Centralized multi-node repair for minimum storage regenerating codes

The repair problem in distributed storage addresses recovery of the data encoded using an erasure code, for instance, a Reed-Solomon (RS) code. We consider the problem of repairing a single node or multiple nodes in RS-coded storage systems using the smallest possible amount of inter-nodal communication. According to the cut-set bound, communication cost of repairing h ě 1 failed nodes for an pn, k " n´rq MDS code using d helper nodes is at least dhl{pd`h´kq, where l is the size of the node. Guruswami and Wootters (2016) initiated the study of efficient repair of RS codes, showing that they can be repaired using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality.In this paper we construct families of RS codes that achieve the cutset bound for repair of one or several nodes. In the single-node case, we present RS codes of length n over the field F q l , l " exppp1`op1qqn log nq that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for scalar MDS codes to achieve the cut-set bound using linear repair schemes. For the case of multiple nodes, we construct a family of RS codes that achieve the cut-set bound universally for the repair of any h " 2, 3, . . . failed nodes from any subset of d helper nodes, k ď d ď n´h. For a fixed number of parities r the node size of the constructed codes is close to the smallest possible node size for codes with such properties.The results of Sections IV and V were presented at the 58th IEEE

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“…. , k´1uztju (˘, which contradicts (34). Thus, our assumption is wrong, and d j does not divide lcmpd i : i P t1, 2, .…”

Section: A Lower Bound On the Sub-packetization Of Scalar Linear Mmentioning

confidence: 85%

The Repair Problem for Reed–Solomon Codes: Optimal Repair of Single and Multiple Erasures With Almost Optimal Node Size

Tamo

Barg

2019

IEEE Trans. Inform. Theory

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“…. , s 2´2 , we have f pi 1 , apg 12 , uqq "`f pi 1 , apg 12 , 0qq`b 1 puq˘mod s, f pi 2 , apg 12 , uqq "`f pi 2 , apg 12 , 0qq`b 2 puq˘mod s.…”

Section: B Optimal Repair Of Any Two Erasuresmentioning

confidence: 99%

“…According to definition of the function f in (20) and the remarks made after it, we have f pi, apg 12 , 0qq " f pi, apg 12 , 1qq " f pi, apg 12 , 2qq, i P rnszti 1 , i 2 u f pi 1 , apg 12 , 0qq " f pi 1 , apg 12 , 2qq ‰ f pi 1 , apg 12 , 1qq, f pi 2 , apg 12 , 0qq " f pi 2 , apg 12 , 1qq ‰ f pi 2 , apg 12 , 2qq.…”

Section: B Repairing Any Two Erasures From Any K`1 Helper Nodesmentioning

confidence: 99%

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Cooperative Repair: Constructions of Optimal MDS Codes for All Admissible Parameters

Barg

2019

IEEE Trans. Inform. Theory

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Two widely studied models of multiple-node repair in distributed storage systems are centralized repair and cooperative repair. The centralized model assumes that all the failed nodes are recreated in one location, while the cooperative one stipulates that the failed nodes may communicate but are distinct, and the amount of data exchanged between them is included in the repair bandwidth.As our first result, we prove a lower bound on the minimum bandwidth of cooperative repair. We also show that the cooperative model is stronger than the centralized one, in the sense that any MDS code with optimal repair bandwidth under the former model also has optimal bandwidth under the latter one. These results were previously known under the additional "uniform download" assumption, which is removed in our proofs.As our main result, we give explicit constructions of MDS codes with optimal cooperative repair for all possible parameters. More precisely, given any n, k, h, d such that 2 ď h ď n´d ď n´k we construct pn, kq MDS codes over the field F of size |F | ě pd`1´kqn that can optimally repair any h erasures from any d helper nodes. The repair scheme of our codes involves two rounds of communication. In the first round, each failed node downloads information from the helper nodes, and in the second one, each failed node downloads additional information from the other failed nodes. This implies that our codes achieve the optimal repair bandwidth using the smallest possible number of rounds.

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“…The codes which achieve the above cut-set bound with equality are termed as minimum storage multi-repair (MSMR) codes. The constructions of MSMR codes based on product-matrix codes and interference-alignment based codes have been investigated in [17].…”

Section: B Repair Of Multiple Erasuresmentioning

confidence: 99%

On Epsilon-MSCR Codes for Two Erasures

Devi

Lalitha

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Cooperative regenerating codes are regenerating codes designed to tradeoff storage for repair bandwidth in case of multiple node failures. Minimum storage cooperative regenerating (MSCR) codes are a class of cooperative regenerating codes which achieve the minimum storage point of the tradeoff. Recently, these codes have been constructed for all possible parameters (n, k, d, h), where h erasures are repaired by contacting any d surviving nodes. However, these constructions have very large sub-packetization. ǫ-MSR codes are a class of codes introduced to tradeoff subpacketization level for a slight increase in the repair bandwidth for the case of single node failures. We introduce the framework of ǫ-MSCR codes which allow for a similar tradeoff for the case of multiple node failures. We present a construction of ǫ-MSCR codes, which can recover from two node failures, by concatenating a class of MSCR codes and scalar linear codes. We give a repair procedure to repair the ǫ-MSCR codes in the event of two node failures and calculate the repair bandwidth for the same. We characterize the increase in repair bandwidth incurred by the method in comparison with the optimal repair bandwidth given by the cut-set bound. Finally, we show the subpacketization level of ǫ-MSCR codes scales logarithmically in the number of nodes.• Consider a function g(i 1 , i 2 ) = i2−1 2

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Centralized multi-node repair for minimum storage regenerating codes

Cited by 17 publications

References 19 publications

The Repair Problem for Reed–Solomon Codes: Optimal Repair of Single and Multiple Erasures With Almost Optimal Node Size

The Repair Problem for Reed–Solomon Codes: Optimal Repair of Single and Multiple Erasures With Almost Optimal Node Size

Cooperative Repair: Constructions of Optimal MDS Codes for All Admissible Parameters

On Epsilon-MSCR Codes for Two Erasures

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