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Locally Repairable Codes with Multiple Repair Alternatives  [PDF]
Lluis Pamies-Juarez,Henk D. L. Hollmann,Frédérique Oggier
Computer Science , 2013,
Abstract: Distributed storage systems need to store data redundantly in order to provide some fault-tolerance and guarantee system reliability. Different coding techniques have been proposed to provide the required redundancy more efficiently than traditional replication schemes. However, compared to replication, coding techniques are less efficient for repairing lost redundancy, as they require retrieval of larger amounts of data from larger subsets of storage nodes. To mitigate these problems, several recent works have presented locally repairable codes designed to minimize the repair traffic and the number of nodes involved per repair. Unfortunately, existing methods often lead to codes where there is only one subset of nodes able to repair a piece of lost data, limiting the local repairability to the availability of the nodes in this subset. In this paper, we present a new family of locally repairable codes that allows different trade-offs between the number of contacted nodes per repair, and the number of different subsets of nodes that enable this repair. We show that slightly increasing the number of contacted nodes per repair allows to have repair alternatives, which in turn increases the probability of being able to perform efficient repairs. Finally, we present pg-BLRC, an explicit construction of locally repairable codes with multiple repair alternatives, constructed from partial geometries, in particular from Generalized Quadrangles. We show how these codes can achieve practical lengths and high rates, while requiring a small number of nodes per repair, and providing multiple repair alternatives.
Repair Locality with Multiple Erasure Tolerance  [PDF]
Anyu Wang,Zhifang Zhang
Computer Science , 2013, DOI: 10.1109/TIT.2014.2351404
Abstract: In distributed storage systems, erasure codes with locality $r$ is preferred because a coordinate can be recovered by accessing at most $r$ other coordinates which in turn greatly reduces the disk I/O complexity for small $r$. However, the local repair may be ineffective when some of the $r$ coordinates accessed for recovery are also erased. To overcome this problem, we propose the $(r,\delta)_c$-locality providing $\delta -1$ local repair options for a coordinate. Consequently, the repair locality $r$ can tolerate $\delta-1$ erasures in total. We derive an upper bound on the minimum distance $d$ for any linear $[n,k]$ code with information $(r,\delta)_c$-locality. For general parameters, we prove existence of the codes that attain this bound when $n\geq k(r(\delta-1)+1)$, implying tightness of this bound. Although the locality $(r,\delta)$ defined by Prakash et al provides the same level of locality and local repair tolerance as our definition, codes with $(r,\delta)_c$-locality are proved to have more advantage in the minimum distance. In particular, we construct a class of codes with all symbol $(r,\delta)_c$-locality where the gain in minimum distance is $\Omega(\sqrt{r})$ and the information rate is close to 1.
Locally Repairable Codes  [PDF]
Dimitris S. Papailiopoulos,Alexandros G. Dimakis
Mathematics , 2012,
Abstract: Distributed storage systems for large-scale applications typically use replication for reliability. Recently, erasure codes were used to reduce the large storage overhead, while increasing data reliability. A main limitation of off-the-shelf erasure codes is their high-repair cost during single node failure events. A major open problem in this area has been the design of codes that {\it i)} are repair efficient and {\it ii)} achieve arbitrarily high data rates. In this paper, we explore the repair metric of {\it locality}, which corresponds to the number of disk accesses required during a {\color{black}single} node repair. Under this metric we characterize an information theoretic trade-off that binds together locality, code distance, and the storage capacity of each node. We show the existence of optimal {\it locally repairable codes} (LRCs) that achieve this trade-off. The achievability proof uses a locality aware flow-graph gadget which leads to a randomized code construction. Finally, we present an optimal and explicit LRC that achieves arbitrarily high data-rates. Our locality optimal construction is based on simple combinations of Reed-Solomon blocks.
Binary Locally Repairable Codes ---Sequential Repair for Multiple Erasures  [PDF]
Wentu Song,Chau Yuen
Mathematics , 2015,
Abstract: Locally repairable codes (LRC) for distribute storage allow two approaches to locally repair multiple failed nodes: 1) parallel approach, by which each newcomer access a set of $r$ live nodes $(r$ is the repair locality$)$ to download data and recover the lost packet; and 2) sequential approach, by which the newcomers are properly ordered and each newcomer access a set of $r$ other nodes, which can be either a live node or a newcomer ordered before it. An $[n,k]$ linear code with locality $r$ and allows local repair for up to $t$ failed nodes by sequential approach is called an $(n,k,r,t)$-exact locally repairable code (ELRC). In this paper, we present a family of binary codes which is equivalent to the direct product of $m$ copies of the $[r+1,r]$ single-parity-check code. We prove that such codes are $(n,k,r,t)$-ELRC with $n=(r+1)^m,k=r^m$ and $t=2^m-1$, which implies that they permit local repair for up to $2^m-1$ erasures by sequential approach. Our result shows that the sequential approach has much bigger advantage than parallel approach.
Optimal Locally Repairable Codes and Connections to Matroid Theory  [PDF]
Itzhak Tamo,Dimitris S. Papailiopoulos,Alexandros G. Dimakis
Mathematics , 2013,
Abstract: Petabyte-scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency. Classical codes like Reed-Solomon are highly sub-optimal for distributed environments due to their high overhead in single-failure events. Locally Repairable Codes (LRCs) form a new family of codes that are repair efficient. In particular, LRCs minimize the number of nodes participating in single node repairs during which they generate small network traffic. Two large-scale distributed storage systems have already implemented different types of LRCs: Windows Azure Storage and the Hadoop Distributed File System RAID used by Facebook. The fundamental bounds for LRCs, namely the best possible distance for a given code locality, were recently discovered, but few explicit constructions exist. In this work, we present an explicit and optimal LRCs that are simple to construct. Our construction is based on grouping Reed-Solomon (RS) coded symbols to obtain RS coded symbols over a larger finite field. We then partition these RS symbols in small groups, and re-encode them using a simple local code that offers low repair locality. For the analysis of the optimality of the code, we derive a new result on the matroid represented by the code generator matrix.
When Locally Repairable Codes Meet Regenerating Codes --- What If Some Helpers Are Unavailable  [PDF]
Imad Ahmad,Chih-Chun Wang
Mathematics , 2015,
Abstract: Locally repairable codes (LRCs) are ingeniously designed distributed storage codes with a (usually small) bounded number of helper nodes participating in repair. Since most existing LRCs assume exact repair and allow full exchange of the stored data ($\beta=\alpha$), they can be viewed as a generalization of the traditional erasure codes (ECs) with a much desired feature of local repair. However, it also means that they lack the features of functional repair and partial information-exchange ($\beta<\alpha$) in the original regenerating codes (RCs). Motivated by the significant bandwidth (BW) reduction of RCs over ECs, existing works by Ahmad et al and by Hollmann studied "locally repairable regenerating codes (LRRCs)" that simultaneously admit all three features: local repair, partial information-exchange, and functional repair. Significant BW reduction was observed. One important issue for any local repair schemes (including both LRCs and LRRCs) is that sometimes designated helper nodes may be temporarily unavailable, the result of multiple failures, degraded reads, or other network dynamics. Under the setting of LRRCs with temporary node unavailability, this work studies the impact of different helper selection methods. It proves, for the first time in the literature, that with node unavailability, all existing methods of helper selection, including those used in RCs and LRCs, are strictly repair-BW suboptimal. For some scenarios, it is necessary to combine LRRCs with a new helper selection method, termed dynamic helper selection, to achieve optimal BW. This work also compares the performance of different helper selection methods and answers the following fundamental question: whether one method of helper selection is intrinsically better than the other? for various different scenarios.
XORing Elephants: Novel Erasure Codes for Big Data  [PDF]
Maheswaran Sathiamoorthy,Megasthenis Asteris,Dimitris Papailiopoulos,Alexandros G. Dimakis,Ramkumar Vadali,Scott Chen,Dhruba Borthakur
Computer Science , 2013,
Abstract: Distributed storage systems for large clusters typically use replication to provide reliability. Recently, erasure codes have been used to reduce the large storage overhead of three-replicated systems. Reed-Solomon codes are the standard design choice and their high repair cost is often considered an unavoidable price to pay for high storage efficiency and high reliability. This paper shows how to overcome this limitation. We present a novel family of erasure codes that are efficiently repairable and offer higher reliability compared to Reed-Solomon codes. We show analytically that our codes are optimal on a recently identified tradeoff between locality and minimum distance. We implement our new codes in Hadoop HDFS and compare to a currently deployed HDFS module that uses Reed-Solomon codes. Our modified HDFS implementation shows a reduction of approximately 2x on the repair disk I/O and repair network traffic. The disadvantage of the new coding scheme is that it requires 14% more storage compared to Reed-Solomon codes, an overhead shown to be information theoretically optimal to obtain locality. Because the new codes repair failures faster, this provides higher reliability, which is orders of magnitude higher compared to replication.
Locally Repairable Codes and Matroid Theory  [PDF]
Antti P?ll?nen
Mathematics , 2015,
Abstract: Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. A traditional approach is to look for codes which simultaneously maximize error tolerance and minimize storage space consumption. However, this tends to yield codes for which error correction requires an unrealistic amount of communication between storage nodes. LRCs solve this problem by allowing errors to be corrected locally. This thesis reviews previous results on the subject presented in [1]. These include that every almost affine LRC induces a matroid such that the essential properties of the code are determined by the matroid. Also, the generalized Singleton bound for LRCs can be extended to matroids as well. Then, matroid theory can be used to find classes of matroids that either achieve the bound, meaning they are optimal in a certain sense, or at least come close to the bound. This thesis presents an improvement to the results of [1] in both of these cases. [1] T. Westerb\"ack, R. Freij, T. Ernvall and C. Hollanti, "On the Combinatorics of Locally Repairable Codes via Matroid Theory", arXiv:1501.00153 [cs.IT], 2014.
An Upper Bound On the Size of Locally Recoverable Codes  [PDF]
Viveck Cadambe,Arya Mazumdar
Computer Science , 2013,
Abstract: In a {\em locally recoverable} or {\em repairable} code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage node as possible. In this paper, we bound the minimum distance of a code in terms of its length, size and locality. Unlike previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence the Simplex codes are the first example of a optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds.
A Family of Erasure Correcting Codes with Low Repair Bandwidth and Low Repair Complexity  [PDF]
Siddhartha Kumar,Alexandre Graell i Amat,Iryna Andriyanova,Fredrik Br?nnstr?m
Mathematics , 2015,
Abstract: We present the construction of a new family of erasure correcting codes for distributed storage that yield low repair bandwidth and low repair complexity. The construction is based on two classes of parity symbols. The primary goal of the first class of symbols is to provide good erasure correcting capability, while the second class facilitates node repair, reducing the repair bandwidth and the repair complexity. We compare the proposed codes with other codes proposed in the literature.
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