Achieving Arbitrary Locality and Availability in Binary Codes

0.37 MB·English

by  Anyu Wang

Achieving Arbitrary Locality and Availability in
Binary Codes
Anyu Wang and Zhifang Zhang
Key Laboratory of Mathematics Mechanization, NCMIS
Academy of Mathematics and Systems Science, CAS, Beijing, China
Email: {wanganyu, zfz}@amss.ac.cn
Abstract—The ith coordinate of an (n,k) code is said to have t = δ −1 in [22]) is investigated further in [14], [19]. This
locality r and availability t if there exist t disjoint groups, each property is particularly useful because it permits access of a
5 containingatmostr othercoordinatesthatcantogetherrecover coordinate from multiple ways in parallel, which is appealing
1 the value of the ith coordinate. This property is particularly
in distributed storage systems with hot data.
0 usefulforcodesfordistributedstoragesystemsbecauseitpermits
2 local repair and parallel accesses of hot data. In this paper, Specifically, the ith coordinate of a code is said to have
for any positive integers r and t, we construct a binary linear locality r and availability t if there exist t disjoint groups,
n code of length (cid:0)r+t(cid:1) which has locality r and availability t for each containing at most r other coordinates that can together
a all coordinates. Tthe information rate of this code attains r ,
J r+t recoverthevalueoftheithcoordinate.Codesthathavelocality
8 wonhliychknisowalnwcaoynsshtriguhcetironthtahnatthcaatnoafcthhieevdeiraercbtitprraordyulcotccaolidtye,atnhde r and availability t for information coordinates (e.g., all
1 availability. systematiccoordinatesinasystematiclinearcode)arestudied
in [14], [22]. More precisely, the authors in [22] give an

] I. INTRODUCTION upperboundontheminimumdistanceforany[n,k,d] linear
T q
Nowadays various forms of data redundancy are used in codes that have locality r and availability t for information
I
. coding for distributed storage systems to insure data integrity coordinates, i.e.,
s
c andtoimproveperformanceefficiency.Amongthosethecode
[ t(k−1)+1
with locality r has become an attractive subject since it was d≤n−k+2−(cid:100) (cid:101).
t(r−1)+1
1 proposed independently by Gopalan et al. [3], Oggier et al.
v [8], and Papailiopoulos et al. [11]. More precisely, the ith They further prove the existence of codes attaining this upper
4 coordinate of a code is said to have locality r if the value at boundwhenn≥k(rt+1).In[14],anupperboundisderived
6
this coordinate can be recovered by accessing at most r other foraspecialclassofcodesofwhichanyrepairgroupcontains
2
4 coordinates. In other words, associating each coordinate with only1paritysymbol,andsomeexplicitconstructionsattaining
0 a storage node in a distributed storage system, an (n,k) code this bound are given there.
1. with repair locality r (r (cid:28) k) can greatly reduces the disk In this paper we focus on codes that have locality r and
0 I/O complexity for node repair. Considering data reliability availability t for all coordinates, for which the only known
5 and storage efficiency, a lot of work studied upper bounds bounds are due to Tamo et al. [19], i.e.
1 on the minimum distance and information rate of such codes
: t
v [1]–[3], [10], [21], [24]. Codes attaining these upper bounds k (cid:89) 1
≤ , (1)
i were constructed in [5], [20], [21], and some have even found n 1+ 1
X i=1 ir
their way into practice [6], [16]. By now, a tight upper bound
ar r+r1 hasbeenprovenfortheinformationrate,whileacomplete and t
descriptionforthetightupperboundontheminimumdistance (cid:88) k−1
d≤n− (cid:98) (cid:99).
remains open. ri
i=0
Things become more complicated when people start con-
sidering locality r in the case of multiple node erasures. First, These bounds are proven for all (linear or nonlinear) (n,k)
locality r that can tolerate up to δ − 1 erasures is realized codes by using graph methods. There remains much work
in [12], [17], [18] by using inner-error-correcting codes of to be done in this field, such as discussing tightness of the
length at most r +δ −1 and minimum distance at least δ. upper bounds, constructing explicit codes which are optimal
Thenanotherwayisdevelopedin[9],[22]byprovidingδ−1 with respect to some upper bound, and etc. By far, the only
disjoint local repair groups. A general framework for these known construction of codes achieving any given locality r
works is built in [23]. Besides specific code structures, the and availability t is the direct product code (see, e.g., [19],
methods of sequential local repair [13] and cooperative local [22]), while other constructions are given for special values
repair [15] are also used for multiple erasures. of r and t. Table 1 lists almost all previous constructions of
Recently, the property of t-availability which is related codesthathavelocalityr andavailabilitytforallcoordinates.
to the structure of δ − 1 disjoint local repair groups (e.g.
n,k,d r,t details can be found in the proof of Theorem 5. Then we also
[19],[22]: n=(r+1)t,k=rt,d=2t ∀r,∀t need some basic concepts of block design.
Directproductcode
n=2m−1, r=2,
[7]:Simplexcode k=m,d=2m−1 t=2m−1−1 Definition 2. Let X be a v-set (i.e. a set with v elements),
[13,Example1]: n=(cid:0)r+2(cid:1),k=(cid:0)r+1(cid:1),d=3 ∀r,t=2 whose elements are called points. A t-(v,k,λ) design is a
completegraph 2 2
collection of distinct k-subsets (called blocks) of X with the
[21,Construction3]: n,k,d=n−m+1† ∀r,t=2
orthogonalpartition property that any t-subset of X is contained in exactly λ
[4,Construction4]: n=2m−1,
tensorproductmatrix k= 3n,d=4 r=2,t=3 blocks.
Constructionin n=(cid:0)r+t(cid:1)7,k=(cid:0)r+t−1(cid:1),
t t ∀r,∀t Given a t-(v,k,λ) design with v points P ,...,P and b
thispaper d=t+1 1 v
blocks B ,...,B , its b × v incidence matrix A = (a ) is
1 b ij
Table1
defined by

 1, if P ∈B
j i
a =
ij
 0, if Pj (cid:54)∈Bi
A. Our Contribution
For any positive integers r and t, we construct a linear where 1≤i≤b and 1≤j ≤v.
code of length (cid:0)r+t(cid:1) which has locality r and availability t
t III. CODECONSTRUCTION
for all coordinates. Besides arbitrary locality and availability
for all coordinates, the following aspects make the code more The code is constructed by defining its parity check matrix.
desirable. For any positive integers r and t, let m = r + t. In the
following, we define a matrix over F , denoted as H(m,t),
(1) The code is over the binary field, which means efficient 2
containing(cid:0) m (cid:1)rowsand(cid:0)m(cid:1)columns.EachrowofH(m,t)
implementation in practice. t−1 t
(2) Its information rate attains r which is always higher is associated with a (t−1)-subset of [m] and each column
r+t
of H(m,t) is associated with a t-subset of [m]. Given the
thanthatofthedirectproductcode.Althoughnospecific
elements in [m] ordered as 1 (cid:31) 2 (cid:31) ··· (cid:31) m, we arrange
boundontheinformationrateisnewlybuiltinthiswork,
the rows (also columns) of H(m,t) in the lexical order of
throughdetailedcomparisonswithpreviousconstructions
the associated subsets of [m]. More precisely, for any two
and some related bounds, we believe our code has near
optimal information rate when t is not too large (say, subsets E,F ⊆ [m] with the elements in each subset sorted
t<r). in the order (cid:31), then E is before F if and only if for the first
elementswhereE andF differ,sayainE andbinF,itholds
B. Organization a (cid:31) b. In this order, for 1 ≤ i ≤ (cid:0) m (cid:1) and 1 ≤ j ≤ (cid:0)m(cid:1),
t−1 t
Section II introduces formal definitions of locality and suppose the ith row is associated with the subset Ei and the
availability,aswellassomebasicconceptsaboutblockdesign. jth column is associated with the subset Fj, then the (i,j)th
Section III presents the code construction and reveals its rela- element hij of H(m,t) is defined as follows:
tionwiththeblockdesign.SectionIVgivescomparisonswith 
previousconstructionsandinformationratebounds.SectionV  1, if Ei ⊆Fj
h =
concludes the paper. ij  0, if Ei (cid:42)Fj
II. LOCALITYANDAVAILABILITY
LetusgiveanexampleofH(m,t).Supposet=3andm=
LetC bean[n,k,d]q linearcodewithgeneratormatrixG= 5. The matrix H(5,3) is given in Fig. 1. Actually, the matrix
(g1,...,gn), where gi is a k-dimensional column vector over H(m,t) is the parity check matrix of the code (denoted as C)
Fq for 1 ≤ i ≤ n. Denote [m] = {1,2,··· ,m} for any we construct in this section. Next we prove some properties
positive integer m. Then the locality r and availability t for of H(m,t) to help understand the code C.
linear codes are formally defined below.
Lemma 3. Form>t>1,thematrixH(m,t)isoftheblock
Definition 1. The ith coordinate, 1 ≤ i ≤ n, of an [n,k,d]q form
linear code C is said to have locality r and availability t if
 
there exist t disjoint subsets R(i),...,R(i) ⊆ [n]\{i} such H(m−1,t−1) 0
1 t
that for 1≤j ≤t, H(m,t)=  (2)
I H(m−1,t)
(1) |R(i)|≤r, and (mt−−11)
j
(2) gi is an Fq-linear combination of {gl}l∈Rj(i). where I(mt−−11) is the unit matrix of size (cid:0)mt−−11(cid:1) and 0 is a
zero matrix. Particularly, for m=t, H(m,t)=H(m,1)τ =
In this paper, we prove locality r and availability t by
(1,...,1)τ which is an all-one column vector of dimension m.
verifying some equivalent conditions on the dual code. The
Proof: It is obvious that H(m,1) = (1,...,1) and
†In [21], the specific values of n,k and m depend on the corresponding H(m,m)=(1,...,1)τ.Form>t>1,accordingtotheorder
orthogonal partition and the encoding map, and the constraints are too
complicatedtobestatedhere. inwhichtherows(andcolumns)ofH(m,t)arearranged,the
1 1 1 1 1 1 2 2 2 3 with {a ,...,a ,b }, which have 1 in that column. Conse-
1 t−2 j
2 2 2 3 3 4 3 3 4 4 quently,thesum(inF )ofrightpartsofrowsinRis0.Since
2
3 4 5 4 5 5 4 5 5 5 therowsinthebottomblockofH(m,t)areobviouslylinearly
1,2  1 1 1 0 0 0 0 0 0 0  independent, it follows rank H(m,t)=(cid:0)m−1(cid:1).
t−1
1,3  1 0 0 1 1 0 0 0 0 0  ItiseasytoverifythatrankH(m,m)=rankH(m,1)=1
 
1,4  0 1 0 1 0 1 0 0 0 0  which coincides with the results in Lemma 4. Then the parity
 
1,5  0 0 1 0 1 1 0 0 0 0  check matrix of the code C can be taken as
 
 
2,3  1 0 0 0 0 0 1 1 0 0  H =(I H(m−1,t)). (3)
  (m−1)
2,4  0 1 0 0 0 0 1 0 1 0  t−1
2,5  0 0 1 0 0 0 0 1 1 0  Therefore C is of length (cid:0)m(cid:1) = (cid:0)r+t(cid:1) and information rate
3,4  0 0 0 1 0 0 1 0 0 1  1−(cid:0)m−1(cid:1)/(cid:0)m(cid:1)= m−t = tr . In thte following, we continue
  t−1 t m r+t
3,5  0 0 0 0 1 0 0 1 0 1  to investigate the locality and availability of the code C.
 
4,5 0 0 0 0 0 1 0 0 1 1 Theorem 5. The code C which has the parity check matrix
H(r + t,t) (or H as defined in (3)) has locality r and
Fig.1. ThematrixH(5,3) availability t for all coordinates.
Proof: It is equivalent to prove that for each coordinate
upper left block (i.e. the former (cid:0)m−1(cid:1) rows and the former i ∈ [(cid:0)r+t(cid:1)] there exist t codewords in the dual code, say
t−2 t
(cid:0)m−1(cid:1) columns) corresponds to the subsets of [m] containing c1,...,ct,suchthat|suppcj|=r+1andsuppcj ∩suppcl =
t−1
1. Actually this block can be regarded as one defined over the {i} for 1 ≤ j (cid:54)= l ≤ t. Actually, we will see the rows of
set{2,3,...,m}withcolumnscorrespondingto(t−1)-subsets H(r+t,r) are exactly these codewords.
and rows corresponding to (t−2)-subsets. Thus this block is First, for any row in H(r+t,r), suppose it is associated
the matrix H(m−1,t−1). The upper right block of H(m,t) with a (t−1)-subset E ⊆ [r +t]. Since E is contained in
isobviously0.Eachrowofthebottomleftblockcorresponds r+t−(t−1)=r+1 t-subsets of [r+t], this row has r+1
a (t−1)-subset of [m]\{1} which is uniquely contained in 1’s. Namely, the support of each row is of size r+1.
a t-subset of [m] containing 1. Moreover, because the subsets Then, for each coordinate i∈[(cid:0)r+t(cid:1)] which corresponds to
t
aresortedinthelexicalorder,thebottomleftblockistheunit a column of H(r +t,t), suppose this column is associated
matrix of size (cid:0)m−1(cid:1). Similar to the upper left block, it can with a t-subset Fi ⊆ [r+t]. Because Fi contains t (t−1)-
t−1
see the bottom right block is H(m−1,t). subsets, there are t rows which have 1 in this column. We
When t>r+1, the matrix H(m=r+t,t) has more rows claimthatexcludingthecoordinatei,supportsofthesetrows
than columns. Actually, the following lemma states that the are pairwise disjoint. Otherwise, assume there are two rows,
rows of H(m,t) are linearly dependent, so some rows can be saythejthrow(denotedashj,associatedwiththesubsetEj)
deleted when regarded as a parity check matrix. andthelthrow(denotedashl,associatedwiththesubsetEl),
suchthat{i,u}⊆supph ∩supph forsomeu∈[r+t]\{i}.
j l
Lemma 4. For the block decomposition of H(m,t) as shown
It implies that
in (2), each row in the upper block (i.e. (H(m−1,t−1) 0))
is a F -linear combination of rows in the bottom block E ⊆F ∩F and E ⊆F ∩F .
2 j i u l i u
(I H(m−1,t)).Consequently,rankH(m,t)=(cid:0)m−1(cid:1).
(mt−−11) t−1 As a result, Ej ∪ El ⊆ Fi ∩ Fu. But the union (resp.
Proof: For any row (denoted as h) in the upper block, intersection) of two different (t−1)-subsets (resp. t-subsets)
suppose it is associated with a (t−1)-subset {1,a ,...,a } is of size at least t (resp. at most t − 1), which leads a
1 t−2
where a ∈ [m] \ {1} for 1 ≤ i ≤ t − 2. Denote [m] = contradiction.Therefore,thetrowsareexactlythecodewords
i
{1,a ,...,a }∪{b ,...,b }.Thentherowhhas1’sin c ,...,c we need to complete the proof.
1 t−2 1 m−t+1 1 t
the columns associated with the subsets {1,a ,...,a ,b }, Finally we determine the minimum distance of the code C
1 t−2 j
1 ≤ j ≤ m−t+1. As a result, the left part of h is a sum from its parity matrix H given in (3). Since each column in
of the left parts of the rows in the bottom block associated therightpartofH,i.e.H(m−1,t),hast1’sandtheleftpart
with the subsets {a ,...,a ,b }, 1 ≤ j ≤ m−t+1. For of H is a unit matrix, there exist t+1 columns in H which
1 t−2 j
simplicity, the collection of these rows is denoted by R. We are linearly dependent. Thus the minimum distance of C is at
only need to show the sum (in F ) of the right parts of rows most t+1. On the other hand, from the availability t it can
2
in R is 0. see any t erasures are recoverable for C, thus the minimum
Let us focus on the right part, it can see only the columns distance is at least t+1. Therefore the minimum distance of
associated with the subsets {a ,...,a ,b ,b }, 1 ≤ i,j ≤ C is t+1.
1 t−2 i j
m − t + 1, have 1’s in the rows in R. Furthermore, for
A. Relation with the block design
each of these columns, say the column associated with
{a ,...,a ,b ,b }, there are exactly two rows in R, i.e. the Actually, the matrix H(r+t,t) can be viewed as an inci-
1 t−2 i j
row associated with {a ,...,a ,b } and the row associated dencematrixofa1-((cid:0)r+t(cid:1),r+1,t)design.Moreover,suppose
1 t−2 i t
theblocksareB ,...,B ,whereb= t ·(cid:0)r+t(cid:1)=(cid:0)r+t(cid:1),then k/n
it holds 1 b r+1 t t−1 0.9 □▼● □
|Bi∩Bj|≤1 for 1≤i<j ≤b (4) 0.8 ▼● □▼● □ □ □ □ □ □ □
0.7 ▼● ▼
In other words, once we find a 1-(n,r + 1,t) design with 0.6 ● ▼ ▼
blocks satisfying the condition (4), it immediately derives a 0.5 ● ● ▼ ▼ ▼
linear code of length n with locality r and availability t by ●
taking its incidence matrix as the parity check matrix of the 0.4 ▼▼ OurCode ● ●
code. To make the resulting code has high information rate, 0.3 ●● Directproductcode
the incidence matrix needs to have low rank comparing with 0.2 □□ Tamoetal.'supperbound
its column size. However, it is difficult to find such 1-designs,
0.1
constructingthosewithincidencematricesoflowrankiseven
t
harder. Our construction of the matrix H(m,t) provides a 1 2 3 4 5 6 7 8 9 10
goodwaytodosuchthings.Besides,someconstructionsfrom
Fig.2. Comparisonoftheinformationrateforr=10,1≤t≤10.
geometry are also feasible. For example, the following matrix
H gives a 1-(9,3,2) design satisfying the property (4), i.e. k/n
0.7
□✶▼●
  ▼▼ OurCode
1 0 0 1 0 0 1 0 0
0.6 ●● Directproductcode
0 1 0 0 1 0 0 1 0 □ □□ Tamoetal.'supperbound
  0.5 ▼
H =0 0 1 0 0 1 0 0 1 . 0.4 ● □✶▼ □ ✶✶ Simplexcode
1 1 1 0 0 0 0 0 0 ▼ □ □ □
0 0 0 1 1 1 0 0 0 0.3 ● ▼ ▼ ✶▼ □ □ □ □ □ □ □ □ □
Then it induces a0bin0ary0line0ar0cod0e w1ith l1oca1lity r = 2 and 00..12 ● ● ● ● ▼ ▼ ▼ ▼ ▼ ▼ ▼ ✶▼ ▼
availability t = 2, but its information rate is 4 which is less ● ● ● ● ● ● ● ● ● t
9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
than r . In fact, this matrix H corresponds to the direct
r+t
product code construction. More details can be found in the Fig.3. Comparisonoftheinformationrateforr=2,1≤t≤16.
next section.
IV. COMPARISONSWITHOTHERCONSTRUCTIONSAND 2) Prakash et al’s construction: Recently, Prakash et al.
INFORMATIONRATEBOUNDS [13]presentedaconstructionoflocally2-reconstructiblecodes
by using Tura´n graphs. It was also shown in [13] that the
Theinformationrateandminimumdistancearetwoimpor-
resultingcodeshave2-availabilitywhenusingcompletegraphs
tant parameters for evaluating an error-correcting code. It is
instead of using Tura´n graphs. Particularly, a complete graph
well known that a tradeoff exists between these two parame-
with v vertices induces a binary code with locality r = v−
ters. Although the minimum distance of the code constructed 1 and availability t = 2. This code has length (r+1)(r+2)
in Section III is t + 1 which is the lowest for a locally 2
and information rate r . In fact, after a permutation of the
repairable code with availability t, we will see it performs r+2
columns of the parity check matrix they constructed there,
well in information rate through the following comparisons.
one can get exactly the matrix H(r+2,2). That is, the code
is equivalent to our construction for the very special case of
A. Comparison with Other Constructions
t = 2, while our work develops a general construction for
1) The direct product code: The direct product code, see arbitrary t.
e.g., [19], [23], is another code that can achieve arbitrary
B. Comparison with Information Rate Bounds
locality and availability. Specifically, the direct product of t
binary (r + 1,r) single-parity-check codes induces a code The only known bound on the information rate of locally
with locality r, availability t and information rate ( r )t. repairable codes with availability t for all coordinates is due
r+1
The code we constructed in Section III also achieves locality to Tamo et al. [19] (see the bound (1) stated in Section I).
r and availability t, but has information rate r . Because There does exist a gap between this upper bound and the rate
r+t
(1 + 1)t > 1 + t for all t > 1, it follows that ( r )t = we have achieved (i.e. r ). But the following remarks help
r r r+1 r+t
1/(1+ 1)t < 1/(1+ t) = r . Thus, when t > 1, our code neutralize this difference.
r r r+t
alwayshashigherinformationratethanthedirectproductcode (1) Toourknowledge,nocodesattainingthebound(1)have
with the same locality r and availability t. Fig. 2 and Fig. 3 been given (except for the special case of t=1). It was
display the t-k curves of the codes. pointed out in [19] that information rate of the direct
n
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not obtained definitely) than the bound (1).
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(4) However, for large t, there do exist codes which have [17] N. Silberstein, A. S. Rawat, O. O. Koyluoglu, and S. Vishwanath,
information rate exceeding r . For example, the binary “Optimal locally repairable codes via rank-metric codes,” in Proc.
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[18] W.Song,S.Dau,C.Yuen,andT.Li,“Optimallocallyrepairablelinear
availability t = 2m−1 −1. Its information rate is m codes,” IEEE J. Sel. Areas Commun., vol. 32, pp. 6925–6934, May
2m−1
greaterthan r = 2 forallm≥3.Acomparison 2014.
r+t 2m−1+1 [19] I. Tamo and A. Barg, “Bounds on locally recoverable codes with
between all these codes and bounds is shown in Fig. 3.
multiplerecoveringsets,” inProc.IEEEInt.Symp.Inf.Theory(ISIT),
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codes with locality and availability, we still have a long [20] I. Tamo, D. S. Papailiopoulos, and A. G. Dimakis, “Optimal locally
repairable codes and connections to matroid theory,” in Proc. IEEE
way to go.
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[21] I.TamoandA.Barg, “Afamilyofoptimallocallyrecoverablecodes,”
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[22] A. Wang and Z. Zhang, “Repair locality with multiple erasure
Inthispaperweconstructabinarylinearcodewitharbitrary
tolerance,” IEEE Trans. on Inform. Theory, vol. 60, pp. 6979–6987,
locality r and availability t. The code can always have higher Nov.2014.
informationratethanthedirectproductcodewhichistheonly [23] A. Wang and Z. Zhang, “Repair locality from a combinatorial
perspective,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Honolulu,
knownconstructionwiththesameproperty.Besides,itattains
2014,pp.1972–1976.
theoptimalinformationrateatt=2.Thisconstructionreveals [24] A. Wang and Z. Zhang, “An integer progamming based bound for
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get more results on the codes with locality and availability.
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