Cooperative local repair in distributed storage

2.1 Cooperative locality from codes with multiple disjoint local repair groups for code symbols

In [11, 17, 18], codes with multiple disjoint local repair groups for all code symbols are studied. These codes allow for multiple ways to recover a particular code symbol by contacting disjoint sets of small number of other code symbols. In particular, the work in [11, 17, 18] study codes with at least t disjoint local repair groups, each comprising of at most \(\tilde {r}\) other code symbols. We claim, according to our definition, these codes also have \((\tilde {r}i, \ell = i)\)-cooperative locality for each i∈ [t]. Without loss of generality, we establish this for i=t, i.e., we argue that a code with t disjoint repairs groups (each of size at most \(\tilde {r}\)) has \((\tilde {r}t, \ell = t)\)-cooperative locality.

Consider a set of t code symbols in failure. For any of these t failed code symbols, each symbol can have at least one failed code symbol in at most t−1 of its t disjoint repair groups. This implies that the code symbol under consideration has at least one of its local repair groups free of any failures. Thus, the code symbol can be repaired with the help of one of its intact local repair groups. This leave us with t−1 code symbols in failure (erasure). Now, for another code symbol in failure, we can have at most t−2 of its disjoint local repair groups with at least one failed code symbol. This leaves at least 2 of its disjoint local groups intact; therefore, this code symbol can be repaired with the help of one of its intact local repair groups. Following the similar argument, we can see that all of the t failed code symbols can be repaired in a code with t disjoint repair groups for all code symbols. In the worst case, we contact at most \(\tilde {r}t\) code symbols to repair all of the t failures. This establishes the \((\tilde {r}t, \ell = t)\)-cooperative locality for the codes under consideration.

Similarly, the codes with availability [19, 20], which enable multiple disjoint repair groups only for information (systematic) symbols in a codeword, can allow for cooperative local repair for certain ranges of system parameters. In particular, ([20] Construction I can give codes with \((\tilde {r}\ell, \ell)\)-cooperative locality and rate \(\frac {\tilde {r}}{\tilde {r} + \ell }\).

2.2 Comparison with the codes with \((\tilde {r}, \delta)\)-locality [9, 10]

In [16], Prakash et al. propose to study codes with \((\tilde {r}, \delta)\)-locality, a generalization that enforces additional requirements which the local repair groups of an LRC need to satisfy. In particular, a code \({\mathcal C}\) is said to have \((\tilde {r}, \delta)\)-locality if there is a set of codes \(\{\mathcal {C}_{i}\}_{i \in {\mathcal L}}\) obtained by puncturing the code \({\mathcal C}\), for some index set \({\mathcal L}\), such that the following three requirements hold: (1) for each \(i \in {\mathcal L}\), the support of \(\mathcal {C}_{i}\) is no more than \(\tilde {r}+\delta -1\); (2) for each \(i \in {\mathcal L}\), the minimum distance of \({\mathcal C}_{i}\) is larger than or equal to δ; and (3) each code symbol is contained in the support of at least one of the punctured codes \(\mathcal {C}_{i}\), \(i \in {\mathcal L}\). The rate vs. distance trade-offs for the codes with \((\tilde {r}, \delta)\)-locality and the constructions attaining these trade-offs are presented in [9, 10].

Note that a code with (r,δ)-locality ensures repair of any δ−1 failures within each punctured code. Here, we would like to highlight that the notion of (r,ℓ)-cooperative locality is different from that of (r,δ)-locality. In particular, codes with (r,ℓ)-cooperative locality are not required to meet the requirement (2) in the aforementioned definition of the codes with (r,δ)-locality. As a result, there are families of codes which satisfy the requirements of (r,ℓ)-cooperative locality but that do not meet the definition of the codes with (r,δ)-locality. We illustrate this with the help of the following example.

We note that the code \({\mathcal C}\) is nothing but a [7,3,4] Simplex code which we study in Section 7. It follows from the analysis presented in Section 7 that this code has (r=3,ℓ=2)-cooperative locality, i.e., any set of ℓ=2 failed code symbols can be recovered by contacting r=3 other code symbols. Let us assume that the code symbols a and a+b are in failure. In this case, we can recover both the failed code symbols from the set of r=3 code symbols (b,b + c,a + b + c). In other words, (a,a + b,b,b + c,a + b + c) form a punctured code of the original code \({\mathcal C}\) at r+ℓ=5 indices. However, this punctured code does not have minimum distance at least ℓ+1=δ=3. This can easily be observed from the fact that the punctured sub-code does not allow the repair of two code symbols b + c and a + b + c from the remaining set of three code symbols (a,a + b,b). Moreover, there is no punctured codes of the original code at at most r+ℓ=5 indices which has minimum distance at least ℓ+1=3. Therefore, \({\mathcal C}\) is an example of a code with (r=3,ℓ=2)-cooperative locality which does not have (r=3,δ=ℓ+1=3)-locality as defined in [16].

This also shows that the definition of (r,ℓ)-cooperative locality is not a strengthening of the definition of (r,δ)-locality. Hence, one cannot directly invoke the impossibility results for the codes with (r,δ)-locality to obtain impossibility results for the codes with (r,ℓ)-cooperative locality. However, as far as the achievability is concerned, a construction for a code with \((\tilde {r}, \delta)\)-locality gives a construction with \((r = \ell \tilde {r}, \ell = \delta - 1)\)-cooperative locality as explained in Section 4.

4.1 Partition code

As it is clear from the construction, partition code has the rate \(\frac {k}{n} = \frac {r}{r + \ell ^{2}}\). Moreover, a code symbol can be recovered from any \(\frac {r}{\ell }\) other code symbols from its local group. In the worst case, when ℓ failed code symbols belong to ℓ distinct local groups, we can recover all ℓ symbols from \(\ell \frac {r}{\ell } = r\) code symbols, downloading \(\frac {r}{\ell }\) symbols from each of the ℓ local groups containing one failed code symbol.

In the above construction of the partition codes, we use an \(\left (\frac {r}{\ell } + \ell, \frac {r}{\ell }\right)\) MDS code to encode disjoint groups of \(\frac {r}{\ell }\) message symbols. Note that the rate of this MDS code governs the rate of the overall code. One can potentially use some other code \({\mathcal C}^{\text {local}}\) of minimum distance at least ℓ+1 to encode disjoint groups of \(\frac {r}{\ell }\) message symbols. Now, we use r(x), x∈ [ ℓ] to denote the number of symbols that needs to be contacted to repair x erasure in one local group. For the case when an \(\left (\frac {r}{\ell } + \ell, \frac {r}{\ell }\right)\) MDS code is used, we have \(r(x) = \frac {r}{\ell }\) for x∈ [ ℓ]. Let r ^∗(x) denote the upper concave envelope of r(x) on the interval \([\!1, \ell ] \in \mathbb {R}\). Assume that we have p disjoint local groups, then a pattern of ℓ erasures can be represented by a vector (l ₁,l ₂,…,l _p ). Here, l _i denotes the number of erasures within the ith local group. Note that we have \(\sum _{i = 1}^{p}l_{i} = \ell \).

Since the rate of the partition code is agnostic to the number of local groups, we can use the value of p which can support k message symbols and minimizes the R.H.S. of (11). This approach optimizes the value of r for a given choice of ℓ and \({\mathcal C}^{\text {local}}\).

4.2 Product code

Here, we would like to note that the difference between the rate achieved by the partition code and the bound in (7) gets smaller as the parameter r becomes large as compared to the parameter ℓ. It is an interesting problem to either tighten the bound in (7) or present a construction for codes with (r,ℓ)-cooperative locality which have higher rate than that of the partition code. In the next two sections, we present two approaches to achieve this goal.

5.1 When ℓ=3

Let us take an \(\left [\frac {r}{3} + 1, \frac {r}{3}, 2\right ]\) MDS code over \(\mathbb {F}_{q}\) as the inner code. This code can repair any one failed code symbol by contacting the remaining \(\frac {r}{3}\) code symbols. For outer code, we employ a code with (r _out,1)-cooperative locality over \(\mathbb {F}_{q^{r/3}}\). This can repair any one super symbol (which consists of \(\frac {r}{3}\) symbols of \(\mathbb {F}_{q}\)) by contacting r _out symbols over \(\mathbb {F}_{q^{r/3}}\), i.e., \(r_{\text {out}}\frac {r}{3}\) symbols over \(\mathbb {F}_{q}\). (Note that in order to repair a super symbol, we can obtain the value of r _out required super symbols by contacting \(\frac {r}{3}\) symbols over \(\mathbb {F}_{q}\) from each of their corresponding codewords of the inner code.)

Hence, for all r<9, the concatenated codes have a higher rate than the partition codes.

5.2 When ℓ=4

5.3 General values of ℓ

Note that the concatenated code obtained in this section has a minimum distance of at least \((x+1)\left (\left \lfloor \frac {\ell }{x+1} \right \rfloor + 1\right)\).

6.1 Bipartite graphs with large girth

The girth of a graph is the number of vertices in the shortest cycle of the graph. In this section, we explore a particular class of codes based on bipartite graphs with high girth. In this construction, the code symbols are associated with the edges of a bipartite graph and both the left and right vertices in the the bipartite graph enforces the constraints on the code symbols associated with the edges incident on these vertices. The analysis of the cooperative locality of the codes obtained in this manner is based on the fact that the underlying bipartite graph has high girth.

Before stating our general result on the cooperative locality of the codes obtained in this manner, we consider small values of ℓ. Note that any two edges in \({\mathcal G}\) (code symbols in a codeword of \({\mathcal C}\)) share at most one vertex (appear together in at most one local constraint). Thus, for ℓ=2 code symbols in erasure, it is possible to find two local constraints that contain exactly one of the two erased symbols. This allows for the repair of both the erased symbols by utilizing these two local constraints. In other words, the code \({\mathcal C}\) has (2(Δ _max−1),2)-cooperative locality, where Δ _max denotes the maximum degree of the underlying bipartite graph \({\mathcal G}\). Similarly, even in the presence of ℓ=3 erasures, one can find at least two local constraints such that there is only one erasure among the code symbols participating in each of these constraints. Figure 2 a, b illustrates this fact by considering two possible patterns of ℓ=3 erasures. Now, using these two constraints, one can repair two erasures, which leaves only one erased symbol which can then be recovered with the help of any of the two local constraints it appears in. The repair of ℓ=3 erasures involves at most 3(Δ _max−1) other code symbols; hence, the code \({\mathcal C}\) has (3(Δ _max−1),3)-cooperative locality. In order to cooperatively repair ℓ>3 erasures in \({\mathcal C}\), we utilize the fact that the underlying bipartite graph \({\mathcal G}\) has high girth.

6.2 Expander graphs

The above analysis of the construction based on bipartite graphs fails to show a high minimum distance on top of the local repair property. However, with the graphical construction, it is also possible to have high distance, and hence protection against catastrophic failures. Next, we show how the expansion property of graphs leads to such conclusion.

6.2.2 Regular expander graph

We now study the cooperative locality of the codes obtained by the double covers of Δ-regular expander graphs [28]. The analysis of the cooperative locality is based on the analysis of the decoding algorithm for these codes presented in [30]. Note that we naturally modify the decoding algorithm from [30] to perform erasure correction in a cooperative manner.

Let \({\mathcal G} = ({\mathcal V}, {\mathcal E})\) be a Δ-regular graph with \(|{\mathcal V}| = N\) and λ as the second (absolute) largest eigenvalue of its adjacency matrix⁵. Given \({\mathcal G}\), we construct a bipartite graph \(\widetilde {{\mathcal G}} = ({\mathcal V}_{0} \cup {\mathcal V}_{1}, \widetilde {{\mathcal E}})\) with \(|{\mathcal V}_{0}| = |{\mathcal V}_{1}| = N\) in the following manner (see Fig. 3):

• Each vertex \(u \in {\mathcal V}\) in the original graph \({\mathcal G}\) corresponds to a left node \(u_{l} \in {\mathcal V}_{0}\) and a right node \(u_{r} \in {\mathcal V}_{1}\) in the graph \(\widetilde {{\mathcal G}}\).

  

• For a pair of vertices \((u_{l}, v_{r}) \in {\mathcal V}_{0} \times {\mathcal V}_{1}\), there exists an edge \((u_{l}, v_{r}) \in \widetilde {{\mathcal E}}\) iff there is an edge between the vertices u and v in the original graph \({\mathcal G}\), i.e., \((u, v) \in {\mathcal E}\).

The bipartite graph \(\widetilde {{\mathcal G}}\) is referred to as the double cover of the graph \({\mathcal G}\). Note that the bipartite graph \(\widetilde {{\mathcal G}}\) is Δ-regular with total n=N Δ edges. Moreover, the following result holds for the bipartite graph \(\widetilde {{\mathcal G}}\).

Lemma 1.

(Expander mixing lemma) [31] Let \(\widetilde {{\mathcal G}}\) be the Δ-regular bipartite graph as described above. Then, for every \({\mathcal S} \subseteq {\mathcal V}_{0}\) and \({\mathcal T} \subseteq {\mathcal V}_{1}\), we have

$$\begin{array}{*{20}l} \left|\widetilde{{\mathcal E}}({\mathcal S} \times {\mathcal T}) - \frac{d |{\mathcal S}| |{\mathcal T}|}{N}\right| \leq \lambda \sqrt{|{\mathcal S}| |{\mathcal T}|}, \end{array} $$

(23)

where \(\widetilde {{\mathcal E}}({\mathcal S} \times {\mathcal T})\)denotes the collection of the edges from the nodes in the set \({\mathcal S}\) to the nodes in the set \({\mathcal T}\).

Given the bipartite graph \(\widetilde {{\mathcal G}}\) and a code \({\mathcal C}_{0}\) with Δ-symbol long codewords, we define a code \({\mathcal C}\) as a slight generalization of the method of Section 6.1. Each edge in \(\widetilde {{\mathcal G}}\) corresponds to a code symbol in the codewords of \({\mathcal C}\). For each node in the bipartite graph \(\widetilde {{\mathcal G}}\), the Δ code symbols associated with the Δ edges incident on the node constitute a codeword in the code \({\mathcal C}_{0}\). Note that we assume the local code \({\mathcal C}_{0}\) to be an MDS code throughout this paper. In Fig. 4, we present an algorithm which corrects any ℓ erasures in \({\mathcal C}\) an cooperative manner by contacting at most \(\ell \Delta \cdot \text {rate}({\mathcal C}_{0})\) code symbols. The algorithm alternates between the left nodes \({\mathcal V}_{0}\) and the right nodes \({\mathcal V}_{1}\) in order to utilize the smaller code \({\mathcal C}_{0}\) associated with the vertices to correct the erasures.

Let \({\mathcal S}^{1} \subseteq {\mathcal V}_{0}\) denote the set of nodes that have erasures among the code symbols associated with their edges and did not attempt to correct those erasures in the first round of the algorithm. This implies that each vertex in \({\mathcal S}^{1}\) has at least \(d_{\min }({\mathcal C}_{0})\) erasures among the code symbols associated with its Δ edges. Therefore, we have

$$\begin{array}{*{20}l} |{\mathcal S}^{1}| \leq \frac{\ell}{d_{\min}({\mathcal C}_{0})}. \end{array} $$

(24)

We use \({\mathcal S}^{i}\) for i≥2 to denote the set of (left or right) vertices that have erasures among the Δ code symbols associated with them in the beginning of ith round and did not attempt to correct those erasures. Note that \({\mathcal S}^{i} \subseteq {\mathcal V}_{0}\) and \({\mathcal S}^{i} \subseteq {\mathcal V}_{1}\) when i is an odd and even round of decoding, respectively. Next, we employ the expander mixing lemma (cf. Lemma 1) to show that \(\big \{|{\mathcal S}^{1}|, |{\mathcal S}^{2}|, |{\mathcal S}^{3}|, \ldots \big \}\) is a strictly decreasing sequence.

Lemma 2.

Let \({\mathcal S}^{1}, {\mathcal S}^{2}, \ldots \) be the sequence of sets of (left or right) vertices in the bipartite graph \(\widetilde {{\mathcal G}}\) as defined above. Assume that the minimum distance of \({\mathcal C}_{0}\) is at least (1+ε)λ and \(\ell \leq \frac {N\lambda \epsilon \delta }{2} = \frac {n\lambda \epsilon \delta }{2\Delta }\). Then, for i≥1, we have

$$\begin{array}{*{20}l} \big|{\mathcal S}^{i+1}\big| \leq \frac{\big|{\mathcal S}^{i} \big|}{1+ \epsilon}. \end{array} $$

(25)

Proof.

We prove the relation in (25) for i=1; the proof for general i involves steps similar to those in the proof of the i=1 case. Note that each code symbol that is in erasures after the first round of decoding is associated with some edge incident on a left node belonging to the set \({\mathcal S}^{1}\). By the definition of the set \({\mathcal S}^{2}\), it has at least \(d_{\min }({\mathcal C}_{0})\) erasures among the Δ code symbols associated with it after the first round of decoding. In other words, this implies that each vertex in the set \({\mathcal S}^{2}\) has at least \(d_{\min }({\mathcal C}_{0})\) edges incident on it which are emanating from the vertices from the set \({\mathcal S}^{1}\). Therefore, we have

$$\begin{array}{*{20}l} |{\mathcal S}^{2}|d_{\min}({\mathcal C}_{0}) &\leq |\widetilde{{\mathcal E}}\left({\mathcal S}^{1} \times {\mathcal S}^{2}\right)| \\ & \overset{(a)}{\leq} \frac{\Delta |{\mathcal S}^{1}| |{\mathcal S}^{2}|}{N}+ \lambda \sqrt{|{\mathcal S}^{1}| |{\mathcal S}^{2}|} \\ & \overset{(b)}{\leq} \frac{\Delta |{\mathcal S}^{1}| |{\mathcal S}^{2}|}{N} + \lambda \frac{|{\mathcal S}^{1}| + |{\mathcal S}^{2}|}{2} \\ & \overset{(c)}{\leq} \frac{\Delta \ell |{\mathcal S}^{2}|}{N \cdot d_{\min}({\mathcal C}_{0})} + \lambda \frac{|{\mathcal S}^{1}| + |{\mathcal S}^{2}|}{2}, \end{array} $$

(26)

where (a) and (c) follows from Lemma 1 and (24), respectively. Note that we employ the AM-GM inequality to obtain (b). It follows from (26) that

$$\begin{array}{*{20}l} \left|{\mathcal S}^{2}\right| \leq \frac{\lambda}{2 \cdot d_{\min}({\mathcal C}_{0}) - \lambda - 2\Delta \ell/(N \cdot d_{\min}({\mathcal C}_{0}))}\left|{\mathcal S}^{1} \right|. \end{array} $$

(27)

By replacing \(d_{\min }({\mathcal C}_{0}) = \delta \Delta \) in (27), we get

$$\begin{array}{*{20}l} \left|{\mathcal S}^{2}\right| \leq \frac{\lambda}{2 \delta \Delta - \lambda - 2\ell/(N\delta)}\left|{\mathcal S}^{1} \right|. \end{array} $$

(28)

Under our assumption that \(\frac {2\ell }{N\delta } \leq \epsilon \lambda \), it follows from (28) that

$$\begin{array}{*{20}l} \left|{\mathcal S}^{2}\right| \leq \frac{\lambda}{2 \delta \Delta - (1 + \epsilon)\lambda}\left|{\mathcal S}^{1} \right|. \end{array} $$

(29)

Now, under the assumption that δ Δ≥(1+ε)λ, we get from (29) that

$$\begin{array}{*{20}l} \left|{\mathcal S}^{2}\right| \leq \frac{\left|{\mathcal S}^{1} \right|}{1 + \epsilon}. \end{array} $$

(30)

It follows from the Lemma 2 that in at most logarithmic (in ℓ) rounds of decoding, the algorithm described in Fig. 4 can correct ℓ erasures.

The codes based on the double covers of Δ-regular expander graphs have been studied in the coding theory literature before (see, e.g., [30]). The rate and the minimum distance of the code \({\mathcal C}\) depends on the rate and the minimum distance of the code \({\mathcal C}_{0}\). Note that \({\mathcal C}_{0}\) characterizes the local constraints associated with the vertices in the bipartite graph \(\widetilde {{\mathcal G}}\). In particular, if \(\text {rate}({\mathcal C}_{0})= R\) and \(d_{\min }({\mathcal C}_{0}) = \delta \Delta \), then we have that \(\text {rate}({\mathcal C}) \geq 2R -1\) and \(d_{\min }({\mathcal C}) \geq \delta (\delta - \frac {\lambda }{\Delta })n\) [28, 30].

As we show in this section, for an ε>0 and local code \({\mathcal C}_{0}\) such that \(d_{\min }({\mathcal C}_{0}) = \delta \Delta \geq (1 + \epsilon)\lambda \), it is possible to correct \(\ell \leq \frac {N\lambda \epsilon \delta }{2}\) erasures using the algorithm described in Fig. 4. Moreover, in the worst correction of each erasure involves contacting at most \(\text {rate}({\mathcal C}_{0})\Delta \le (1\,+\,\text {rate}({\mathcal C}))\Delta /2\) other intact code symbols (assuming that the local code \({\mathcal C}_{0}\) is an MDS code). Therefore, the codes based on the double cover of a Δ-regular expander graph and a local code \({\mathcal C}_{0}\) have (r,ℓ)-cooperative locality for any \(\ell \leq \frac {N\lambda \epsilon \delta }{2} = \frac {n\lambda \epsilon \delta }{2\Delta }\) and \( r = \ell (1+\text {rate}({\mathcal C}))\Delta /2\), that is,

$$ \text{rate}({\mathcal C}) \ge \frac{2r}{\ell \Delta} -1. $$

In the next section, we show an explicit family of algebraic codes that exhibit very strong cooperative local repair property, as well as a very high minimum distance.