] )) = dim(w +2 (W [] ) = β. (34) Poof: W hav dim(w [+2] ) = dim(w [] ) + dim(w +1 + W +2 ) dim(w [] (W +1 + W +2 )) = α + 2α 2β, wh th scond quality follows fom Lmma 8, Lmma 11 and th fact that any st of nods a linaly indpndnt. On th oth hand, w wit dim(w [+2] ) = dim(w [] ) + ( dim(w +1 ) dim(w +1 W [] ) ) + ( dim(w +2 ) dim(w +2 W [+1] ) = α + 2α β dim(w +2 W [+1] ). Th lmma follows fom quating both quations. Thom 13. Exact lina gnating BR cods do not xist whn 2 < k and k.

8 Poof: Assuming th xists an xact-pai gnating cod satisfying th constaints, w consid th fist nods. Thn, ths nods sto linaly indpndnt vctos. W wit, fo i = 1,...,, W i = ( ) V i1 V i2 wh Vi1 contains β linaly indpndnt columns and V i2 contains th maining (α β) basis vctos fo nod i. Now, consid nod + 1. W hav dim(w +1 W1 ) = β. That mans that nod + 1 contains β columns, linaly dpndnt on th columns fom th fist nods. Sinc any st of nods among th fist +1 nods should b linaly indpndnt, w.l.o.g, w can assum that th β dpndnt nods of nod + 1, V +1,1 is of th fom V +1,1 = V i,1 x i, (35) such that x i 0 β 1 i = 1,...,. Now, consid nod + 2. Fom Lmma 12, nod + 2 contains (α β) vctos linaly indpndnt fom vctos in nods 1 though + 1. Th maining basis vctos of nod + 2 (which a linaly indpndnt of th (α β) vctos) a containd in V +2,1. Now, to pai any st of nods fom th st of fist + 1 nods, nod +2 can only pass V +2,1. Othwis, Lmma 11 will b violatd. Thn, this implis that V +2,1 W J, wh J {1,..., + 1} such that J =. Thm it can b sn that V +2,1 can only b of th sam fom in (35) V +1,1 = V i,1 y i, such that y i 0 β 1 i = 1,...,. Simila asoning applis to nod i fo i = + 3,..., k + 1 to conclud that V i,1 can b wittn as in (35). Now, assum th fist nods fail. Thn, nod i can only pass V i,1 fo i = + 1,... k + 1. W call fom Lmma 11 that S [] i = W i W []. Th total numb of vctos passd by ths nods is (k + 1)β ( + 1)β. On th oth hand, fom (35), all V i,1 a gnatd by β nods. Thus, th st {V i,1, i = + 1,..., k + 1} must b linaly dpndnt, which contadicts th lina indpndnc popty of th passd subspacs passd fo pai, as statd by Lmma 10. C. Cas k In this cas, all points on th tadoff satisfy = min(α, dβ) + min(α, (d i)β), (36) d k + d + a a β α β. (37) Poptis satisfis by BR xact gnating cods dvlopd in th pvious sction xtnd to th cas k with slight modifications. W stat th poptis without poofs as th tchniqus a th sam. Lmma 14. Fo an abitay st R of stoag nods of siz, and a st A such that A = j + < k fo som intg j a 1, fo all xact-gnating cods opating on th functional tadoff, it holds that dim(w R ) = α, (38) dim(w E ) dim(w E W A ) = min(α, (d j)β). (39) Rmak 4. In cas k, a st of a no long linaly indpndnt. This is xpctd as α > dβ. Instad, it can b sn fom Lmma 14 that any st of nods a linaly indpndnt. Rcall that fom th analysis of Thom 2, at th BR point, two scnaios gnat th sam minimum cut: u 1 = [,,..., ] and u 2 = [,...,, ]. Equivalntly, w hav wh f(u) is dfind as in (3). = f(u 1 ) = f(u 2 ), (40) Lmma 15. Fo xact-gnating cods opating at th BR point, givn sts E, A, R and B such that E =, E and A a disjoint, R and B a disjoint, A = j with j a 1, R = and B = a, it holds that dim(w E) = dβ, (41) dim(w E) dim(w E W A) = (d j)β, (42) dim(w R) dim(w R W B) = (d a)β. (43) Poof: Th sult can b divd by pocding as in Lmma 14 and using th fact that = f(u 2 ) fom (40). Equations (42) and (43) follow by noticing that α dβ. Lmma 10 and Lmma 11 hold tu cas k. Th following lmma is usd to div th contadiction. Lmma 16. lt k = a +, thn at th BR point, fo any st of + 1 nods, it holds that Poof: W hav dim(w +1 W [] ) = β. (44) dβ = dim(w [] ) (45) = dim(w i) dim(w i W [i 1] ) (46) = α i=+1 dim(w i W [i 1] ), (47) wh th last quality follows fom th fact that th fist nods a linaly indpndnt. Thus, it follows that dim(w i W [i 1] ) = α dβ = ( )(α aβ). (48) i=+1 Now w wit ( )(α aβ) = i=+1 i=+1 dim(w i W [i 1] ) (49) dim(w i W [] ) (50) = ( ) dim(w +1 W [] ), (51) wh th last quality follows using symmty. Thn, it follows that dim(w +1 W [] ) α aβ. (52) Combining (48) and (52), w obtain dim(w i W [i 1] ) ( 1)(α aβ). (53) i=+2

9 On th oth hand, w hav dim(w i W [i 1] ) i=+2 i=+2 dim(w i W Ei ) (54) = ( 1)β, (55) wh E i is a st of nods containing th fist i 1 nods and abitay i + 1 nods, xcluding nod i, and th quality follows fom Lmma 11. Combining (53) and (55), it follows ( 1)(α aβ) dim(w i W [i 1] ) ( 1)β. i=+2 (56) It follows that α aβ = d a β β. Th last quality holds only whn d = k. Othwis, α aβ > β. Thfo, w only consid th cas d = k fo which α = (a + 1)β and α aβ = β. oov, it follows fom (56) that dim(w i W [i 1] ) = ( 1)β. (57) i=+2 Using (48), w obtain dim(w +1 W [] ) = β. Thom 17. Exact lina gnating BR cods do not xist whn < k and k. Poof: Consid pai of th st of nods E containing nods 1 though. Consid hlp nod i. As dim(w i W [] ) = dim(w i W [] ) = β, it follows that W i W [] = W i W [] = Si E. Thn, ach hlp nod snds vctos in th span of W []. Thus, th span of all sub-spacs S [] i is includd in th span of W [] : Si E W []. This implis that i dim( Si E) dim(w []). Namly, w should hav dβ α: i this is a contadiction as dβ > α. VI. NON-FEASIBILITY OF EXACT-INTERIOR POINT In this sction, w study th non-fasibility of th intio points fo k, d, similaly to [23]. W not that all intio points satisfy (d k + )β α dβ. This can b wittn as (d a + 1)β α d β, wh d = d and a = k. This is simila to th singl asu cas with ducd paamts. a) Paamtization of th intio points: Lt α = (d p)β θ, namly α = (d p)β θ with p {0, 1,..., } with θ [0, β) such that θ = 0 if p = a 1. Points at th tadoff satisfy = min(α, (d i)β). A. Poptis of Exact-Rpai Cods W psnt a st of poptis that xact-pai cods, satisfying th functional tadoff, must satisfy. Lmma 18. Fo a st A of abitay nods of siz j, a st L of nods of siz such that L A =, w hav 0 j p, I(W L, W A ) = ((j p)β θ) p < j < a (58) α j a. Poof: Fist, w not that whn j a, I(W L, W A ) = H(W L ) H(W L A) = H(W L ) = α. In th following, w assum j < a. W wit I(W L, W A) = H(W L) H(W L A) (59) = α min(α, (d j)β) (60) = (α min(α, (d j)β)) = (α (d j)β) + (61) = ((j p)β θ) +. (62) Coollay 19. Fo an abitay st L siz, and a disjoint st A such that A = m < k fo som intg m, w hav H(W L S L A) = H(W L W A ) = min(α, (d j)β). (63) Poof: Fom Lmma 7, w hav H(W L S L A ) min(α, (d j)β). On th oth hand, fom Lmma 8, H(W L S L A) H(W L W A ) = min(α, (d j)β). (64) Thus, H(W L S L A ) = H(W L W A ) = min(α, (d j)β). Lmma 20. In th situation wh nod m is an abitay hlp nod assisting in th pai of a scond st of abitay nods L of siz, w hav H(S L m) = β, (65) ispctiv of th idntity of th oth d 1 hlp nods. oov, fo st B of siz B d k + with B L =, w hav H(S L B) = B β. (66) Poof: Patition th st of d hlps into A and B such that A = k and B = d k +, such that m B. W hav H(W L SA L ) = min(α, (d k + )β) = (d k + )β, as α (d k + )β fo all points on th tadoff. oov, xact pai quis H(W L SA L, SL B ) = 0. Thus, H(SL B ) (d k + )β. This implis H(SB L ) = (d k + )β. oov, it must hold that H(Sm) L = β in addition to Sm L and Sm L bing indpndnt if m m. oov, by choosing B, on obtains H(S L ) = β. a) Hlp Nod Pooling: Consid a st F consisting of a collction of f d + nods (f is a multipl of ), and a subst R of th st F consisting of nods. A hlp nod pooling scnaio is a scnaio wh on failu on any nods L R, th d hlp nods assisting in its pai includ all th f maining nods in F. Th maining hlp nods a dnotd by V(L). Lt R =. Lmma 21. In th hlp nod pooling scnaio wh min(a, f ) > p + 2, fo any st of abitay nod F R, w hav H(S R ) (2β θ). (67) Poof: Th statmnt holds tu fo all f f and. Thn, fo th poof, consid = p + 2 and F = R, f = (p + 3). Consid pai of an abitay nod L R, wh th st of hlps includ and th (p + 1) maining nods in R. Thn, w wit

10 I(S L ; W R) = I(S L ; W L, W R L) (68) Thn, w obtain = I(S L ; W R L) + I(S L ; W L W R L) (69) I(S L ; W L W R L) (70) = H(W L W R L) H(W L W R L, S L ) (71) H(W L W R L) H(W L SR L, L S L ) (72) = min(α, (d (p + 1))β) min(α, (d (p + 2))β) (73) = (d (p + 1))β (d (p + 2))β = β. (74) H(S L W R) = H(S L ) I(S L ; W R) β β = 0. (75) Hnc, H(S L W R) = 0. Sinc, th choic of th st L fom R was abitay, it follows H(S R W R) = 0. It follows fom Lmma 18 that H(S R ) = I(S R ; W R ) I(W ; W R ) = (2β θ). Lmma 22. In th hlp nod scnaio wh min{a, f } > p + 1, fo an abitay st of nods F R, and an abitay pai of st of nods L 1 and L 2, it must b that and hnc H(S L1 SL2 ) θ, (76) H(S R ) (β + ( 1)θ). (77) Poof: Th st is R assumd to consist of = (p + 1) nods, and th st F is such that F = R {}. I(S L ; W R) = I(S L ; W R L, W L) (78) Thn, it must b that = I(S L ; W R L) + I(S L ; W L W R L) (79) I(S L ; W L W R L) (80) = H(W L W R L) H(W L W R L, S L ) (81) H(W L W R L) H(W L S L R L, S L ) (82) = min(α, (d ( 1))β) min(α, (d )β) (83) = (d p)β θ (d (p + 1))β (84) = (β θ). (85) H(S L W R) = H(S L ) I(S L ; W R) β (β θ) = θ. (86) Not that th last inquality holds fo any st L R. Nxt, consid L 1, L 2 R. Fo this, consid H(S L 1, SL 2 ) = I(W R; S L 1, SL 2 ) + H(SL 1, SL 2 WR) (87) I(W R; W ) + H(S L 1 WR) (88), SL 2 = I(W R; W ) + H(S L 1 WR) + H(SL 2 WR, SL 1 ) (89) (β θ) + θ + θ = (β + θ), (90) wh th last inquality follows fom Lmma 18. Thn, w hav H(S L1 SL2 ) = H(SL1, SL2 ) H(SL2 ) (91) = H(S L1 ) β (92), SL2 (β + θ) β = θ, (93) wh th fist quality follows fom (66). Finally, patitioning th nods in R in abitay sts L 1, L 2,..., L, it follows H(S R ) H(S L1 ) + H(S Li B. Non-xistnc poof i=2 SLi 1 ) β + ( 1)θ. (94) Fist, w consid th intio points that a multipl of β. That is: α = (d p)β, θ = 0, with p lying in th ang 1 p a 2. Thom 23. Exact-pai cods do not xist fo th intio points with θ = 0. Poof: Consid a sub-ntwok F consisting of d + nods. Not that fo any st L F, H(W L SF L L ) = 0. oov, fo distinct, L 1, L 2 F, with θ = 0, w hav H(S L1 SL2 ) = 0. W patition th nods in F into goups of siz, dnotd L i. Thn, w wit H(W F ) H({S L F L} Li ) (95) = H({S F L L } Li ) (96) L i H(S F Li L i ) (97) L i β = (d + )β, (98) wh th inquality follows fom Lmma 22. On th oth hand, = i = i min(α, (d i)β) (99) min((d p)β, (d i)β) (100) = 2(d p)β + i 2 min((d p)β, (d i)β) (101) 2(d p)β + (a 2)β (102) 2β + (d p)β + (a 2)β (103) 2β + (d p)β + (a 2)β (104) = (d 2)β + (k 2 p)β (d 2)β, (105) wh w assum p a 2 (Non SR point). Thus, p + 2 k d. Both bounds a contadictoy, thus poving th impossibility sult in cas θ = 0. Thom 24. Fo any givn valus of, xact-pai gnating cods do not xist fo th paamts lying in th intio of th stoag-bandwidth tadoff whn θ 0, xcpt possibly fo th cas p + 2 = a and θ d p d p β. Poof: S Appndix VIII-C. VII. CONCLUSION W studid th poblm of cntalizd pai of multipl asus in distibutd stoag systms. W xplicitly chaactizd th optimal functional tadoff btwn th pai

11 bandwidth and th stoag siz p nod. Fo instanc, w obtaind th xpssions of th xtm points on th tadoff, namly th minimum stoag multi-nod pai (SR) and th minimum bandwidth multi-nod pai (BR) points. In cas k, w showd that th tadoff ducs to a singl point, fo which w hav povidd a cod constuction achiving it. Futhmo, w povd that th functional BR point is not achivabl fo lina xact pai cods. Similaly, w hav shown that th functional pai tadoff is not achivabl und xact pai, xcpt fo mayb a small potion na th SR point. Opn poblms in this topic includ achivability of non-lina xact BR cods, ducing th subpacktization siz fo xact SR cods, and chaactization of intio points fo xact pai. A. poof of Lmma 5 VIII. APPENDICES W fist stat th following lmma which will b usful in th poof. Lmma 25. Th scnaio u = [,...,, ] achivs th lowst final valu of minimum cut: lim f(u) lim f([,...,, ]), u P, (106) α + α + wh f(u) and P a dfind in (3) and (4), spctivly. Poof: fo a spcific cut u, w hav lim f(u) α + g i 1 = (d u j )β = dβg β g i 1 g 1 u j = gdβ β u i (g i) g 1 g 1 = β(dg g u i + iu i ) = β((d k)g + g iu i ). (107) To obtain th smallst minimum cut valu, w nd to solv th following poblm g minimiz (d k)g + iu i u,g subjct to 1 u i, g u i = k. (108) It can b sn that th solution to (108) is givn by u = [,...,, ]. W now study th diffnt functions C j (α) fo j = 0,..., a. a) j=0: w hav C 0 (α) = min(α, dβ) + min(α, (d i)β) = min(α, dβ ) + min(α, (d i)β ). C 0 (α) is a picwis lina function with bakpoints givn by { d () β, d (a 2) β,..., d β, d β}. C 0 incass fom 0 at a slop of k. Its slop is thn ducd by by th succssiv bakpoints and thn finally by until it lvls off. b) 1 j a: fo ach j, w hav j 1 C j (α) = min(α, (d i)β) + min(α, (d j)β) + min(α, (d i)β) i=j j 1 = min(α, + min(α, (d i)β ) (d j)β ) + i=j min(α, (d i)β ). C j (α) is also picwis-lina function with nonincasing succssiv slops. Its bakpoints a givn by { d () β,..., d j β, d (j 1) β,..., d d j β} { β}. d j Th xact lativ position of th bakpoint β with spct to th oth bakpoints of C j (α) dpnds on th systm s paamts. Howv, w giv a low bound on d j β. d j d (j 1) = d d + 2 j( 2 ) ( )d + 2 a( 2 ) ( )k + 2 a( 2 ) = 0, wh th fist inquality follows by noticing that th xpssion is dcasing in j and ltting j = a, and th scond inquality follows as th cosponding xpssion is incasing d. Figu 1 illustats th lativ positions of all th bakpoints of C 0 (α) and C j (α), j 1, wh fo xampl d j [ d (j 1) C j ( ) = lim C j(α). α +, d (j 2) ]. W dnot by Lmma 26. Fo 1 j a, th xists a point α c (j) [ d, d ] such that C 0 (α c (j)) = C j (α c (j)), C 0 (α) C j (α) C 0 (α) C j (α) C j (α) = C j ( ) if α α c (j), if α α c (j), if α α c (j). Poof: W.l.o.g, assum β = 1. Fist, w not that (109) d (j 1) C 0 (α) = C j (α) = kα fo α. Nxt, w analyz th bhavio of ach of th functions C 0 (α) and C j (α) ov th succssiv intvals I i

12 > 1 α 0 d () d (a 2) d j d (j 1) d (j 2) d d d α 0 d () d (a 2) d j d j d (j 1) d (j 2) d d Fig. 1: lativ positions of th bakpoints (with β = 1) ( d i, d (i 1) x i = d i ] fo i {j 1, j 2,..., 1}. Lt and dfin s j (I i ) as th slop of C j (α) just bfo α = x i. Consid a givn intval I i = (x i, x i 1 ], w hav C 0 (α) has no bakpoint insid I i. Thus, C 0 (α) incass by C 0 (x i 1 ) C 0 (x i ) = s 0 (I i ). C j (α) has ith on o two bakpoints insid I i. 1) In cas C j (α) has a singl bakpoint insid I i (at α = d i ), C j (α) incass by C j (x i 1 ) C j (x i ) = s j (I i ) + (s j(i i ) ) = s j (I i ) +. 2) In cas C j (α) has two bakpoints insid I i, namly at α = d j and α = d i. Lt = d j d i (c.f. Figu 1). Assuming d j d i, thn, C j (α) incass by C j(x i 1) C j(x i) = (s j(i i) )(1 ) + (s j(i i) ) + s j(i i) = s j(i i) +. Assuming d j C j(x i 1) C j(x i) d i, thn, C j (α) incass by = sj(ii) + (k )( ) + (sj(ii) )(1 ) = s j(i i) +, which shows that th incas dos not dpnd on th lativ position of th two bakpoints. Now that w hav computd th incas incmnt of ach C j ov I i, w pocd to compa C 0 (α) and C j (α) fo 1 j a. W discuss two cass: Cas 1: Assum d j I j0 fo som j 0 [1, j 1]. j 0 may not xist, which will b discussd in th scond cas. Basd on th abov discussion, it can b sn that C j (α) C 0 (α), fo α x j0. This can sn by noticing that s 0 (I i ) = s j (I i ) and that (C j (x i 1 ) C j (x i )) (C 0 (x i 1 ) C 0 (x i )) = 0, i < j 0. Ov I j0, C j also dominats C 0 at vy point as s 0 (I j0 ) = s j (I j0 ) and (C j (x i 1 ) C j (x i )) (C 0 (x i 1 ) C 0 (x i )) = 0. Fo i > j 0, w hav s 0 (I i ) s j (I i ) =. oov, ov ach I i, i > j 0, w hav (C j (x i 1 ) C j (x i )) (C 0 (x i 1 ) C 0 (x i )) = (s j (I i ) + ) (s 0 (I i ) ) = 0. Combining th last quation and th obsvation that C j (x j0 1) C j (x j0 1), it follows that C j continu to dominat C 0 ov th succssiv intvals I i, i > j 0. So fa, w hav shown that C j (α) C 0 (α), fo α d. Fo α d, w obsv that C j incass with a slop of and lvls off at d whil C 0 incass at small slop givn by and lvls off at d > d. oov, w know fom Lmma 25 that C 0 lvls off at a high valu than that of C j. Thus, th xists α c (j) [ d, d ] that satisfis (109). Cas 2: Assum d < d j d, thn, using simila agumnts as in th fist cas, it follows that fo α d, C j ( d ) C 0( d d ). At α =, C j(α) has a slop of +, which is high than that of C 0, givn by. Thus, th slop of C j mains high than than of C 0 until C j lvls off. Combining ths obsvations with th fact that C 0 lvls off at a high valu, it follows that both cuvs will intsct only onc. oov, th intsction at a point at which C j has lvld off i.., w hav α c (j) max( d, d j ). Thfo, (109) holds also in this cas. Using Lmma 26 and th fact that C a achivs th smallst final valu fom Lmma 25, that is C a ( ) C j ( ), j [0, ], it follows that (8) holds fo any j [0, a]. oov, as α c (a) [ d, d ], α c(a) satisfis α c (a) + (d i)β = (a + 1)βd β a2 + a, 2

13 which implis that α c (a) + a(d a ) = (a + 1)βd β a2 + a. 2 Simplifying th last quation yilds (9). B. Stoag-bandwidth tadoff xpssion W stat with th cas k = a+. Th optimization tad-off is minimiz α α (110) subjct to C(α). Th constaint is a pic-wis lina function C(α) is givn by (a + 1)βd βa(a + 1)/2, α α c, α + b j, α [ b 0, α c], j=0 C(α) = ( + i)α + b j, α [ b i, b i 1 ], kα, j=i with α c = d+a a β, b i = (d i)β and b j = β(a i)(d j=i fo i = 1,..., a 1, α b, (111) (a 1 + i) ) 2 (a i)( d a i) = γ γg (i), 2d such that (a i)( d a i) g (i) =. 2d Th xpssion C(α) incass fom 0 to a maximum valu givn by β((a + 1)d ( ) a+1 2 ). To solv (110), w lt α = C 1 () und th condition β((a + 1)d ( ) a+1 2 ). Thfo, w obtain, α =, k b j j=i +i b j j=0 with b i + ibi + j=i b j [0, kb ], [( + i) b i, [ b 0 + j=i fo i = a 1,... 1, + j=0 b j, α c + b j, ( + i) b i 1 j=0 b j], + j=i (112) = a2 2 + a 2 2a + 2da 2 i 2 2 i 2i d) γ 2d = k2 2 + (k ) + 2kd 2 (i 2 + i) 2i γ 2d γ f(i), such that f (i) = 2d k (k ) + 2kd 2 (i 2 + i) 2i. b j] Thfo, fixing and vaying γ, w wit, kg()γ [0, ], α k γg = (i), [ γ +i f (i), γ f (i 1) ], fo i = a 1,... 1, γg (0), [ γ d+a a f (0), (g(0) + )γ]. d (113) As a function of γ, aft simplifications, w obtain th xpssion of α as in Thom 6. W not that th a a pic-wis lina potions on th cuv. oov, th minimum bandwidth point γ BR is givn by d γ BR = g (0) + d+a a = d d(a + 1) ( a+1). 2 Th xpssion of α BR is givn by α BR = γ BRg(0) d + a a = γ BR. d In cas k, w hav = 0. Th xpssion of th tadoff is obtaind fom (113) by stting = 0 and liminating th last lin. W not that in this cas, th a a 1 pic-wis lina potions on th tad-off cuv. C. Poof of Thom 24 Poof: Tak a subntwok of d+ nods. Lt L and b two goups of nods. Patition th d maining nods into two sts, A of cadinality p and B of cadinality d p. Exact pai quis It follows that H(W L S L A, S L B, S L ) = 0, (114) H(W S A, S B, S L ) = 0. (115) H(W L, W W A, S L B, S B, S L ) (116) = H(W L W A, S L B, S B, S L ) + H(W W L, W A, S L B, S B, S L ) (117) = 0. (118) Thfo, w hav H(SB, L SB, S L ) H(W L, W W A) (119) = H(W L W A) + H(W W AW L) (120) = H(W L) I(W L; W A) + H(W ) I(W ; W AW L) (121) = α 0 + α (β θ) (122) = 2α β + θ (123) = 2((d p)β θ) β + θ (124) = (2d 2p )β θ. (125) Th low bound dos not dpnd on whth d is a multipl of. Nxt, w obtain an an upp bound on th sam quantity cas: p + 2 < a: H(SB, L SB, S L ) H(SL L i, SL i ) + H(S L ) (126) L i B (2β θ) + β (127) L i B = (d p )(2β θ) + β (128) = (2d 2p )β (d p )θ, (129)

14 wh th fist inquality is obtaind using Lmma 21. Equations (125) and (129) a in contadiction if d p > d > (p + 2), which is tu as d k > p + 2 and θ 0. cas: p+2 = a: In this cas, Lmma 22 is usd to div an upp bound on H(SB L, S B, SL ). Lmma 22 dos not hold if a = 2. It holds fo a > 2 k > 2. Thus, w consid k > 2. W hav H(SB, L SB, S L ) H(SL L i, SL i ) + H(S L ) (130) L i B L i B (β + θ) + β (131) = (d p)β + (d p )θ. (132) Equations (125) and (129) a in contadiction whn θ < d p β. (133) d p REFERENCES [1] S. Ghmawat, H. Gobioff, and S.-T. Lung, Th googl fil systm, in AC SIGOPS opating systms viw, vol. 37, no. 5. AC, 2003, pp [2] A. G. Dimakis, P. Godfy, Y. Wu,. J. Wainwight, and K. Ramchandan, Ntwok coding fo distibutd stoag systms, IEEE Tans. Inf. Thoy, vol. 56, no. 9, pp , [3] N. B. Shah, K. Rashmi, P. V. Kuma, and K. Ramchandan, Intfnc alignmnt in gnating cods fo distibutd stoag: Ncssity and cod constuctions, IEEE Tans. Inf. Thoy, vol. 58, no. 4, pp , [4] C. Suh and K. Ramchandan, Exact-pai mds cod constuction using intfnc alignmnt, IEEE Tans. Inf. Thoy, vol. 57, no. 3, pp , [5] Y. Wu and A. G. Dimakis, Rducing pai taffic fo asu codingbasd stoag via intfnc alignmnt, in 2009 IEEE Intnational Symposium on Infomation Thoy. IEEE, 2009, pp [6] D. S. Papailiopoulos, A. G. Dimakis, and V. R. Cadamb, Rpai optimal asu cods though hadamad dsigns, IEEE Tans. Inf. Thoy, vol. 59, no. 5, pp , [7] Z. Wang, I. Tamo, and J. Buck, Explicit DS cods fo optimal pai bandwidth, axiv ppint axiv: , [8] A. S. Rawat, O. O. Koyluoglu, and S. Vishwanath, Pogss on highat SR cods: Enabling abitay numb of hlp nods, axiv ppint axiv: , [9] S. Gopaaju, A. Fazli, and A. Vady, inimum stoag gnating cods fo all paamts, axiv ppint axiv: , [10] V. R. Cadamb, C. Huang, S. A. Jafa, and J. Li, Optimal pai of DS cods in distibutd stoag via subspac intfnc alignmnt, axiv ppint axiv: , [11] I. Tamo, Z. Wang, and J. Buck, Zigzag cods: DS aay cods with optimal building, IEEE Tans. Inf. Thoy, vol. 59, no. 3, pp , ach [12] K. Rashmi, N. Shah, and P. Kuma, Optimal xact-gnating cods fo distibutd stoag at th SR and BR points via a poduct-matix constuction, IEEE Tans. Inf. Thoy, vol. 57, no. 8, pp , Aug [13] R. Bhagwan, K. Tati, Y. Chng, S. Savag, and G.. Volk, Total call: Systm suppot fo automatd availability managmnt. in NSDI, vol. 4, 2004, pp [14] A. S. Rawat, O. O. Koyluoglu, and S. Vishwanath, Cntalizd pai of multipl nod failus with applications to communication fficint sct shaing, axiv ppint axiv: , [15] P. Hu, C. W. Sung, and T. H. Chan, Boadcast pai fo wilss distibutd stoag systms, in th Intnational Confnc on Infomation, Communications and Signal Pocssing (ICICS). IEEE, 2015, pp [16] A.-. Kmac, N. L Scouanc, and G. Staub, Rpaiing multipl failus with coodinatd and adaptiv gnating cods, in Intnational Symposium on Ntwok Coding (NtCod), IEEE, 2011, pp [17] K. W. Shum and Y. Hu, Coopativ gnating cods, IEEE Tans. Inf. Thoy, vol. 59, no. 11, pp , Nov [18] J. Li and B. Li, Coopativ pai with minimum-stoag gnating cods fo distibutd stoag, in INFOCO, 2014 Pocdings IEEE. IEEE, 2014, pp [19] A. Wang and Z. Zhang, Exact coopativ gnating cods with minimum-pai-bandwidth fo distibutd stoag, in INFOCO, 2013 Pocdings IEEE. IEEE, 2013, pp [20] V. R. Cadamb, S. A. Jafa, H. alki, K. Ramchandan, and C. Suh, Asymptotic intfnc alignmnt fo optimal pai of DS cods in distibutd stoag, IEEE Tans. Inf. Thoy, vol. 59, no. 5, pp , [21] Z. Wang, I. Tamo, and J. Buck, Optimal building of multipl asus in DS cods, axiv ppint axiv: , [22]. Y and A. Bag, Explicit constuctions of high-at DS aay cods with optimal pai bandwidth, axiv ppint axiv: , [23] N. B. Shah, K. V. Rashmi, P. V. Kuma, and K. Ramchandan, Distibutd stoag cods with pai-by-tansf and nonachivability of intio points on th stoag-bandwidth tadoff, IEEE Tans. Inf. Thoy, vol. 58, no. 3, pp , ach [24] R. Ahlswd, N. Cai, S.-Y. Li, and R. W. Yung, Ntwok infomation flow, IEEE Tans. Inf. Thoy, vol. 46, no. 4, pp , [25] T. Ho,. édad, R. Kott, D. R. Kag,. Effos, J. Shi, and B. Long, A andom lina ntwok coding appoach to multicast, IEEE Tans. Inf. Thoy, vol. 52, no. 10, pp , [26] Y. Wu, A. G. Dimakis, and K. Ramchandan, Dtministic gnating cods fo distibutd stoag, in Allton Confnc on Contol, Computing, and Communication. Cits, 2007, pp [27] V. R. Cadamb, S. A. Jafa, H. alki, K. Ramchandan, and C. Suh, Asymptotic intfnc alignmnt fo optimal pai of DS cods in distibutd stoag, IEEE Tans. Inf. Thoy, vol. 59, no. 5, pp , ay [28]. Elyasi, S. ohaj, and R. Tandon, Lina xact pai at gion of (k + 1, k, k) distibutd stoag systms: A nw appoach, in IEEE Intnational Symposium on Infomation Thoy 2015(ISIT 15), Jun 2015, pp