On the Achievability Region of Regenerating Codes for Multiple Erasures

Zorgui, Marwen; Wang, Zhiying

doi:10.1109/isit.2018.8437474

Cited by 6 publications

(8 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To illustrate this technique, let us start from the simplest case of repairing single erasure. Returning to the pn, k, s " d`1´kq MDS code defined by the parity-check equations in (14), we observe that the proof of Lemma 3 gives a repair scheme of the first node relying on downloading a 1 s proportion of symbols from each of the d helper nodes (it also gives the µ i 's which at this point we ignore). Moreover, as already remarked, with straightforward changes to the construction we can obtain a code with optimal repair of the ith node for any given i " 1, .…”

Section: Optimal Repair Of Two Erasures From Arbitrary Number Of Hmentioning

confidence: 96%

“…These equations have the same structure as the equations in (14): v 2 here plays the role of u in (14). Only the coefficients of c 2,sv2`v1 vary with the value of v 2 while the coefficients of c i,sv2`v1 are independent of the value of v 2 for all i P rnszt2u.…”

Section: A Optimal Repair Of the First Two Nodesmentioning

confidence: 97%

“…The problem of repairing multiple erasures comes in two variations. One of them is the centralized model, where a single data center is responsible for the repair of all the failed nodes [4], [9]- [14], and the other is the cooperative model, where the failed nodes may communicate but are distinct, and the amount of data exchanged between them is included in the repair bandwidth [15]- [18]. The cut-set bounds on the repair bandwidth for multiple erasures under these two models were derived in [9] and [16] respectively.…”

Section: A Centralized and Cooperative Repair Modelsmentioning

confidence: 99%

“…Proof: This lemma follows immediately from Lemma 2. Indeed, take d " k`1 and s " 2, then there are only two groups of equations in (14), namely those for u " 0, 1. To prove the first statement of Lemma 3, consider the equations in (16) and (17).…”

Section: A Repairing the First Two Nodes From Any K`1 Helper Nodesmentioning

confidence: 99%

See 3 more Smart Citations

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

Barg

2019

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

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.

show abstract

Section: Optimal Repair Of Two Erasures From Arbitrary Number Of Hmentioning

confidence: 96%

Section: A Optimal Repair Of the First Two Nodesmentioning

confidence: 97%

Section: A Centralized and Cooperative Repair Modelsmentioning

confidence: 99%

Section: A Repairing the First Two Nodes From Any K`1 Helper Nodesmentioning

confidence: 99%

See 2 more Smart Citations

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

Barg

2019

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

show abstract

“…The ability to repair multiple failures is also obviously of interest, and this may also be studied under different models, for example, [29,30] study centralised repair (where repair is carried out in one location) and cooperative repair (where failed nodes may communicate) for multiple failures.…”

Section: Performance Measures For Frcsmentioning

confidence: 99%

Functional repair codes: a view from projective geometry

Paterson

2019

Des. Codes Cryptogr.

View full text Add to dashboard Cite

Storage codes are used to ensure reliable storage of data in distributed systems. Here we consider functional repair codes, where individual storage nodes that fail may be repaired efficiently and the ability to recover original data and to further repair failed nodes is preserved. There are two predominant approaches to repair codes: a coding theoretic approach and a vector space approach. We explore the relationship between the two and frame the later in terms of projective geometry. We find that many of the constructions proposed in the literature can be seen to arise from natural and well-studied geometric objects, and that this perspective gives a framework that provides opportunities for generalisations and new constructions that can lead to greater flexibility in trade-offs between various desirable properties. We also frame the cut-set bound obtained from network coding in terms of projective geometry.We explore the notion of strictly functional repair codes, for which there exist nodes that cannot be replaced exactly. Currently only one known example is given in the literature, due to Hollmann and Poh. We examine this phenomenon from a projective geometry point of view, and discuss how strict functionality can arise.Finally, we consider the issue that the view of a repair code as a collection of sets of vector/projective subspaces is recursive in nature and makes it hard to visualise what a collection of nodes looks like and how one might approach a construction. Here we provide another view of using directed graphs that gives us non-recursive criteria for determining whether a family of collections of subspaces constitutes a function, exact, or strictly functional repair code, which may be of use in searching for new codes with desirable properties.

show abstract

On the Sub-Packetization Size and the Repair Bandwidth of Reed-Solomon Codes

Wang

Jafarkhani

2019

IEEE Trans. Inform. Theory

Self Cite

View full text Add to dashboard Cite

Reed-Solomon (RS) codes are widely used in distributed storage systems. In this paper, we study the repair bandwidth and sub-packetization size of RS codes. The repair bandwidth is defined as the amount of transmitted information from surviving nodes to a failed node. The RS code can be viewed as a polynomial over a finite field GF (q ℓ ) evaluated at a set of points, where ℓ is called the sub-packetization size. Smaller bandwidth reduces the network traffic in distributed storage, and smaller ℓ facilitates the implementation of RS codes with lower complexity. Recently, Guruswami and Wootters proposed a repair method for RS codes when the evaluation points are the entire finite field. While the sub-packization size can be arbitrarily small, the repair bandwidth is higher than the minimum storage regenerating (MSR) bound. Tamo, Ye and Barg achieved the MSR bound but the sub-packetization size grows faster than the exponential function of the number of the evaluation points. In this work, we present code constructions and repair schemes that extend these results to accommodate different sizes of the evaluation points. In other words, we design schemes that provide points in between. These schemes provide a flexible tradeoff between the sub-packetization size and the repair bandwidth. In addition, we generalize our schemes to manage multiple failures.

show abstract

On the Achievability Region of Regenerating Codes for Multiple Erasures

Cited by 6 publications

References 30 publications

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

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

Functional repair codes: a view from projective geometry

On the Sub-Packetization Size and the Repair Bandwidth of Reed-Solomon Codes

Contact Info

Product

Resources

About