A Repair Framework for Scalar MDS Codes

Transcription

1 axv:32235v [csit] 7 Dec 203 A Repa Famewo fo Scala MDS Codes Katheyan Shanmugam Student Membe, IEEE, Dmts S apalopoulos Student Membe, IEEE, Alexandos G Dmas Membe, IEEE, and Guseppe Cae Fellow, IEEE Depatment of Electcal and Compute Engneeng The Unvesty of Texas at Austn, Austn, TX-7872 {athsh,dmts}@utexasedu,dmas@austnutexasedu Depatment of Electcal Engneeng Unvesty of Southen Calfona, Los Angeles, CA cae@uscedu Abstact Seveal wos have developed vecto-lnea maxmum-dstance sepaable MDS) stoage codes that mnmze the total communcaton cost equed to epa a sngle coded symbol afte an easue, efeed to as epa bandwdth BW) Vecto codes allow communcatng fewe sub-symbols pe node, nstead of the ente content Ths allows non tval savngs n epa BW In shap contast, classc codes, le Reed- Solomon RS), used n cuent stoage systems, ae deemed to suffe fom nave epa, e downloadng the ente stoed message to epa one faled node Ths manly happens because they ae scala-lnea In ths wo, we pesent a smple famewo that teats scala codes as vecto-lnea In some cases, ths allows sgnfcant savngs n epa BW We show that vectozed scala codes exhbt popetes that smplfy the desgn of epa schemes Ou famewo can be seen as a fnte feld analogue of eal ntefeence algnment Usng ou smplfed famewo, we desgn a scheme that we call clque-epa whch povably dentfes the best lnea epa stategy fo any scala 2-paty MDS code, unde some condtons on the sub-feld chosen fo vectozaton We specfy optmal epa schemes fo specfc 5, 3)- and 6, 4)-Reed- Solomon RS) codes Futhe, we pesent a epa stategy fo the RS code cuently deployed n the Faceboo Analytcs Hadoop cluste that leads to 20% of epa BW savngs ove nave epa whch s the epa scheme cuently used fo ths code Index Tems Scala MDS Codes; Reed Solomon; clqueepa; algnment I INTRODUCTION Lage-scale dstbuted stoage systems employ easue codng to offe data elablty aganst hadwae falues Typcally, the easue codes employed ae n, ) MDS maxmum dstance sepaable) codes An mpotant popety that ensues data elablty aganst falues, s that encoded data fom any nodes suffce to ecove the data stoed Howeve, a cental ssue that ases n coded stoage s the Ths pape was pesented n pat at 50th Annual Alleton Confeence on Communcaton, Contol and Computng 202 [] Ths eseach was patally suppoted by NSF Awads , and eseach gfts by Google, Intel and Mcosoft Repa oblem: how to mantan the encoded epesentaton when a sngle node easue occus To mantan the same edundancy posteo to an easue, a new node has to on the stoage aay and egeneate the lost contents by downloadng and pocessng data fom the emanng stoage nodes Classc codes, le Reed-Solomon ae scala MDS codes Cuently used epa scheme fo these codes s nave epa Ths nvolves downloadng all the contents of any of the emanng nodes to econstuct the ente fle and then eplacng the coded sub-symbols of a sngle faled node Dung epa pocess of an easue, thee ae seveal metcs that can be optmzed, namely epa bandwdth BW) and localty [3] [4] [5] [6] [7] [8] Cuently, the most well undestood one s the total numbe of bts communcated n the netwo, e epa bandwdth BW) Ths was chaactezed n [2] as a functon of stoage pe node Codes wth mnmum stoage that offe optmal bandwdth ae MDS, and ae called mnmum stoage egeneatng MSR) codes Buldng on the wo n [2], a geat volume of studes have developed MSR codes [3] [2] In ths pape, we deal wth the followng specfc epa scenao fo systematc MSR codes: a fle consstng of M sub-symbols, ove some feld, s stoed n n nodes usng an n, ) vecto systematc MDS code Evey node contans α = M sub-symbols ove the feld The fst systematc nodes stoe uncoded sub-symbols n goups of M The paty nodes contan the coded data An MDS code can toleate n easues Suppose one of the systematc nodes fal and ths needs to eplaced Fo such a epa, β sub-symbols ae downloaded fom evey emanng paty node, though sutable lnea combnatons of the α symbols pesent n each paty node Fom evey emanng systematc node, at least β symbols ae downloaded The downloaded symbols must be suffcent to geneate the contents of the faled node though lnea opeatons fo successful epa Accodng to the cut-set lowe bounds of [2], the optmum pe-node download achevable by any code and epa scheme s β = M n ) = α n when exactly β symbols ae downloaded fom all the emanng

2 2 n nodes fo epa It s easly seen that the optmum epa stategy fo MSR codes has mmense beneft ove nave epa fo constant ate codes and fo lage n The ey popety of MSR codes that enables the non-tval epa s that they ae vecto codes, e data n a node s a collecton of smalle sub-symbols ove a feld and few lnea combnatons fom evey node suffce fo epa of a sngle falue MSR codes, wth effcent encodng and decodng schemes, that meet the mnmum cut-set BW bounds, deved n [2], exst fo ate /n /2 [7] In the hgh ate egme, [5], [6], [8], [9], [] have pesented constuctons that acheve the optmal epa bandwdth Howeve, the amount of subpacetzaton sub-symbols) equed s exponental n the paametes n and Code constuctons n [0] ectfy ths poblem by constuctng hgh ate codes, fo specfc ates, that have polynomal subpacetzaton Fo moe detals on egeneatng codes fo othe scenaos, we efe the eade to the suveys [9] [20] [3] An nteestng poblem s developng epa stateges fo exstng systematc scala lnea MDS codes that ae cuently used n easue coded stoage systems A mao lmtng ssue of these codes s that they lac the fundamental ngedent of epa optmal ones: the vecto-code popety Nave epa s cuently the only nown stategy fo these codes In ths wo, we focus on epang a faled systematc node of a systematc scala lnea MDS code, defned ove a lage extenson feld The focus s not on desgnng α n codes that acheve the cut-set bound of n ) fo epa bandwdth but on analyzng the epa effcency of exstng ones We show that any scala lnea MDS code, can be vectozed ove a sutable smalle sub-feld When a systematc scala lnea MDS code s vectozed, the poblem of desgnng the ght lnea combnatons of stoed symbols, fom a suvvng paty node, to be used fo epa also called as epa vecto desgn) can be equvalently seen as the poblem of desgnng epa feld elements Instead of desgnng a epa vecto fo each equaton downloaded, a feld element belongng to the extenson feld s chosen fo evey epa equaton of a vectozed scala code These feld elements satsfy some lnea ndependence constants Ths equvalent fomulaton s the man techncal contbuton of the pape Ths gves some analytcal nsghts fo epa of 2-paty codes when vectozed ove specfc sub-felds We summaze ou contbutons below Ou contbutons: In ths wo, we develop a famewo to epesent scala lnea MDS codes n a vecto fom, when they ae constucted ove extenson felds The vecto fom povdes moe flexblty n desgnng non-tval epa stateges We pose the poblem of desgnng epa vectos the best lnea combnatons to download) fo epang a systematc node, as a poblem of desgnng epa feld elements satsfyng some algebac lnea dependence popetes Usng ths famewo, we develop an algothm, called clque-epa, that outputs an optmal epa scheme fo a gven 2-paty scala lnea MDS code, vewed as a vecto code ove a sutably chosen sub-feld Ths s based on an analytcal condton, obtaned though the epa feld elements appoach, that dectly elates the code s geneato matx entes to the epa bandwdth We show that, fo a specfc 6, 4) Reed Solomon code, the clque epa scheme obtans nontval gans n tems of epa bandwdth Fo ths 6,4) RS code and anothe specfc 5,3) RS code, the gans can be bought close to the optmal cut-set bound of [2], by vectozng ove a smalle sub-feld Futhe, we pesent numecal esults egadng the epa of the 4, 0) RS code cuently used n poducton [8] by Faceboo Hadoop Analytcs cluste Thee, we obseve a 20% savngs n tems of epa BW compaed to nave epa II REAIR OF MDS STORAGE CODES In ths secton, we fst state the epa BW mnmzaton poblem fo systematc vecto MDS codes to clafy the mplcatons of stong vectos pe node nstead of scalas Thoughout the pape, we consde the case of downloadng sub-symbols fom all the emanng n nodes to epa a sngle faled node We see that scala-lnea MDS codes have an nheent defct when assumng ndvsble coded symbols A Vecto MDS Codes Let a fle x be subpacetzed nto M = n )β p-ay nfomaton symbols such that x F M and pattoned n pats x = [ x T T ] xt, wth x F M, whee M denotes the fle sze and F GFp) Hee, the numbe of sub-symbols ove GFp) stoed n a node s α = M = n )β Let us defne the degee of subpacetzaton to be β Z + We want to stoe ths fle wth ate n acoss systematc and n paty stoage unts wth stoage capacty M p-ay symbols each The encodng s gven by: y = y y n = I α I α +) n) n) +) x x ) whee ) F α α epesents a matx of codng coeffcents used by the th node + and hence a paty node) to mx the symbols of the th fle pece x y denotes the vecto of coded sub-symbols stoed n node I α denotes an α α dentty matx The MDS popety s guaanteed f the fle can be econstucted fom fom any subset of sze of the n nodes stong the codewod y

3 3 Rema: The choce of M beng a multple of n ) s due to the followng easons: he lowest pe node epa bandwdth n tems of sub-symbols ove GFp)) possble s β = α n subsymbols accodng to the cut-set bound n [2] 2) Futhe, fo epa of all systematc vecto codes occung n ths wo, we assume that the numbe of subsymbols that wll be downloaded fom evey suvvng paty node wll be β = α n = M n ) Howeve, note that t may not be possble to download onlyβ symbols fom each suvvng systematc ones unless optmal epa s feasble fo that code In fact, the goal of effcent epa wll be to download as close to β subsymbols as possble fom each suvvng systematc node B Repa Vecto Desgn oblem Let [] denote the set {,2,3} To mantan the same edundancy when a sngle systematc node [] fals, a epa pocess taes place to egeneate the lost data n a newcome stoage node Ths pocess s caed out as lnea opeatons on the content of the n emanng nodes, namely, each paty node { + n} sends data of sze β = M e, β equatons) to the newcome n the n ) fom of lnea equatons: d ) = R = [ R ) ) ) x + + R ) ) )x ) ] x, 2) whee R Fn )β β s a epa matx, whch s to be desgned In the same manne, all paty nodes poceed n tansmttng a total of M lnea equatons e, the sze of what was lost) to the newcome, whch eventually eceves the followng system of lnea equatons ) R + T +) ) R + T +) u d = x + xu, 3) R n n) u=,u R n n) u }{{}}{{} useful data ntefeence byx u whee d F M Solvng fo x s not possble due to the ) addtve ntefeence components n the eceved equatons To eteve the lost pece of data, we need to ease the ntefeence tems by downloadng addtonal equatons fom the emanng systematc nodes and the esultng system has to be full-an To ease the ntefeence geneated by the undesed symbols x u ), we need to download fom systematc node u the mnmum numbe of equatons that can e-geneate the ntefeence due to x u, e, we need to download data of sze equal to γ u = an R + +) u R n )T n) u 4) Desgnng R to acheve the followng: mn γ u subect to γ = M u,u s the epa vecto matx) desgn poblem γ = M means that the useful data matx must have full an The cut-set bound of [2] states that β equatons fom each of the emanng systematc nodes s the mnmum one could acheve, e, the mnmum an of each ntefeence space s β Ths esults n a mnmum download bound of n M n = n )β Obseve that the above benefts can only be unloced f we teat each stoed symbol as a bloc of smalle n )β sub-symbols C Scala MDS Codes When we consde scala n, )-MDS codes, we assume that nfomaton symbols x = [x x ] F p m) ae used to geneate n coded symbols y = [y y n ] F p m) n unde the lnea geneato map y = y y n = ) n) n) +) x x 5) 6) whee s the n ) matx that geneates the paty symbols of the code and F p m GFp m ) Smla to the pevous secton, let ) denote the paty coeffcent, dawn fom GFp m ), used by the th paty node to multply symbolx Instead of matces and vectos n the the pevous case, hee we have scalas dawn fom the extenson feld GFp m ) The MDS popety s equvalent to the equement that the nfomaton symbols can be econstucted fom any subset of sze When a node, o a coded symbol s lost, f we wsh to epa t usng lnea methods ove the extenson feld GFp m ), we can pefom nave epa Scala-lnea opeatons on ths code bnds us to ths wost case epa bandwdth cost Movng away fom scala-lnea methods, we could nstead download pats of each symbol defned ove GFp m ) Obseve that, ove GFp m ), each symbol conssts of m sub-symbols defned ove GFp) and GFp m ) s somophc to a vecto space of dmenson m ove GFp) In the followng secton, we descbe how an extenson feld can be used to allow decomposton of each coded symbol nto sub-symbols, such that a scala lnea MDS codes s ntepeted as a a vecto-lnea MDS code The ey deas used ae the followng: ) Each element of the geneato matx s vewed as a squae matx wth dmensons m m ove the feld GFp)

4 4 2) Evey data symbol x and evey coded symbol y ove GFp m ) ae vewed as vectos x and y, espectvely, of dmenson m ove the feld GFp) III VECTORIZING SCALAR CODES We evew some esults [2] [22] egadng epesentatons of fnte feld elements Let the educble pmtve polynomal x) of degee m ove the base feld GFp) that geneates GFp m ) be: x) = a 0 +a x+a m x m +x m, 7) whee a 0,,a m GFp) Let ζ be any oot of the polynomal x) Hence, ζ s a pmtve element Thee may be moe than one oot of the pmtve polynomal All pmtve elements ae somophc to each othe extenson felds obtaned by settng one of the oots to be the pmtve element s somophc to the one obtaned though othe ootshen, any feld element b GFp m ) can be wtten as a polynomal of ζ ove GFp) of degee at most m b = b 0 +b ζ +b m ζ m 8) whee b GFp), {0,m } Defnton : The companon matx of the pmtve polynomal x) = a 0 +a x+a m x m +x m s a m m matx gven by: C = a a a a a m ) Vecto Repesentaton: Any, b GFp m ) can be ntepeted as a vecto that belongs to a vecto space of dmenson m ove GFp) wth the followng vecto epesentaton fb) = [b 0 b b m ] T 9) 2) Matx Repesentaton: Any nonzeo feld element n GFp m ) can be wtten as ζ n, 0 n p m 2 The mappng gζ l ) = C l s an somophsm between GFp m ) and the set of m m matces {0,C 0,C,,C pm 2 } ove GFp) that peseves the feld multplcaton and addton n tems of matx multplcaton and addton ove the space of matces GFp)) m m We efe to gb) = B as the multplcaton opeato coespondng to b GFp m ) and to fb) = b as the vecto epesentaton of b GFp m ) Let MF p m) = {0,C 0,C C pm 2 } be the set of multplcaton opeatos Then clealy, Addtvty: Fo any c, d GFp) and A, B MF p m), we have ca+db MF) 2 Commutatvty: Fo any A,B MF p m), we have AB = BA MF p m) Lemma : [22] If c = ab whee c,a,b GFp m ), then c = Ab whee fc) = c, fb) = b and ga) = A A Vectozaton of the code n 6) he nfomaton symbols x and the coded symbols y can be ewtten as m-dmensonal vectos, x and y ove GFp)) m by settng x = fx ) and y = fy ) 2) Evey enty of the geneato matx, e ), can be epesented n tems of the multplcaton opeato ) MF p m) by settng ) = g ) ) 3) By Lemma, evey ) x s epesented by the matx-vecto multplcaton ) x Settng m = n )β, we obseve that we have changed the scala code n 6) nto the vecto code gven by ) The eason fo the choce of m has been gven n II-A The only dffeence between ths and a genec vecto code s that the matces ) ae multplcaton opeatos that have specfc stuctue Note that the same constucton has been ecently used n [23] Also, ths constucton can be taen to be the fnte feld analogue of the pocedue fo geneatng atonal dmensons out of a eal dmenson [24] that plays an mpotant ole n eal ntefeence algnment IV REAIR FIELD ELEMENTS At ths pont, one could consde the vectozed code obtaned to be a genec vecto lnea code and desgn epa matcesr to solve 5) by seachng ove all possble epa matces Any such desgn seems to depend on the stuctue of the multplcaton opeatos ) Howeve, we use the followng techncal lemma to llustate that desgnng epa matces o epa vectos) as n 5) can be cast as a poblem of desgnng epa feld elements, when t comes to epang a vectozed scala code Ths lets us bypass the need fo loong nto the stuctue of the multplcaton opeatos and the need fo checng all possble epa matces Ths s the man techncal dea behnd the pape Lemma 2: Fo any two nonzeo vectos a T,b T GFp) m, thee always exsts a multplcaton opeato o a matx) M MF) m m) such that b T M = a T oof: The poof s povded n the appendx Rema: We have epesented the multplcaton of c = ab ove the extenson feld as c = Ab, whee A s a multplcaton opeato and b s the vecto epesentaton But ths coesponds to ght multplcaton only Clealy, c T = b T A s not tue as the matx A n geneal s not symmetc Hence, we eque Lemma 2 to establsh

5 5 popetes when the matx s multpled by a vecto fom the left Consde the epa poblem fo the vectozed code as n Secton II-A and use Lemma 2 We download β equatons fom evey node snce we have vectozed ove the feld GFp) and m = n )β Wthout loss of genealty, let us consde the epa of node = As n Secton II-A, epa matces whch multply the n pates ae denoted R +,R n GFp)) βn ) β, doppng the subscpt snce we wll state eveythng fo epa of node Let l denote the -th column of R l Ths coesponds to the th equaton downloaded fom node l Now, due to Lemma 2, we can fx an abtay nonzeo as a efeence vecto Then l,, M l MF p m), l = M l 0) Hence, all the epa vectos can be eplaced by the epa feld elements M l coespondng to the opeatos Ml In tems of the epa feld elements, we have the followng mpotant theoem that gves an altenate chaactezaton of any epa scheme Theoem : Consde a epa scheme wth epa matces R l, + l n fo epang node fo any node n geneal) whee the an of the column of vectos coespondng to the uth data vecto as n 4) s ) R + T +) u γ u = an R n n) u Ths s possble f and only f one can fnd epa feld elements M l F pm fo evey equaton [,β] downloaded fom evey node l [ +,n], such that u: γ u = an p M + u +,M + β u +,M n u n), ),Mβ n u n) ) whee an p a,a m ) wth a GFp m ) s defned accodng to the followng estcted defnton of lnea ndependence: A set of feld elements A GFp m ) ae lnealy ndependent ove a sub-feld GFp) e, have an p equal to the cadnalty of ths set of elements), when one cannot fnd non zeo scalas v GFp) such that v A = 0 oof: The poof uses Lemma 3, and s elegated to the appendx We call the above fomulaton of the epa poblem as epa feld element desgn, as one needs to desgn one epa feld element e M l s) fo evey epa vecto l o equvalently fo evey downloaded equaton The epa bandwdth n bts) of any such scheme wth epa feld elements s popotonal to the sum of the ans, e = log 2 p) bts whee the code s subpacetzed ove GFp) Repa fo node Repa fo node 2 Repa fo node 3 g ω) = ω 3 + g 2 ω) = ω 2 + f ω) = ωg ω) f 2 ω) = g 2 ω)ω 2 +) g ω) = ω + g 2 ω) = ω f ω) = ω 2 +ω +)g ω) f 2 ω) = g 2 ω)ω 2 +) g ω) = ω + g 2 ω) = ω f ω) = ω 2 +ω +)g ω) f 2 ω) = g 2 ω)ω TABLE I REAIR FIELD ELEMENTS FOR THE REAIR OF SYSTEMATIC NODES FOR THE 5,3) RS CODE A Illustaton of epa of a 5,3) Reed-Solomon Code Consde a 5,3)-Reed Solomon code ove F = GF2 4 ) Let ω be the ffth oot of unty Usng the explct fomula fo the geneato matx of the systematc Reed Solomon code gven n [25], we obtan the followng stuctue fo the geneato matx G G = ω 4 ω 2 )ω 4 ω 3 ) ω ω 2 )ω ω 3 ) ω 4 ω)ω 4 ω 3 ) ω 2 ω)ω 2 ω 3 ) ω 4 ω)ω 4 ω 2 ) ω 3 ω)ω 3 ω 2 ) ω 5 ω 2 )ω 5 ω 3 ) ω ω 2 )ω ω 3 ) ω 5 ω)ω 5 ω 3 ) ω 2 ω)ω 2 ω 3 ) ω 5 ω)ω 5 ω 2 ) ω 3 ω)ω 3 ω 2 ) Fo the epa poblem, wthout loss of genealty, the geneato matx gven above can be smplfed by factong out some coeffcents along evey ow and enomalzng so that we can wo on the followng equvalent geneato matx G = 0 0 ω2 +ω ω 0 0 ω 2 + 2) Let ζ be the pmtve element of F coespondng to the pmtve polynomal x) = +x+x 4 Then, ζ 5 = and ω = ζ 3 Now, we consde the epa poblem unde the vecto epesentaton of the code ove GF2) Fo ths case, β = 2 Hence, β = 2 equatons ae downloaded fom the 2 paty nodes The cut-set bound fom the optmal epa bound) fo ths scenao s downloadng 8 equatons n total The polynomals n Table I coespond to epa by downloadng 0 equatons n the event of a falue of any systematc node fo the 5, 3) Reed Solomon code ove GF6) Now, we llustate ths usng the famewo of epa feld elements Each paty node stoes fou equatons ove the bnay feld Let us consde the epa of node f ω) denotes the epa feld element coespondng to the bt downloaded fom the fst paty node node 4) and g ω) denotes the epa feld element fo the second paty node node 5) Let us assume that the coespondng epa vectos ae 4, 4 2, 5 and 5 2 As an llustaton of the esults n ths secton, we show how the epa feld elements n Table I coespond to a epa

6 6 bandwdth of 0 equatons ove GF2) fo epa of node Fst, we show how the column of vectos coespondng to data node n 3) s full an, f we use the epa vectos obtaned though the epa feld elements n Table I We need to vefy the followng: γ = an 2 f ω),f 2 ω),g ω)ω 2 +ω +), g 2 ω)ω 2 +ω +) ) = 4 By expessng eveythng as a polynomal n ζ of degee at most 3 wth coeffcents fom GF2) usng the educble polynomal, we have f ω) f 2 ω) g ω)ω 2 +ω +) g 2 ω)ω 2 +ω +) = ζ 3 ζ 2 ζ 3) We see that they ae lnealy ndependent ove GF2) full an) Hence, γ = 4 Now, we set ) 4 T = f W), ) 4 T 2 = f 2 W), ) 5 T = g W) and ) 5 T 2 = g 2 W) whee W s the multplcaton opeato fo ω By applyng Theoem, the column of vectos coespondng to data fom node as n 3) gven by: ) ) 4 T T 4) 4 2 4) 5 4) 5 2 4) = T f W) T f 2 W) T g W)W 2 +W+I) T g 2 W)W 2 +W+I) s full an ove GF2), wth the above assgnment of epa vectos because of 3) Hee, s any abtay non-zeo efeence vecto Now, the an of ntefeence tems follows fom smla obsevatons egadng epa feld elements: ωg ω) = f ω) mples γ 2 = 3 mang column fo data vecto 3 to have an 3) and ω 2 +)g 2 ω) = f 2 ω) mples γ 3 = 3 Theefoe, γ = 0 equatons ove GF2) needs to be downloaded fo epa fo node Repa bandwdth fo epa of nodes 2 and 3 can be vefed smlaly Now, We ague that 0 bts s the optmal lnea epa bandwdth achevable fo ths code We consde the case whee node 2 fals and we wll assume that 8 epa equatons ae suffcent To ecove the lost data, accodng to Eq ), we eque an 2 [ f ζ) f 2 ζ) g ζ)ζ 3 g 2 ζ)ζ 3] = 4 4) If 8 equatons ae suffcent then thee must exst polynomals such that the followng condtons ae tue: an [ f ζ) f 2 ζ) g ζ)ζ 6 +ζ 3 +) g 2 ζ)ζ 6 +ζ 3 +)) ] = 2, an [ f ζ) f 2 ζ) g ζ)ζ 6 +) g 2 ζ)ζ 6 +)) ] = 2 5) Then the only possblty s that g ζ)ζ 6 + ζ 3 + ) = v f ζ+v 2 f 2 ζ) andg 2 ζ)ζ 6 +ζ 3 +) = v 3 f ζ)+v 4 f 2 ζ) Smlaly, g ζ)ζ 6 +) = v 5 f ζ)+v 6 f 2 ζ) and g 2 ζ)ζ 6 + ) = v 7 f ζ)+v 8 f 2 ζ) Hee, all v GF2) Theefoe, g ζ)ζ 3 ) = g 2 ζ)ζ 3 ) = v +v 5 )f ζ)+v 2 +v 6 )f 2 ζ), v 3 +v 7 )f ζ)+v 4 +v 8 )f 2 ζ) Ths volates the full an condton of 4) Smla aguments hold fo epa of systematc nodes and 3 Futhe, vey smla aguments can be made to show that 9 equatons ae not enough fo epa of the nodes The aguments ae lengthy but follow a smla style to the one above The cucal popety that s used n these convese esults s the followng popety: In the ffth ow [+ζ 6 +ζ 3,ζ 3,ζ 6 +] of the geneato matx, two coeffcents add up to gve the thd coeffcent B Dffeent degees of subpacetzaton Note that we made no assumpton about GFp) So ths could be an extenson feld by tself So, fo a gven extenson feld GFp n ) ), whee p s pme, one could do the vectozaton ove GFp ), so that the effectve degee of subpacetsaton s β = The lowest degee s and the hghest possble degee s and any ntemedate degee would be any s that dvdes The followng ntutve esult shows that any epa scheme fo a lowe degee of subpacetzaton β can be mplemented usng an equvalent epa scheme wth a hghe degee of subpacetzaton β and wth the same benefts wth espect to the epa bandwdth Lemma 3: Consde a scala systematc n,) MDS code ove a feld GFp an ) ), vectozed ove GFp a ), wth β = Let the assocated epa feld elements be M +,M +2 M n that opeate on the paty nodes numbeng fom + to n Consde the an of the column of vectos [ n ) coespondng to node gven by: = an p a M + + M ]) M n n Defne new epa feld [ elements, fo the code subpacetzed ove GFp) to be: M M 2 M a] = [ M, M ζ,m ζ a ] coespondng to a equatons beng dawn fom node, whee + n Hee, ζ GFp a ) s the pmtve element The system of new epa elements { M } have the same epa bandwdth as {M } oof: The poof s elegated to the appendx V CLIQUE REAIR In ths secton, we use the epa feld elements famewo to pove the followng theoem that gves an optmal epa scheme when β = fo any n,n 2) scala MDS code We call ths scheme Clque Repa Fom now on, wthout

7 7 loss of genealty, we assume that +) =, e all paty coeffcents fo paty node + ae Ths can be ustfed as t does not affect the MDS epa poblem Theoem 2: Consde a systematc n,n 2)-MDS code ove GF p 2) and an undected gaph GV,E) such that ) V = and,) E ff +2) +2) GFp ) Then, wth lnea epa schemes, node cannot be epaed wth BW less than M C M 2 when vectozed ovegfp ), whee C s the sze of the lagest clque of G not contanng node oof: The poof s elegated to the appendx Rema: Thee s an altenatve way to see the above theoem GFp )\{0} s a multplcatve subgoup of GFp 2 )\{0} Consde the set of cosets fomed by the subgoupgfp )\{0} Consde the epa of node Among all cosets that do no contan the feld element +2, pc the coset that contans the lagest numbe of elements fom { +2 } Let the numbe of elements fom { +2 } whch le n ths coset be C Then, the epa bandwdth s no less than M C M 2 n tems of GFp ) symbols Although the theoem above only specfes a lowe bound, one can come up wth an algothm to acheve the optmum pefomance It s easy to chec that the followng algothm wos The algothm Geneate Clque dentfes the Algothm Geneate Clque whle = do whle = do f +2) E,) end f end whle end whle +2) ) GFp ) then dsont clques o cosets) Let us assume that thee s a lst {C[]} m such that C[] contans all vetces contaned n clque o coset The algothm Fnd Repa fnds the optmal epa feld element µ fo epang node Notce Algothm 2 Fnd Repa Fnd N : C[N] max agmax C[] N c some node l C max µ +2) l ) that the algothm uns usng On 2 ) feld multplcaton opeatons The scheme gves an analytcal connecton between the epa BW and the coeffcents of the geneato matx see ema afte Theoem 2) Consde the vecto epesentaton of the 5, 3) RS code n Secton IV-A ove GF2 2 ) Then by applyng Theoem 2, we fnd that all the thee nodes le n the same clque In othe wods, ω 2 + ω + ) ω, ωω 2 + ) belong to GF2 2 ) Hence, fo ths code, clque epa does not gve any gan n tems of epa bandwdth Now, we pesent examples of bandwdth savngs that ae possble fo a 6,4) Reed Solomon code and fo the 4,0) Reed Solomon code employed n HDFS open souce module As we wll see, clque epa gves nontval bandwdth savngs ove nave epa fo the 6,4) Reed Solomon code consdeed below Ths can be mpoved futhe by gong to a hghe degee of subpacetzaton VI ANALYSIS OF REAIR OF 6,4) REED SOLOMON CODES Hee, we consde a 6,4)-RS code ove GF2 4 ) Let ζ be the pmtve element of F coespondng to the pmtve polynomal x) = +x+x 4 Usng the fomula n [25], we obtan the followng systematc geneato matx G = ζ 3 +ζ 2 +ζ ζ ζ 2 6) We consde the vecto epesentaton of the code ove GF2 2 ) β = ) If we apply Theoem 2 to ths code, thee ae 3 clques that ae fomed The fst clque o coset) contans nodes and 4 whle the second one contans 2 and the thd one contans node 3 By the clque epa algothm pesented n Secton V, the epa of nodes 2 and 3 eque 6 epa equatons ove GF4) to be downloaded Fo the epa of nodes and 4, 7 equatons ove GF4) need to be downloaded whch s close to the fle sze M = 8, whle the cut-set bound s n n M = 5 equatons Now, consde a hghe degee of subpacetzaton, e each node stoes 4 elements ove GF2) Now, M = 6 elements We get a good epa scheme by Lemma 3) fo nodes 2 and 3 ove GF2) that eques 2 equatons by convetng the clque epa scheme fo these nodes ove GF2 2 ) Hence, the epa bandwdth fo 2 and 3 s 6 2 = 2 equatons fo epa ove GF2) The cut set bound s 5 2 = 0 equatons ove GF2) Fo ths case, the epa scheme wth epa feld elements gven n Table II mpoves the epa BW fo nodes and 4 to 2 equatons compaed to the clque epa equvalent that eques7 2 = 4 equatons{f } epesent epa feld elements fo the fst paty node and {g } epesent epa feld elements fo the second paty node It s possble to show that 2 equatons s the optmal lnea epa bandwdth fo ths code The agument s lengthy, but smla n style to the one n Secton IV-A fo the 5,3) Reed-Solomon code and hence we sp t

8 8 Repa fo node Repa fo node 4 g ζ) = ζ 2 g 2 ζ) = f ζ) = f 2 ζ) = ζ 2 g ζ) = g 2 ζ) = ζ f ζ) = f 2 ζ) = ζ TABLE II REAIR FIELD ELEMENTS FOR REAIR OF NODES AND 4 FOR THE SYSTEMATIC 6,4) RS CODE WITH SUBACKETIZATION OVER GF2) VII NUMERICAL RESULTS ON THE 4, 0) REED-SOLOMON CODE IMLEMENTED IN THE HADOO FILE SYSTEM The Apache Hadoop Dstbuted Fle System HDFS) eles by default on bloc eplcaton fo data elablty A module called HDFS RAID [8], [26]) was ecently developed fo HDFS that allows the deployment of Reed- Solomon and also moe sophstcated dstbuted stoage codes HDFS RAID s cuently used n poducton clustes ncludng Faceboo analytcs clustes stong moe than 30 B of data In ths secton, we pesent numecal esults on mpovng the epa pefomance of the specfc 4, 0) Reed-Solomon code mplemented n HDFS-RAID [26] HDFS RAID mplements a systematc Reed Solomon code ove the extenson feld GF2 8 ) Let ζ be the oot of the pmtve polynomal +x 2 +x 3 +x 4 +x 8 that geneates the extenson feld The geneato matx used s: [ ] I0 G = whee s a 4 0 matx gven by: ζ 6 ζ 78 ζ 249 ζ 75 ζ 8 ζ 59 ζ 89 ζ 63 ζ 69 ζ 62 ζ 98 ζ 3 ζ 37 ζ 253 ζ 49 ζ 43 T = ζ 49 ζ 77 ζ 96 ζ 205 ζ 2 ζ 7 ζ 57 ζ 34 ζ 40 ζ 236 ζ 54 ζ 43 ζ 49 ζ 23 ζ 2 ζ 88 ζ 94 ζ 7 ζ 38 ζ 95 ζ 0 ζ 3 ζ 48 ζ 73 Snce the numbe of pates s 4 n = 4), the clque epa technque s not applcable We consde epa wth the hghest possble subpacetzaton, e β = 2 and each node stoesn )β = 8 elements ovegf2) andm = 80 The epa eques downloadng 2 equatons fom evey paty node We povde a epa scheme n tems of the eght epa feld elements M,M 2,M4,M4 2, belongng to GF2 8 ), as n Theoem The epa scheme, gven n Table III, lsts the epa feld elements fo epa of each node and the total numbe of equatons to be downloaded fo epa n each case The aveage numbe of equatons to be downloaded s 642 equatons The nave epa nvolves downloadng80 equatons and the lowe bound n M n gves 26 equatons We note that the epa scheme that we povde s not the optmal fo the code because an exhaustve seach nvolves checng a huge numbe of combnatons about2 64 combnatons) of the epa feld elements We have seached ove about andom combnatons of the epa feld elements to poduce ths epa scheme that saves about 20 pecent bandwdth ove nave epa Rema: In [8], a new mplementaton of a locally epaable code based on the 4,0) code s used to optmze epa It saves 50 pecent bandwdth ove nave epa but ncus a cost of 4 pecent n addtonal stoage ovehead We have demonstated that the 4,0) code used as s wthout any stoage ovehead can gve non-tval savngs VIII CONCLUSION We ntoduced a famewo fo epang scala codes by teatng them as vectos ove a smalle feld Ths s acheved by teatng multplcaton of scala feld elements n the ognal feld as a matx-vecto multplcaton opeaton ove the smalle feld Intefeence algnment condtons map to desgnng epa feld elements n the lage feld Futhe usng the condtons on desgnng epa feld elements, we ntoduced the clque epa scheme fo two pates when the degee of subpacetzaton s, whch establshes a connecton between the coeffcents of the geneato matx and the epa schemes possble We exhbted good epa schemes fo a few Reed-Solomon codes ncludng the one cuently deployed n Faceboo Ths wo hnts at the exstence of scala MDS codes wth good epa popetes An nteestng poblem would be to come up wth easly testable analytcal condtons, smla n spt to the clque epa scheme, fo codes wth lage numbe of pates and fo hghe degees of subpacetzaton Suffcent condtons fo a specfc class of codes le Reed Solomon would be also nteestng Moe geneally, t seems that scala MDS codes wth nea optmal epa could be desgned usng ths famewo IX ACKNOWLEDGEMENT We than the anonymous evewes fo the helpful comments and suggestons that helped us mpove the pape mmensely REFERENCES [] K Shanmugam, D S apalopoulos, A G Dmas, and G Cae, A epa famewo fo scala MDS codes, n 50th Annual Alleton Confeence on Communcaton,Contol and Computng Alleton), 202 IEEE, 202, pp [2] A G Dmas, B Godfey, Y Wu, M J Wanwght, and K Ramchandan, Netwo codng fo dstbuted stoage systems, IEEE Tansactons on Infomaton Theoy, vol 56, no 9, pp , 200

9 9 TABLE III REAIR SCHEME FOR4, 0) REED SOLOMON CODE THAT SAVES 20 ERCENT IN REAIR BANDWIDTH THE NAIVE REAIR INVOLVES DOWNLOADING 80 BITS FOR REAIR Systematc node epaed Repa feld elements [ M M2 M4 M2 4 ] [ Repa Bandwdth bts downloaded) ζ [ 69 ζ 203 ζ 89 ζ 64 ζ 70 ζ 73 ζ 64 ζ 74] 65 2 [ ζ 8 ζ 9 ζ 75 ζ 248 ζ 8 ζ ζ 69 ζ 26] 64 3 [ ζ 53 ζ 5 ζ 0 ζ 3 ζ 223 ζ 79 ζ 4 ζ 4] 64 4 [ ζ 92 ζ 86 ζ 3 ζ 29 ζ 67 ζ 23 ζ 67 ζ 44] 64 5 [ ζ 46 ζ 23 ζ 86 ζ 5 ζ 28 ζ 69 ζ 69 ζ 46] 63 6 ζ [ 83 ζ 64 ζ 82 ζ 6 ζ 04 ζ 85 ζ 245 ζ 78] 64 7 [ ζ 57 ζ 48 ζ 4 ζ ζ 95 ζ 60 ζ 22 ζ 32] 64 8 [ ζ 5 ζ 74 ζ 206 ζ 224 ζ 04 ζ 00 ζ 52 ζ 43] 65 9 [ ζ 84 ζ 250 ζ 43 ζ 76 ζ 2 ζ 225 ζ 207 ζ 05] 65 0 ζ 6 ζ 80 ζ 3 ζ 89 ζ 69 ζ 37 ζ 5 ζ 77] 64 [3] A G Dmas, K Ramchandan, Y Wu, and C Suh, A suvey on netwo codes fo dstbuted stoage, oceedngs of the IEEE, vol 99, no 3, pp , 20 [4] Dstbuted stoage w [5] I Tamo, Z Wang, and J Buc, MDS aay codes wth optmal ebuldng, n IEEE Intenatonal Symposum on Infomaton Theoy oceedngs ISIT), 20 IEEE, 20, pp [6] V R Cadambe, C Huang, S A Jafa, and J L, Optmal epa of MDS codes n dstbuted stoage va subspace ntefeence algnment, axv pepnt axv:06250, 20 [7] K Rashm, N B Shah, and Kuma, Optmal exact-egeneatng codes fo dstbuted stoage at the MSR and MBR ponts va a poduct-matx constucton, IEEE Tansactons on Infomaton Theoy,, vol 57, no 8, pp , 20 [8] C Suh and K Ramchandan, Exact-epa MDS code constucton usng ntefeence algnment, IEEE Tansactons on Infomaton Theoy, vol 57, no 3, pp , 20 [9] K Rashm, N B Shah, V Kuma, and K Ramchandan, Explct constucton of optmal exact egeneatng codes fo dstbuted stoage, n 47th Annual Alleton Confeence oncommuncaton, Contol, and Computng, 2009 Alleton IEEE, 2009, pp [0] V R Cadambe, C Huang, J L, and S Mehota, olynomal length MDS codes wth optmal epa n dstbuted stoage, n Confeence Recod of the Foty Ffth Asloma Confeence on Sgnals, Systems and Computes ASILOMAR), 20 IEEE, 20, pp [] D S apalopoulos, A G Dmas, and V R Cadambe, Repa optmal easue codes though hadamad desgns, n 49th Annual Alleton Confeence on Communcaton, Contol, and Computng Alleton), 20 IEEE, 20, pp [2] K Rashm, N B Shah, and K Ramchandan, A pggybacng desgn famewo fo ead-and download-effcent dstbuted stoage codes, axv pepnt axv: , 203 [3] D S apalopoulos and A G Dmas, Locally epaable codes, n IEEE Intenatonal Symposum on Infomaton Theoy oceedngs ISIT), 202 IEEE, 202, pp [4] G M Kamath, N aash, V Laltha, and V Kuma, Codes wth local egeneaton, axv pepnt axv:2932, 202 [5] Gopalan, C Huang, H Smtc, and S Yehann, On the localty of codewod symbols, 20 [6] A S Rawat, O O Koyluoglu, N Slbesten, and S Vshwanath, Optmal locally epaable and secue codes fo dstbuted stoage systems, axv pepnt axv:206954, 202 [7] C Huang, H Smtc, Y Xu, A Ogus, B Calde, Gopalan, J L, S Yehann et al, Easue codng n wndows azue stoage, n USENIX confeence on Annual Techncal Confeence, USENIX ATC, 202 [8] M Sathamoothy, M Astes, D apalopoulos, A G Dmas, R Vadal, S Chen, and D Bothau, Xong elephants: Novel easue codes fo bg data, axv pepnt axv:30379, 203 [9] F Ogge and A Datta, Codng technques fo epaablty n netwoed dstbuted stoage systems, 202 [20] A Datta and F Ogge, An ovevew of codes talo-made fo netwoed dstbuted data stoage, axv pepnt axv:09237, 20 [2] R Ldl and H Nedeete, Fnte felds Cambdge Unvesty ess, 996, vol 20 [22] F MacWllams and N Sloan, The Theoy of eo-coectng codes Noth-Holland, 2006 [23] S-N Hong and G Cae, Stuctued lattce codes fo MIMO ntefeence channel, axv pepnt axv:306453, 203 [24] A S Motaha, S O Ghaan, and A K Khandan, Real ntefeence algnment wth eal numbes, axv pepnt axv: , 2009 [25] D J Vesfeld, J N Rdley, H C Feea, and A S Helbeg, On systematc geneato matces fo Reed Solomon codes, IEEE Tansactons on Infomaton Theoy, vol 56, no 6, pp , 200 [26] HDFS-W: A oof of Lemma 2 AENDIX oof: We note that multplcaton of the matx M fom the left by a T does not epesent feld multplcaton Hence, wth espect to left multplcaton, the matxm does not necessaly act as a multplcaton opeato The theoem mples that gven any two abtay non zeo epa vectos, one can fnd a multplcaton matx that connects both Snce, non zeo feld elements n F ae fnte, thee ae fntely many opeatos n MF) Let them be denoted by M,M 2,M pm All these matces o opeatos) have full an We consde the poducts, a T M We show that all of them ae dstnct Suppose fo some,a T M = a T M, then a T M M ) = 0 7) But M M s anothe multplcaton opeato by addtvty popety It s non zeo and has full an snce M M Ths means all the p m poducts ae dffeent Snce thee ae only p m non zeo epa vectos b T, gven any b T, one can always fnd a M such that a T M = b T B oof of Theoem oof: Consde a epa scheme wth epa vectos l, l [ +,n], [,β] Tang an abtay efeence vecto, evey othe epa vecto can

10 0 be wtten n the fom gven by 0) usng Lemma 2 Consde the epa feld elements M l that coespond to the multplcaton opeatos M l obtaned fom 0) All multplcaton matces ae full an matces If thee exsts scalas v l GFp), l {l,l 2 l q }, {, 2 q }, wth at least one nonzeo v l, such that: ) l T l) u = 0 8) l, v l Then ths s equvalent to v l M l l) u = 0 9) l, Usng the fact that the efeence vecto s non zeo and opety : v l Ml l) u = 0 20) l, Ths gves the an condton ove sub-feld GFp) as stated n ) Ths poves the fowad decton Fo the convese, gven a set of epa feld elements t s possble to constuct a set of epa multplcaton opeatos and togethe wth an abtay choce of a non-zeo efeence epa vecto, one can constuct epa vectos satsfyng the same an condtons C oof of Lemma 3 oof: It s enough to show that b feld elements of the fom {M }, b ae lnealy dependent ove GFp a ) f and only f ab feld elements {M ζ s }, b, 0 s a ae also lnealy dependent ove GFp) Hee, M coespond to the epa feld elements and coespond to the coeffcents of the geneato matx coespondng to the paty node Lnea dependence ove GFp a ) mples that thee exsts scalas v GFp a ), wth at least one of them non-zeo, such that v M = 0 Let us ewte feld elements v n GFp a ) as polynomals n ζ wth coeffcents v s fom GFp) Hence, the lnea dependency elaton becomes v s M ζ s = 0 Hence s ove GFp), {M ζ s }, b, 0 s a ae lnealy dependent The convese s also tue snce some scala set ζ s GFp) wth one non zeo element detemnes a scala set v GFp a ) wth one non zeo element Hence, the clam follows GFp ) The tanstvty popety pattons the gaph G nto dsont clques Usng Theoem we have that thee ae two epa feld elements, e, µ GFp 2 ) coespondng to the two epa vectos that wll be used to multply the contents of the two pates espectvely The epa feld elements fo paty s because the coespondng epa vecto acts as the efeence vecto Then, the an of -th bloc s 2 f and µ +2) ae lnealy ndependent ove subfeld GFp ) Smlaly, the an would be f they ae lnealy dependent, eµ +2) GFp ) Now we establsh the followng popety: f and ae n the same clque, then ethe both columns of elements ae smultaneously lnealy dependent o lnealy ndependent ove GFp ) Ths s due to the fact that µ +2 µ +2 ) GFp ) GFp ) foces Smlaly, f and ae n dffeent clques, then the coespondng columns of vectos cannot be lnealy dependent ) smultaneously Suppose ) they ae, then µ GFp ) and µ GFp ) Theefoe, +2) µ +2) = +2) +2) ) µ ) GFp ) But,) / E and theefoe a contadcton Fo epa of node, µ s chosen n such a way that µ +2) / GFp ), so that the coespondng column of vectos ae lnealy ndependent Ths selecton of µ foces all blocs coespondng to the nodes n the same clque to be lnealy ndependent and t can at most mae columns coespondng to exactly one othe clque lnealy dependent Hence, the educton n numbe of equatons to be downloaded comes fom the dependent clque Fom ths, the last clam n the theoem follows +2) +2) D oof of Theoem 2 ) oof: If x +2) y +2) GFp ) ) and y +2) z +2) GFp ), then y +2) ) +2 ) z =x z +2 x ) +2 y