Optimal Fractional Repetition Codes and Fractional Repetition Batch Codes

0.79 MB·English

by  Natalia Silberstein

Optimal Fractional Repetition Codes
and Fractional Repetition Batch Codes
Natalia Silberstein and Tuvi Etzion
Computer Science Department
Technion-Israel Institute of Technology
Haifa 32000, Israel
Email: {natalys, etzion}@cs.technion.ac.il
Abstract—Fractional repetition (FR) codes is a family of codes repetition code based on a complete graph. El Rouayheb and
5 for distributed storage systems (DSS) that allow uncoded exact Ramchandran [15] generalized the construction of [13] and
repairs with minimum repair bandwidth. In this work, we
1 defined a new family of codes for DSS which allow exact
consider a bound on the maximum amount of data that can
0 repairsbytransferforawiderangeofparameters.Thesecodes,
be stored using an FR code. Optimal FR codes which attain
2
this bound are presented. The constructions of these FR codes called DRESS (Distributed Replication based Exact Simple

n are based on families of regular graphs, such as Tura´n graphs Storage) codes [11], consist of the concatenation of an outer
a and graphs with large girth; and on combinatorial designs, such MDS code and the inner repetition code called fractional
J as transversal designs and generalized polygons. In addition,
repetition (FR) code. However, in contrast to MBR codes,
1 based on a connection between FR codes and batch codes, we
where a random set of size d of available nodes is used for a
2 proposeanewfamilyofcodesforDSS,calledfractionalrepetition
batchcodes,whichallowuncodedefficientexactrepairsandload noderepair,therepairswithDRESScodesaretablebased.This

] balancing which can be performed by several users in parallel. usually allows to store more data compared to MBR codes.
T
Constructions of FR codes based on some regular graphs
I
. I. INTRODUCTION and combinatorial designs can be found for example in [7],
s
c In distributed storage systems, data is stored across a [9], [10], [15]. However, the optimality of the constructed FR
[ codesregardingtheFRcapacity,i.e.themaximalityofthesize
network of nodes, which can unexpectedly fail. To provide
of the stored file, was not considered.
1 reliability, data redundancy based on coding techniques is
v introduced in such systems. Moreover, existing erasure codes Inthiswork,weaddresstheproblemofconstructingoptimal
7 FR codes and hence, optimal DRESS codes. Moreover, based
allow to minimize the storage overhead. In [4] Dimakis et al.
7 on a connection between FR codes and combinatorial batch
introduced a new family of erasure codes, called regenerating
1
codes, we propose a new family of codes for DSS, called
5 codes, which allow efficient single node repairs. In particular,
fractionalrepetitionbatch(FRB)codes,whichenableuncoded
0 they presented two families of regenerating codes, called
. minimum storage regenerating (MSR) codes and minimum repairs and load balancing that can be performed by several
1
users in parallel.
0 bandwidthregenerating(MBR)codes,whichcorrespondtothe
The rest of the paper is organized as follows. In Section II
5 two extreme points on the storage-bandwidth trade-off [4]. An
1 (n,k,d,α,β) regenerating code C, where k ≤ d ≤ n−1, wedefineDRESScodesandFRcodesbasedonregulargraphs
q
: and combinatorial designs. In Section III we present optimal
v β ≤ α, is used to store a file in n nodes; each node stores α
i symbolsfromF ,thefinitefieldwithq elements,suchthatthe FR codes based on Tura´n graphs and on graphs with large
X q
girth. In Section IV we consider optimal FR codes based on
storedfilecanberecoveredbydownloadingthedatafromany
r transversal designs and on generalized polygons. In Section V
a set of k nodes. When a single node fails, a newcomer node
we define FRB codes and present some examples for their
which substitutes the failed node contacts with a random set
constructions. Conclusion is given in Section VI. We point
of d other nodes and downloads β symbols of each node in
out that,throughout thispaper, proofsare oftenomitted dueto
this set to reconstruct the failed data. This process is called a
spacelimitations.Detailsofalltheproofscanbefoundin[16].
node repair, and the amount of data downloaded to repair a
failed node, βd, is called the repair bandwidth. II. PRELIMINARIES
In [13], [14] Rashmi et al. presented a construction for
An (n,α,ρ) FR code C is a collection of n subsets
MBRcodeswhichhavetheadditionalpropertyofexactrepair
def
N ,...,N of [θ]={1,2,...,θ}, nα=ρθ, such that
by transfer, or exact uncoded repair. In other words, the 1 n
(n,k,d = n − 1,M = kα − (cid:0)k(cid:1),α = n − 1,β = 1) • |Ni|=α for each i, 1≤i≤n;
2
code proposed in [13], [14] allows efficient exact node repairs • each symbol of [θ] belongs to exactly ρ subsets in C,
where no decoding is needed. Every node participating in a where ρ is called the repetition degree of C.
node repair process just passes one symbol which will be A [(θ,M),k,(n,α,ρ)] DRESS code is a code obtained by
directly stored in the newcomer node. This construction is the concatenation of an outer (θ,M) MDS code and an inner
based on a concatenation of an outer MDS code with an inner (n,α,ρ) FR code C. To store a file f∈FM in a DSS, f is first
q
𝑛,𝛼,𝜌 FR code FR code k-optimal if a file stored by using this code is the
maximumpossibleforthegivenk.WecallanFRcodeoptimal
Node 1 𝑐𝑖1,𝑐𝑖2,…,𝑐𝑖𝛼 if for any k ≤α it is k-optimal.
Let C be an (n,α,ρ) FR code. C can be described by an
file 𝐟∈ 𝔽𝑞𝑀(𝑘) 𝜃,𝑀(𝑘) 𝑀𝐷𝑆 𝑐1,𝑐2,…,𝑐𝜃 Node 2 𝑐𝑗1,𝑐𝑗2,…,𝑐𝑗𝛼 winhcoidseencroewmsatirnidxeIx(eCd),bwyhtihcheinsoadnens×anθdbicnoalruymmnastriinxd,θex=ednρbαy,
the symbols of the corresponding MDS codeword, such that
(I(C)) = 1 if and only if node i contains symbol j. Note
i,j
that every row of I(C) has α ones and every column of I(C)
Node n 𝑐𝑠1,𝑐𝑠2,…,𝑐𝑠𝛼 has ρ ones.
LetG=(V,E)beanα-regulargraphwithn=|V|vertices.
We say that an (n,α,ρ = 2) FR code C is based on G if
I(C)=I(G), where I(G) is the |V|×|E| incidence matrix of
G. Such a code will be denoted by C .
G
Fig. 1: The encoding scheme for a DRESS code
Let D = (P,B) be a design with |P| = n points such that
each block B ∈ B contains ρ points and each point p ∈ P
encoded by using the MDS code; next, the θ symbols of the is contained in α blocks. We say that an (n,α,ρ) FR code C
codeword c from the MDS code, which encodes the file f, are is based on D if I(C) = I(D), where I(D) is the |P|×|B|
f
placed in the n nodes defined by C, as follows: node i ∈ [n] incidence matrix of D. Such a code will be denoted by CD.
of the DSS stores α symbols of c, indexed by the elements
f III. OPTIMALFRCODESWITHREPETITIONDEGREEρ=2
of the subset N . The encoding scheme for a DRESS code is
i
shown in Fig. 1. InthissectionweconsideroptimalFRcodeswithrepetition
Each symbol of c is stored in exactly ρ nodes. It should be degree 2. First, we present the following useful lemma which
f
possible to reconstruct the stored file f of size M from any set shows a connection between the problem of finding the max-
of k nodes, and hence, imum file size of an FR code based on a graph and the edge
isoperimetric problem on graphs [2].
M ≤ min |∪ N |. (1)
i∈I i
|I|=k Lemma 1. Let G = (V,E) be an α-regular graph and let
C be the FR code based on G. We denote by G the family
Since we want to maximize the size of a file that can be G k
of induced subgraphs of G with k vertices. Then the file size
stored by using a DRESS code, in the sequel we will always
M(k) of C is given by
assume that M = min | ∪ N |. Note, that the same G
|I|=k i∈I i
FR code can be used in different DRESS codes, with different M(k)=kα− max |E(cid:48)|.
k’s as reconstruction degrees, and different MDS codes. The G(cid:48)=(V(cid:48),E(cid:48))∈Gk
file size M, which is the dimension of the chosen MDS code, Proof. For each induced subgraph G(cid:48) = (V(cid:48),E(cid:48)) ∈ G we
k
depends on the value of k and hence in the sequel we will defineE(cid:48) tobethesetofalltheedgesofE inthecutbetween
cut
use M(k) to denote the size of the file. An (n,α,ρ) FR V(cid:48) and V \V(cid:48), i.e.,
code is called universally good [15] if for any k ≤ α the
[(θ,M(k)),k,(n,α,ρ)] DRESS code satisfies Ec(cid:48)ut ={{v,u}∈E :v ∈V(cid:48),u∈V \V(cid:48)}.
(cid:18)k(cid:19) Clearly, kα = 2|E(cid:48)|+|Ec(cid:48)ut| for every G(cid:48) ∈ Gk. Note that
M(k)≥kα− 2 , (2) M(k)=minG(cid:48)∈Gk{|E(cid:48)|+|Ec(cid:48)ut|} and hence
where the righthand side of equation (2) is the maximum file M(k)= min {|E(cid:48)|+αk−2|E(cid:48)|}=αk− max {|E(cid:48)|}.
size (called MBR capacity) that can be stored using an MBR G(cid:48)∈Gk G(cid:48)∈Gk
code [4]. Note also that if an FR code C is universally good
then |Ni∩Nj| ≤ 1, for Ni,Nj ∈ C, i (cid:54)= j ∈ [n] [13]. In the The following lemma directly follows from Lemma 1.
sequel, we will consider only universally good FR codes.
Lemma 2. Let G be an α-regular graph with n vertices, and
An upper bound on the maximum file size M(k) of a
let M(k) be the file size of the corresponding code C . The
[(θ,M(k)),k,(n,α,ρ)] DRESS code (nα = ρθ), called the G
graph G contains a k-clique if and only if M(k)=kα−(cid:0)k(cid:1).
FR capacity and denoted in the sequel by A(n,k,α,ρ), was 2
presented in [15]: Corollary 3. The file size M(k) of an FR code C , where G
G
is a graph which does not contain a k-clique, is strictly larger
A(n,k,α,ρ)≤ϕ(k), where ϕ(1)=α, (3)
than the MBR capacity.
(cid:24) (cid:25)
ρϕ(k)−kα
ϕ(k+1)=ϕ(k)+α− . OneofthemainadvantagesofanFRcodeisthatitsfilesize
n−k
usuallyexceedstheMBRcapacity.Hence,asaconsequenceof
Note that for any given k, the function A(n,k,α,ρ) is deter- Corollary 3, we consider regular graphs which do not contain
mined by the parameters of the inner FR code. We call an a k-clique for a given k. In particular, we consider a family
By Lemma 1, to obtain a large value for M(k), every
6,3,2 FR code
induced subgraph with k vertices should be as sparse as
𝑐 𝑐 𝑐
A 1 2 3
possible.Hence,fortherestofthissectionweconsidergraphs
A 1 D
2 𝑐 𝑐 𝑐 where the induced subgraphs with k vertices, 1≤k ≤α, will
3 B 4 5 6
𝒇∈𝔽𝑴(𝒌) 4 becycle-free.Thesearegraphswithgirthatleastk+1,where
𝟗,𝑴(𝒌) 𝐌𝐃𝐒 B 5 E C 𝑐7 𝑐8 𝑐9 the girth of a graph is the length of its shortest cycle.
𝒄 ,𝒄 ,…,𝒄 6
𝑘 M 𝑘 𝟏 𝟐 𝟗 C 7 98 F D 𝑐1 𝑐4 𝑐7 LleetmMm(ka)6b.eLtheet Gfilebseizaenofαt-hreegcuolrarresgproanpdhinwgitFhRncovdeertiCceGs.aTnhde
1 3 E 𝑐2 𝑐5 𝑐8 girthofGisatleastk+1ifandonlyifM(k)=kα−(k−1).
2 5
3 7 F 𝑐3 𝑐6 𝑐9 Corollary7. Foreachk ≤g−1,anFRcodeCG basedonan
α-regular graph G with girth g attains the bound in (3), and
hence it is k-optimal. C also attains the bound of Lemma 5.
G
Corollary 8. An FR code C based on an α-regular graph G
G
Fig.2:The((9,M(k)),k,(6,3,2))DRESScodewiththeinner with girth g ≥α+1 is optimal.
FR code based on the complete bipartite graph K
3,3
The proof of the following theorem follows from Lemma 6
and the fact that any two cycles in a graph with girth g have
of regular graphs, called Tura´n graphs, which do not contain at most (cid:98)g/2(cid:99)+1 common vertices.
a clique of a given size and also have the smallest number of
Theorem 9. If G is a graph with girth g, then the file size
vertices [6]. Let r,n be two integers such that r divides n. An
M(k) of an FR code C based on G satisfies
(n,r)-Tura´n graph is defined as a regular complete r-partite G
(cid:26)
graph,i.e.,agraphformedbypartitioningthesetofnvertices kα−k+1 if k ≤g−1
M(k)=
into r parts of size n and connecting each two vertices of kα−k if g ≤k ≤g+(cid:100)g(cid:101)−2.
r 2
different parts by an edge. Clearly, an (n,r)-Tura´n graph does
A(d,g)-cageisad-regulargraphwithgirthg andminimum
not contain a clique of size r+1 and it is an (r−1)n-regular
r number of vertices. Let N(d,g) be the minimum number of
graph.
verticesina (d,g)-cage.AlowerboundonN(d,g),knownas
The following theorem shows that FR codes obtained from
Moore bound [3, p. 180], is given by
Tura´n graphs attain the upper bound in (3) for all k ≤ α and
hence they are optimal FR codes. The proof of this theorem (cid:40) 1+d(cid:80)g−23(d−1)i if g is odd
follows from Lemma 1 and by Tura´n’s theorem [6, p. 58]. n0(d,g)= 2(cid:80)g−2i2=(0d−1)i if g is even .
i=0
Theorem 4. LetT =(V,E)bean(n,r)-Tura´ngraph,r <n,
Lemma 10. The bound in (3) is not tight for ρ=2 if
α = (r−1)n, and let k be an integer such that 1 ≤ k ≤ α.
r
Then the (n,α,2) FR code CT based on T has file size given αk−α−k+3≤n<N(α,k+1).
by
(cid:22)r−1 k2(cid:23) As a consequence of Lemma 10 we have that the bound
M(k)=kα− · (4) in (3) is not always tight and hence we have a similar better
r 2
bound on A(n,k,α,ρ):
which attains the upper bound in (3).
A(n,k,α,ρ)≤ϕ(cid:48)(k), where ϕ(cid:48)(1)=α,
Note that an (n − 1)-regular complete graph K is an
n (cid:24) (cid:25)
ρA(n,k,α,ρ)−kα
(n,n)-Tura´n graph. Hence, the construction of MBR codes ϕ(cid:48)(k+1)=A(n,k,α,ρ)+α− .
n−k
from [13], [14] can be considered as a special case of our
constructionoftheDRESScodeswithaninnerFRcodebased IV. OPTIMALFRCODESWITHREPETITIONDEGREEρ>2
onaTura´ngraph.Notealsothatanα-regularcompletebipartite
In this section, we consider FR codes with repetition degree
graph K is a (2α,2)-Tura´n graph. The following example
α,α ρ > 2. Note, that while codes with ρ = 2 have the maximum
illustrates Theorem 4 for such a graph.
data/storage ratio, codes with ρ > 2 provide multiple choices
Example 1. The (6,3,2) FR code based on K and its file for node repairs. In other words, when a node fails, it can be
3,3
size for 1≤k ≤3 are shown in Fig. 2. repaired from different d-subsets of available nodes.
We present generalizations of the constructions from the
The proof of the following lemma can be easily verified
previoussectionwhichwerebasedonTura´ngraphsandgraphs
from Lemma 1.
with a given girth. These generalizations employ transversal
Lemma 5. Let C be an (n,α,2) FR code. Then the file size designs and generalized polygons, respectively.
M(k) of C for any 1≤k ≤α satisfies A transversal design of group size h and block size (cid:96),
denoted by TD((cid:96),h) is a triple (P,G,B), where
M(k)≤kα−k+1. 1) P is a set of (cid:96)h points;
2) G is a partition of P into (cid:96) sets (groups), each one of
1 2 3 4 1 5 9 13 1 6 12 15
size h;
Node 1 Node 5 Node 9
3) B is a collection of (cid:96)-subsets of P (blocks);
4) each block meets each group in exactly one point;
5 6 7 8 2 6 10 14 2 5 11 16
5) any pair of points from different groups is contained in
Node 2 Node 6 Node 10
exactly one block.
The properties of a transversal design TD((cid:96),h) which will be
9 10 11 12 3 7 11 15 3 8 10 13
useful for our constructions are summarized in the following
Node 3 Node 7 Node 11
lemma [1].
Lemma 11. Let (P,G,B) be a transversal design TD((cid:96),h). 13 14 15 16 4 8 12 16 4 7 9 14
The number of points is given by |P| = (cid:96)h, the number of Node 4 Node 8 Node 12
groups is given by |G| = (cid:96), the number of blocks is given by
|B|=h2, and the number of blocks that contain a given point Fig. 3: The (12,4,3) FR code based on TD(3,4)
is equal to h.
Let TD be a transversal design TD(ρ,α), ρ ≤ α+1, with SimilarlytoanFRcodeC withρ=2basedonagraphG
G
block size ρ and group size α. Let C be an (n,α,ρ) FR with girth g, one can consider an FR code C based on a
TD GP
code based on TD (see Section II). By Lemma 11, there are generalized g-gon (generalized polygon GP [3]) for ρ > 2.
ρα points in TD and hence n=ρα. Note, that all the symbols One can prove that the file size of C is identical to the file
GP
stored in node i correspond to the set Ni of blocks from TD size of CG for k ≤g+(cid:100)g2(cid:101)−2 given in Theorem 9. However,
thatcontainthepointi.SincebyLemma11thereareαblocks a generalized g-gon is known to exist only for g ∈{3,4,6,8}.
that contain a given point, it follows that each node stores α This observation also holds for a general biregular bipartite
symbols. graphofgirth2g,notonlytheincidencegraphofageneralized
polygon.
Theorem 12. Let k = bρ+t, for integers b,t ≥ 0 such that
t ≤ ρ−1. For an (n = ρα,α,ρ) FR code CTD based on a Remark 3. Note that the problem of constructing FR codes
transversal design TD(ρ,α) we have with ρ > 2 also can be considered in terms of bipartite
(cid:18)k(cid:19) (cid:18)b(cid:19) expander graphs (see e.g [5]). Let GEx = (L ∪ R,E) be
M(k)≥kα− +ρ +bt.
a bipartite expander and let C be the FR code such that
2 2 Ex
the subset N , 1 ≤ i ≤ n, corresponds to the ith vertex in
Remark 1. Note, that for all k ≥ρ+1, the file size of the FR i
L and the symbol j, 1 ≤ j ≤ θ, corresponds to the jth
code C is strictly larger than the MBR capacity.
TD vertex in R, |L| = n and |R| = θ. Then calculating M(k)
Note that the incidence matrix of the transversal design can be described by calculating the number of neighbours of
TD(2,α) is equal to the incidence matrix of the (2α,2)-Tura´n any subset of L of size k. In other words, for an FR code with
graph, and hence in this case C =C . file size M(k) it should hold that |Γ(A)| ≥ M(k) for every
TD T
A ⊆ L of size k, where Γ(A) denotes the set of neighbours
Example2. LetTDbeatransversaldesignTD(3,4)definedas
of A. Hence, to have an FR code with file size M(k), one
follows: P ={1,2,...,12}; G ={G ,G ,G }, where G =
1 2 3 1 need to construct a (k,M(k)) expander graph, where M(k) is
{1,2,3,4}, G2 = {5,6,7,8}, and G3 = {9,10,11,12}; B = k k
its expansion factor [5].
{B ,B ,...,B }, where B = {1,5,9}, B = {1,6,10},
1 2 16 1 2
B3 = {1,7,11}, B4 = {1,8,12}, B5 = {2,5,10}, B6 = V. FRACTIONALREPETITIONBATCHCODES
{2,6,9}, B = {2,7,12}, B = {2,8,11}, B = {3,5,12},
7 8 9 In this section we propose a new type of codes for DSS,
B = {3,6,11}, B = {3,7,10}, B = {3,8,9},
10 11 12 called fractional repetition batch (FRB) codes, which enable
B = {4,5,11}, B = {4,6,12}, B = {4,7,9}, and
13 14 15 uncoded efficient exact node repairs and load balancing which
B ={4,8,10}.
16 can be performed by several users in parallel. An FRB code
The placement of symbols from a codeword of the corre-
is a combination of an FR code and an uniform combinatorial
sponding MDS code of length 16 is shown in Fig. 3. The
batch code.
values of a file size M(k) for 1 ≤ k ≤ 4 are given in the
The family of codes called batch codes was proposed in [8]
following table.
for load balancing in distributed storage. A batch code stores
k M(k)
θ (encoded) data symbols in n system nodes in such a way
1 4
that any batch of t data symbols can be decoded by reading at
2 7
mostonesymbolfromeachnode.Inaρ-uniformcombinatorial
3 9
batchcode,proposedin[12],eachnodestoresasubsetofdata
4 11
symbols and no decoding is required during retrieval of any
Remark 2. The conditions on the parameters of TD such that batch of t symbols. Each symbol is stored in exactly ρ nodes
theboundonthefilesizeofanFRcodeC fromTheorem12 andhenceitisalsocalledareplicationbased batchcode.Aρ-
TD
attains the recursive bound in (3) can be found in [16]. uniformcombinatorialbatchcodeisdenotedbyρ−(θ,N,t,n)-
CBC, where N =ρθ is the total storage over all the n nodes. To construct an FRB code, one need a bipartite expander with
These codes were studied in [8], [12], [17]. two different expansion factors, M(k)/k and 1, for two sides
Next, we provide a formal definition of FRB codes. This L and R of a graph, respectively.
definition is based on the definitions of a DRESS code and a
uniformcombinatorialbatchcode.Letf ∈FM beafileofsize VI. CONCLUSION
q
M and let c ∈ Fθ be a codeword of an (θ,M) MDS code WeconsideredtheproblemofconstructingoptimalFRcodes
f q
which encodes the data f. Let {N ,...,N } be a collection and as a consequence, optimal DRESS codes. We presented
1 n
of α-subsets of the set [θ]. A ρ − (n,M,k,α,t) FRB code constructions of FR codes based on Tura´n graphs, graphs with
C, k ≤ α, t ≤ M, represents a system of n nodes with the a given girth, transversal designs, and generalized polygons.
following properties: Based on a connection between FR codes and batch codes,
we proposed a new family of codes for DSS, FRB codes,
1) Everynodei,1≤i≤n,storesαsymbolsofc indexed
f
which have the properties of batch codes and FR codes
by N ;
i
simultaneously. These are the first codes for DSS which allow
2) Every symbol of c is stored in ρ nodes;
f
uncoded efficient exact repairs and load balancing.
3) From any set of k nodes it is possible to reconstruct the
stored file f, in other words, M =min|I|=k|∪i∈I Ni|; ACKNOWLEDGMENT
4) Any batch of t symbols from c can be retrieved by
f This research was supported in part by the Israeli Science
downloading at most one symbol from each node.
Foundation (ISF), Jerusalem, Israel, under Grant no. 10/12.
Note that the retrieval of any batch of t symbols can be
Natalia Silberstein was also supported in part at the Technion
performed by t different users in parallel, where each user
by a Fine Fellowship.
gets a different symbol.
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2
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CTD is a 3 − (12,11,4,4,11) FRB code, which stores [14] K. V. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran, “Dis-
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Remark 4. Similarly to FR codes, the problem of construct- [16] N. Silberstein and T. Etzion, “Optimal Fractional Repetition Codes”,
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