MDS Array Codes with Optimal Rebuilding

Transcription

1 0 IEEE Intenational Symposium on Infomation Theoy Poceedings MDS Aay Codes with Optimal Reuilding Itzhak Tamo, Zhiying Wang and Jehoshua Buck Electical Engineeing Depatment, Califonia Institute of Technology, Pasadena, CA 95, USA Electical and Compute Engineeing, Ben-Guion Univesity of the Negev, Bee Sheva 8405, Isael {tamo, zhiying, Astact MDS aay codes ae widely used in stoage systems to potect data against easues. We addess the euilding atio polem, namely, in the case of easues, what is the the faction of the emaining infomation that needs to e accessed in ode to euild exactly the lost infomation? It is clea that when the nume of easues equals the maximum nume of easues that an MDS code can coect then the euilding atio is (access all the emaining infomation). Howeve, the inteesting (and moe pactical) case is when the nume of easues is smalle than the easue coecting capaility of the code. Fo example, conside an MDS code that can coect two easues: What is the smallest amount of infomation that one needs to access in ode to coect a single easue? Pevious wok showed that the euilding atio is ounded etween and 4, howeve, the exact value was left as an open polem. In this pape, we solve this open polem and pove that fo the case of a single easue with a -easue coecting code, the euilding atio is. In geneal, we constuct a new family of -easue coecting MDS aay codes that has optimal euilding atio of in the case of a single easue. Ou aay codes have efficient encoding and decoding algoithms (fo the case = they use a finite field of size ) and an optimal update popety. I. INTRODUCTION Easue-coecting codes ae the asis of the uiquitous RAID schemes fo stoage systems, whee disks coespond to symols in the code. Specifically, RAID schemes ae ased on MDS (maximum distance sepaale) aay codes that enale optimal stoage and efficient encoding and decoding algoithms. With edundancy symols, an MDS code is ale to econstuct the oiginal infomation if no moe than symols ae eased. An aay code is a two dimensional aay, whee each column coesponds to a symol in the code and is stoed in a disk in the RAID scheme. We ae going to efe to a disk/symol as a node o a column intechangealy, and an enty in the aay as an element. Examples of MDS aay codes ae EVENODD [], [], B-code [], X-code [4], RDP [5], and STAR-code [6]. Suppose that some nodes ae eased in an systematic MDS aay code, we will euild them y accessing (eading) some infomation in the suviving nodes, all of which ae assumed to e accessile. The faction of the accessed infomation in the suviving nodes is called the euilding atio. If nodes ae eased, then the euilding atio is since we need to ead all the emaining infomation. Howeve, is it possile to lowe this atio fo less than easues? Fo example, Figue shows the euilding of the fist systematic (infomation) node fo an MDS code with 4 infomation elements and edundancy This wok was suppoted in pat y the NSF gant ECCS and y an NSF-NRI awad. nodes, which equies the tansmission of elements. Thus the euilding atio is /. In a stoage system thee is a diffeence etween easues of systematic and paity nodes. Easue of the fome will affect the infomation access time since pat of the aw infomation is missing, howeve easue of the latte does not have such effects, since the entie aw infomation is accessile. Moeove in most stoage systems the nume of paity nodes is negligile compaed to the systematic ones. Theefoe ou constuction focuses on optimally euilding the systematic nodes. Souce File a c d Encode to (4,) MDS aay code a c z c a d a z a d c d z c Figue. Reuilding of a (4, ) MDS aay code ove F. Assume the fist node (column) is eased. In [7], [8], a elated polem is discussed: the nodes ae assumed to e distiuted in a netwok, and the epai andwidth is defined as the minimum amount of data needed to tansmit in the netwok in ode to etain the MDS popety. Note that one lock of data tansmitted can e a function of seveal locks of data. In addition, etaining MDS popety does not imply euilding the oiginal eased node, wheeas we estict ou polem to exact euilding. Theefoe, the epai andwidth is a lowe ound of the euilding atio. An (n, k) MDS code has n nodes in each codewod and contains k nodes of infomation and = n k nodes of edundancy. A lowe ound fo the epai andwidth was shown as [7] M k n n k, () whee M is the total amount of infomation, and all the suviving nodes ae assumed to e accessile. It can e veified that Figue matches this lowe ound. A nume of woks addessed the epai andwidth polem [7] [6], and it was shown y intefeence alignment in [], [] that this ound is asymptotically achievale fo exact epai. Instead of tying to constuct MDS codes that can e easily euilt, a diffeent appoach [7], [8] was used y tying to find ways to euild existing families of MDS aay codes. The atio of euilding a single systematic node was shown to e 4 + o() fo //$ IEEE 40

2 EVENODD o RDP codes. Howeve, fom the lowe ound of () the atio is /. The main contiution of this pape is the fist explicit constuction of systematic (n, k) MDS aay codes fo any constant = n k, which achieves optimal euilding atio of. We call them intesecting zigzag sets codes (IZS codes). The code has low encoding and decoding complexity (fo =, the code uses finite field of size ). In addition, the code has optimal update: when an infomation element is updated, only + elements in the aay need update. The est of pape is oganized as follows: Section II gives definitions and notations, Section III constucts the MDS aay codes with optimal euilding atio, and genealizations of the codes ae given in Section IV. Due to space limitation, a lot of details ae omitted, and the eade is efeed to [9]. II. DEFINITIONS AND PROBLEM SETTINGS In the est of the pape, we ae going to use [i, j] to denote {i, i +,..., j} fo integes i j. And denote the complement of a suset X M as X = M\X. Let A = (a i,j ) e an aay of size p k ove a finite field F, each enty of which is an infomation element. We add to the aay two paity columns and otain an (n = k +, k) MDS code of aay size p n. Each element in these paity columns is a linea comination of elements fom A. Moe specifically, let the two paity columns e ( 0,,..., p ) t and (z 0, z..., z p ) t. Then fo l [0, p ], l = a Rl α a a and z l = a Zl β a a, fo some susets R l, Z l of elements in A, and some coefficients {α a }, {β a } F. We will assume that each infomation element appeas exactly once in each paity column, which implies optimal update fo the code. Then it can e shown that the sets R l s (o Z l s), l [0, p ], ae patitions of A into p equally sized sets of size k. Moeove, y the MDS popety, any set R l (o Z l ) does not contain two infomation elements fom the same column. Theefoe, fo the j-th systematic column (a 0,..., a p ) t, its p elements ae contained in p distinct sets R l, l [0, p ]. In othe wods, the memeship of the j- th column s elements in the sets {R l } defines a pemutation g j : [0, p ] [0, p ], such that g j (i) = l iff a i R l. Similaly, we can define a pemutation f j coesponding to the second paity column, whee f j (i) = l iff a i Z l. Fo example, Figue shows a (5, ) code. Each element in the paity column Z is a linea comination of elements with the same symol. Fo instance the in column Z is a linea comination of all the elements in columns 0,, and. And each systematic column coesponds to a pemutation of the fou symols. Oseving that thee is no impotance of the elements odeing in each column, w.l.o.g. we can assume that the fist paity column contains the sum of each ow of A and g j s = k j=0 α i,ja i,j. coespond to identity pemutations, i.e. i We efe to the fist and second paity columns as the ow column and the zigzag column espectively, likewise R l and Z l, l [0, p ], ae efeed to as ow sets and zigzag sets espectively. By assuming that the fist paity column 0 R Z 0 Figue. Pemutations fo zigzag sets in a (5, ) code with 4 ows. Columns 0,, and ae systematic nodes and columns R, and Z ae paity nodes. Each element in column R is a linea comination of the systematic elements in the same ow. Each element in column Z is a linea comination of the systematic elements with the same symol. The shaded elements ae accessed to euild column. contains the ow sums, the code is uniquely defined y (i) the pemutations { f j } of zigzags, and (ii) the coefficients in the linea cominations. Due to the following theoem, thee exist coefficients such that the code is MDS, and thus we will focus on finding pope pemutations { f j } fist. The poof is omitted. Theoem Let A = (a i,j ) e an aay of size p k and the zigzag sets e Z = {Z 0,..., Z p }, then thee exists a (k+, k) MDS aay code fo A with Z as its zigzag sets ove the field F of size geate than p(k ) +. The idea ehind choosing the zigzag sets is as follows: assume a systematic column (a 0, a,..., a p ) t is eased. Each element a i is contained in exactly one ow set and one zigzag set. Fo euilding of element a i, access the paity of its ow set o zigzag set. Moeove access the values of the emaining elements in that set, except a i. We say that an element a i is euilt y a ow (zigzag) if the paity of its ow set (zigzag set) is accessed. Fo example, in Figue supposing column is eased, one can access the shaded elements and euild its fist two elements y ows, and the est y zigzags. It can e veified that all the thee systematic columns can e euilt y accessing half the emaining elements. Fo a fixed nume of ows p define the euilding atio fo an (k +, k) MDS code C as R(C) = k i=0 accesses to euild node i, pk(k + ) which denotes the aveage faction of accesses in the suviving aay fo euilding one systematic node. Define the atio function fo all (k +, k) MDS codes with p ows as R(k) = min R(C). C By () fo any p and k, R(k). Fo example, the code in Figue achieves the lowe ound /, and theefoe R() = /. Fom this example, we can see that in ode to get low atio, we want the union of the ow sets and zigzag sets used to euild a column to e as small as possile. In othe wods, fo an eased column i, if elements in ows X i [0, p ] ae euilt y ows and X i = p (p even), then k i=0 j =i f i (X i ) f j (X i ) = k i=0 j =i ( p + f i(x i ) f j (X i ) ) should e small, whee f(x) = { f(x) : x X}. () 4

3 III. (k +, k) MDS ARRAY CODE CONSTRUCTIONS The pevious section gave us a lowe ound fo the atio function. The question is can we achieve it? If so, how? We know that each (k+, k) MDS aay code with ow and zigzag columns is defined y a set of pemutations f 0,..., f k and thei susets X i s. The following constuction constucts a family of such MDS aay codes. Fom any set T F m, T = k, we constuct an (k +, k) MDS aay code of size m (k + ). The atio of the constucted code will e popotional to the sum in (), and thus we will ty to constuct such pemutations and susets that minimize (). We will show that some of these codes have the optimal atio of. In this section all the calculations ae done ove F. By ause of notation we use x [0, m ] oth to epesent the intege and its inay epesentation. It will e clea fom the context which meaning is in use. Constuction Let A = (a i,j ) e an aay of size m k fo some integes k, m and k m. Let T F m e a suset of vectos of size k. Fo v T we define the pemutation f v : [0, m ] [0, m ] y f v (x) = x + v, whee x is epesented in its inay epesentation. One can check that this is actually a pemutation. Fo example when m =, v = (, 0) f (,0) () = (, ) + (, 0) = (0, ) =, and the coesponding pemutation of v is [,, 0, ]. In addition, we define X v as the set of integes x in [0, m ] such that the inne poduct etween thei inay epesentation and v satisfies x v = 0, e.g., X (,0) = {0, }. The constuction of the two paity columns is as follows: The fist paity column is simply the ow sums. The zigzag sets Z 0,..., Z m ae defined y the pemutations { f vj : v j T} as a i,j Z l if f vj (i) = l. We will denote the pemutation f vj as f j. Assume column j needs to e euilt, and denote S = {a i,j : i X j } and S z = {a i,j : i / X j }. Reuild the elements in S y ows and the elements in S z y zigzags. Recall that y Theoem this code can e an MDS code ove a field lage enough. The following theoem gives the atio fo Constuction and the poof is omitted. Theoem The code descied in Constuction and geneated y the vectos v 0, v,..., v k is a (k +, k) MDS aay code with atio R = + k i=0 j =i f i (X i ) f j (X i ) m. () k(k + ) The following lemma will help us to calculate the sum in (), ut fist we associate to any vecto v = (v,..., v m ) F m the suset of integes B v [m] whee i B v if v i =. Lemma fo any v, u T { Xv, B v \B u mod = 0 f v (X v ) f u (X v ) = 0, B v \B u mod =. Poof: Conside the goup (F m, +). Recall that f v(x) = X + v = {x + v : x X}. The sets f v (X v ) = X v + v and (4) f u (X v ) = X v + u ae cosets of the sugoup X v = {w F m : w v = 0}, and they ae eithe identical o disjoint. Moeove, they ae identical iff v u X v, namely (v u) v = i:vi =,u i =0 = 0. Howeve, B v \B u mod = i:vi =,u i =0, and the esult follows. This constuction enales us to constuct an MDS aay code fom any suset of vectos in F m. Howeve, it is not clea which suset of vectos should e chosen. The following is an example of a code constuction fo a specific set of vectos. Example Let T = {v F m : v = } e the set of vectos with weight and length m. Notice that T = ( m ). Constuct the code C y T accoding to Constuction. Given v T, {u T : B v \B u = } = ( m ), which is the nume of vectos with s in diffeent positions as v. Similaly, {u T : B v \B u = } = ( m ) and {u T : B v \B u = } = (m ). By Theoem and Lemma, fo lage m the atio is + m ( m )(m m ( m )((m ) + ) + 9 m. ) Note that this code eaches the lowe ound of the atio as m tends to infinity. In the following we will constuct codes that each the lowe ound exactly. Let { f 0,..., f k } e a set of pemutations ove the set [0, m ] with associated susets X 0,..., X k [0, m ], whee each X i = m. We say that this set is a set of othogonal pemutations if fo any i, j [0, k ] f i (X i ) f j (X i ) m = δ i,j, whee δ i,j is the Konecke delta. Let {e i } i= m e the standad vecto asis of F m and set e 0 to e the zeo vecto. The following theoem constucts a set of othogonal pemutations of size m +. Theoem 4 The pemutations f 0,..., f m and sets X 0,..., X m constucted y the vectos {e i } i=0 m and Constuction whee X 0 is modified to e X 0 = {x F m : x (,,..., ) = 0} is a set of othogonal pemutations. Moeove the (m +, m + ) MDS aay code of aay size m (m + ) defined y these pemutations has optimal atio of. Hence, R(m + ) =. The poof is omitted. Note that the optimal code can e shotened y emoving some systematic columns and still etain an optimal atio, i.e., fo any k m + we get R(k) =. Actually this set of othogonal pemutations in Theoem 4 is optimal in size, as the following theoem suggests. Theoem 5 Let F e an othogonal set of pemutations ove the integes [0, m ], then the size of F is at most m +. Poof: We will pove it y induction on m. Fo m = 0 thee is nothing to pove. Let F = { f 0, f,..., f k } e a set of othogonal pemutations ove the set [0, m ]. We only need to show that F = k m +. It is tivial to see that fo any g, h S m the set hfg = 4

4 f f 0 f 0 Othogonal set of pemutations 0 (a) Systematic nodes C0 C C C C4 a 0,0 a a 0 a0,0 a0, a 0, 0, 0, z0 a0,0 a, a, Encoding y the othogonal 0 a,0 a a a,0 a, a, z a,0 a, a pemutations,, 0, a,0 a, a, a,0 a, a, a,0 a, a, Paity nodes a,0 a, a, () z z a,0 a0, a, a,0 a, a, Figue. (a) The set of othogonal pemutations as in Theoem 4 with sets X 0 = {0, }, X = {0, }, X = {0, }. () A (5, ) MDS aay code geneated y the othogonal pemutations. The fist paity column C is the ow sum and the second paity column C 4 is geneated y the zigzags. Fo example, zigzag z 0 contains the elements a i,j that satisfy f j (i) = 0. {h f 0 g, h f g,..., h f k g} is also a set of othogonal pemutations with sets g (X 0 ), g (X ),..., g (X k ). Thus w.l.o.g. we can assume that f 0 is the identity pemutation and X 0 = [0, m ]. Fom the othogonality we get that k i= f i(x 0 ) = X 0 = [ m, m ]. Note that fo any i = 0, X i X 0 = X 0 = m. Assume the contay, thus w.l.o.g we can assume that X i X 0 > m, othewise X i X 0 > m. Fo any j = i = 0 we get that f j (X i X 0 ), f i (X i X 0 ) X 0, (5) f j (X i X 0 ) = f i (X i X 0 ) > m = X 0. (6) Fom equations (5) and (6) we conclude that f j (X i X 0 ) f i (X i X 0 ) =, which contadicts the othogonality popety. Define the set of pemutations F = { fi }k i= ove the set of integes [0, m ] y fi (x) = f i(x) m, which is a set of othogonal pemutations with sets {X i X 0 } k i=. By induction k m and the esult follows. The aove theoem implies that the nume of ows has to e exponential in the nume of columns in any systematic code with optimal atio and optimal update. Notice that the code in Theoem 4 achieves the maximum possile nume of columns, m +. An exponential nume of ows is pactical in stoage systems, since they ae composed of dozens of nodes (disks) each of which has size in an ode of gigaytes. In addition, inceasing the nume of columns can e done using duplication (Theoem 7) o a lage set of vectos (Example ) with a cost of a small incease in the atio. Example Let A e an aay of size 4. We constuct a (5, ) MDS aay code fo A as in Theoem 4 that accesses of the emaining infomation in the aay to euild any systematic node (see Figue ). Fo example, X = {0, }, and fo euilding of node (column C ) we access the elements a 0,0, a 0,, a,0, a,, and the following fou paity elements 0 = a 0,0 + a 0, + a 0, = a,0 + a, + a, z f () = z 0 = a 0,0 + a, + a, z f () = z = a,0 + a, + a 0,. It is tivial to euild node fom the accessed infomation. Note that each of the suviving node accesses exactly of its elements. It can e easily veified that the othe systematic nodes can e euilt the same way. Reuilding a paity node is easily done y accessing all the infomation elements. The following theoem shows that only a small field is needed to make the constuction in Theoem 4 an MDS code. Theoem 6 The field F suffices fo the code C constucted in Theoem 4 to e an (m +, m + ) MDS code. Poof: Let the infomation element of C in ow i, column j e a i,j, and let its zigzag coefficient e β i,j. Set u i = i j=0 e i. Assign all ow coefficients to e and assign zigzag coefficient β i,j = if i u j = and β i,j = othewise. In an easue of two systematic columns i, j [0, m], i < j, we access the entie emaining infomation in the aay. Set = + e i + e j, and ecall that a,i Z l if l = + e i, thus a,i, a,j Z +ei and a,j, a,i Z +ej. Fom the two paity columns we need to solve the following equations (fo some y, y, y, y 4 F ) β,i 0 0 β,j 0 β,j β,i 0 This set of equations is solvale if a,i a,j a,i a,j = y y y y 4. β,i β,i = β,j β,j. (7) Note that β,i = β,i and β,j = β,j and each of the β s is eithe o, thus (7) is satisfied ove F and the esult follows. It is clea that F does not suffice fo an MDS code, theefoe this is the optimal field size. The coefficients in Figue ae assigned y the poof of Theoem 6. IV. GENERALIZATIONS In this section, fist we ae going to incease the nume of columns in the constucted (k +, k) MDS codes, such that the atio is appoximately. Second we will constuct codes with > paity nodes that has optimal atio. Let C e a (k +, k) aay code whee the zigzag sets {Z l } p l=0 ae defined y the set of pemutations { f i} k i=0 S p and p is the nume of ows in the aay. Fo an intege s, an s-duplication code C is an (sk +, sk) MDS code with zigzag pemutations defined y duplicating the k pemutations s times each. Moeove, the fist paity column is the ow sums. The coefficients in the paities may e diffeent fom the code C. Suppose in the euilding algoithm of C, fo column i, elements of ows J = {j, j,..., j t } ae euilt y zigzags, and the est y ows. In C, all the s columns coesponding to f i ae euilt in the same way: the elements in ows J ae euilt y zigzags. Theefoe we otain the following theoem. Theoem 7 If a (k +, k) code C has atio R(C), then its s- duplication code C has atio at most R(C)( + sk+ s ). Hence the s-duplication of the code in Theoem 4 has atio at most fo lage s. + (m+) 4

5 Using duplication we can have aitaily lage nume of columns, independent of the nume of ows. Moeove the aove theoem shows that it also has an almost optimal atio. A constuction in the full vesion of the pape [9] uses a finite field of size at least s +, fo an MDS s-duplication of the code in Theoem 4. Fo example, in an s-duplication code fo m = 0, the aay is of size 04 (s + ). Fo s = and s = 6, the atio is 0.5 and 0.57 y Theoem 7, the code length is 4 and 68, and the field size needed is 4 and 8, espectively. Both of these two codes ae suitale fo pactical applications. Next we genealize Constuction into aitay nume of paity nodes. Let n k = e the nume of paity nodes, we will constuct an (n, k) MDS aay code with m ows fo some m > 0, i.e., it can euild up to easues. We assume that each systematic node stoes /k of the infomation and is stoed in column i, fo some i [0, k ]. The l-th paity node is stoed in column k + l, whee 0 l. Simila to the case of =, each paity column coesponds to k pemutations. Constuction Let T e a suset of vectos of Z m. We will constuct fo each intege l, 0 l, a set of pemutations F l T = { f l v : v T}, whee each pemutation acts on the set [0, m ]. By ause of notation we use x [0, m ] oth to epesent the intege and its -ay epesentation, and all the calculations ae done ove Z. Define f l v(x) = x + lv. Fo example, fo m =, =, x = 4, l =, v = (0, ), f(0,) (4) = 4 + (0, ) = (, ) + (0, ) = (, 0) =, and the pemutation f l v is [, 0,, 5,, 4, 8, 6, 7]. Fo x [0, m ], we define the zigzag set Z x in paity node l as the elements a i,j such that thei coodinates satisfy f l v j (i) = x. In euilding of systematic node i the elements in ows X l i = {v [0, m ] : v v i = l} ae euilt y paity node l, l [0, ]. Note that simila to Theoem, these paity nodes descied fom an (n, k) MDS aay code unde appopiate selection of coefficients in the linea cominations of the zigzags. The next theoem is a genealization of Theoem 4. Theoem 8 Let { fi l : i [0, m], l [0, ]} e the set of pemutations defined y the set T = {e 0, e,..., e m } togethe with the coesponding sets {Xi l}, and modify Xl 0 to Xl 0 = {x : x (,,..., ) = l}, fo 0 l. Then the (m + +, m + ) code defined y these pemutations has atio /, which matches the lowe ound in (). Supisingly, one can infe fom Theoem 8 that changing the nume of paities fom to adds only one node to the system, ut educes the atio fom / to / in the euilding of any systematic column. Simila to the paity case, the aove theoem achieves the optimal nume of columns. In othe wods, the nume of ows has to e exponential in the nume of columns in any systematic code with optimal atio, optimal update, and paities. A natual genealization of the euilding polem is the euilding of multiple easues. In ode to simultaneously euild e easues in a stoage system with paities (e ), one needs to access at least e/ of the suviving infomation. 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