Asymptotic Interference Alignment for Optimal Repair of MDS Codes in Distributed Storage

Cadambe, Viveck R.; Jafar, Syed A.; Maleki, Hamed; Ramchandran, Kannan; Suh, Changho

doi:10.1109/tit.2013.2237752

Cited by 194 publications

(268 citation statements)

References 28 publications

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“…The first MSR code constructions appeared in [7], [16], which roughly correspond to the family of parameters {n, k, d} with rate k/n 1/2. The asymptotic existence of MSR codes for all triples {n, k, d} was eventually shown in [11] using interference alignment techniques developed for a wireless interference channel; these codes achieve optimality as a regenerating code (as well as approach the MSR point) only when α → ∞, i.e.,…”

Section: A Previous Workmentioning

confidence: 99%

Minimum Storage Regenerating Codes for All Parameters

Goparaju

Fazeli

Vardy

2017

IEEE Trans. Inform. Theory

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Abstract-Regenerating codes for distributed storage have attracted much research interest in the past decade. Such codes trade the bandwidth needed to repair a failed node with the overall amount of data stored in the network. Minimum storage regenerating (MSR) codes are an important class of optimal regenerating codes that minimize (first) the amount of data stored per node and (then) the repair bandwidth. Specifically, an [n, k, d]-(α) MSR code C over F q stores a file F consisting of αk symbols over F q among n nodes, each storing α symbols, in such a way that:• the file F can be recovered by downloading the content of any k of the n nodes; and • the content of any failed node can be reconstructed by accessing any d of the remaining n − 1 nodes and downloading α/(d−k+1) symbols from each of these nodes. In practice, the file F is typically available in uncoded form on some k of the n nodes, known as systematic nodes, and the defining node-repair condition above can be relaxed to requiring the optimal repair bandwidth for systematic nodes only. Such codes are called systematic-repair MSR codes.Unfortunately, finite-α constructions of [n, k, d] MSR codes are known only for certain special cases: either low rate, namely k/n 0.5, or high repair connectivity, namely d = n − 1. Our main result in this paper is a finite-α construction of systematic-repair [n, k, d] MSR codes for all possible values of parameters n, k, d. We also introduce a generalized construction for [n, k] MSR codes to achieve the optimal repair bandwidth for all values of d simultaneously.

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Section: A Previous Workmentioning

confidence: 99%

Minimum Storage Regenerating Codes for All Parameters

Goparaju

Fazeli

Vardy

2017

IEEE Trans. Inform. Theory

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“…Towards constructing high-rate exact-repairable MDS codes with optimal repair-bandwidth, Cadambe et al [3] show the existence of such codes when subpacketization level approaches infinity. Motivated by this result, the problem of designing high-rate exactrepairable MDS codes with finite sub-packetization level and optimal repair bandwidth is explored in [2,8,17,22,23,28,32,34,36] and references therein.…”

Section: Background and Related Workmentioning

confidence: 99%

“…The problem of constructing MDS codes with optimal repair-bandwidth (cf. (1.1)) has been explored in [3,8,17,18,22,23,28,36] and references therein. Note that as the number of the nodes contacted during the repair process d gets larger, the optimal repair bandwidth defined by the cut-set bound becomes significantly smaller than the naive repair bandwidth of k symbols (over F) or k symbols (over B).…”

Section: Introductionmentioning

confidence: 99%

MDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth

Rawat

Tamo

Guruswami

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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An (n, M ) vector code C ⊆ F n is a collection of M codewords where n elements (from the field F) in each of the codewords are referred to as code blocks. Assuming that F ∼ = B , the code blocks are treated as -length vectors over the base field B. Equivalently, the code is said to have the sub-packetization level . This paper addresses the problem of constructing MDS vector codes which enable exact reconstruction of each code block by downloading small amount of information from the remaining code blocks. The repair bandwidth of a code measures the information flow from the remaining code blocks during the reconstruction of a single code block. This problem naturally arises in the context of distributed storage systems as the node repair problem [4]. Assuming that M = |B| k , the repair bandwidth of an MDS vector code is lower bounded by (n − 1)/(n − k) · symbols (over the base field B) which is also referred to as the cut-set bound [4]. For all values of n and k, the MDS vector codes that attain the cut-set bound with the sub-packetization level = (n − k) n/(n−k) are known in the literature [23,36].This paper presents a construction for MDS vector codes which simultaneously ensures both small repair bandwidth and small sub-packetization level. The obtained codes have the smallest possible sub-packetization level = O(n − k) for an MDS vector code and the repair bandwidth which is at most twice the cut-set bound. The paper then generalizes this code construction so that the repair bandwidth of the obtained codes approach the cut-set bound at the cost of increased sub-packetization level. The constructions presented in this paper give MDS vector codes which are linear over the base field B.

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“…In (8,9), information is measured in units of log 2 (|F|) bits. The collection of random variables {S…”

Section: A the Repair Matrix And The Constraints Imposed By Exact-rementioning

confidence: 99%

Outer bounds on the storage-repair bandwidth trade-off of exact-repair regenerating codes

Sasidharan

Prakash

Krishnan

et al. 2016

IJICOT

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In this paper three outer bounds on the storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set {(n, k, d), (α, β)} under the exact-repair (ER) setting are presented. The tradeoff under the functionalrepair (FR) setting was settled in the seminal work of Dimakis et al. that introduced the framework of regenerating codes as well as a subsequent paper by Wu. While it is known that the ER tradeoff coincides with the FR tradeoff at the extreme points of the tradeoff, known respectively as the minimum-storage-regenerating (MSR) and minimumbandwidth-regenerating (MBR) points, its characterization on the interior points remains open.The first outer bound presented here termed as the repair-matrix bound, in conjunction with a recent code construction known as improved layered codes characterizes the normalized ER tradeoff for the case of (n, k = 3, d = n − 1). The repair-matrix bound is derived by building on top of the techniques introduced by Shah et al. and applies to every parameter set (n, k, d). It was earlier proved by Tian that the ER tradeoff lies strictly away from the FR tradeoff for the specific case (n = 4, k = 3, d = 3). The repair-matrix bound shows that a non-vanishing gap exists between the ER and FR tradeoffs for every parameter set (n, k, d).The second outer bound builds upon a bound due to Mohajer and Tandon and improves the bound using the very same techniques introduced in the Mohajer-Tandon paper and for this reason, is termed here as the improved Mohajer-Tandon bound. While for d = k the improved Mohajer-Tandon bound performs on par with the MohajerTandon bound, for d > k there is a significant improvement in the region of the tradeoff away from the MSR point. In the vicinity of the MSR point however, the repair-matrix bound outperforms the improved Mohajer-Tandon bound.In the third and final result, we restrict our focus to linear codes, and present an outer bound for the normalized ER tradeoff applicable to linear codes for the case k = d. In conjunction with the well-known class of layered codes, our third outer bound characterizes the normalized ER tradeoff in the case of linear codes for the case k = d = n−1. This bound is derived by analyzing the rank-structure of a parity-check matrix for a linear ER code.

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Asymptotic Interference Alignment for Optimal Repair of MDS Codes in Distributed Storage

Cited by 194 publications

References 28 publications

Minimum Storage Regenerating Codes for All Parameters

Minimum Storage Regenerating Codes for All Parameters

MDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth

Outer bounds on the storage-repair bandwidth trade-off of exact-repair regenerating codes

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