TY - JOUR
T1 - The Repair Problem for Reed-Solomon Codes
T2 - Optimal Repair of Single and Multiple Erasures with Almost Optimal Node Size
AU - Tamo, Itzhak
AU - Ye, Min
AU - Barg, Alexander
N1 - Publisher Copyright:
© 1963-2012 IEEE.
PY - 2019/5
Y1 - 2019/5
N2 - 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\geqslant 1 failed nodes for an (n,k=n-r) maximum distance separable (MDS) code using d helper nodes is at least dhl/(d+h-k) , 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 the families of RS codes that achieve the cut-set bound for repair of one or several nodes. In the single-node case, we present the RS codes of length n over the field {\mathbb F}-{q^{l}}, l=\exp ((1+o(1))n\log n) 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=1,2, {\dots },r failed nodes from any subset of d helper nodes, k\leqslant d\leqslant 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.
AB - 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\geqslant 1 failed nodes for an (n,k=n-r) maximum distance separable (MDS) code using d helper nodes is at least dhl/(d+h-k) , 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 the families of RS codes that achieve the cut-set bound for repair of one or several nodes. In the single-node case, we present the RS codes of length n over the field {\mathbb F}-{q^{l}}, l=\exp ((1+o(1))n\log n) 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=1,2, {\dots },r failed nodes from any subset of d helper nodes, k\leqslant d\leqslant 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.
KW - Distributed storage
KW - MSR codes
KW - Reed-Solomon codes
KW - cut-set bound
KW - multiple-node repair
KW - regenerating codes
UR - http://www.scopus.com/inward/record.url?scp=85057827586&partnerID=8YFLogxK
U2 - 10.1109/TIT.2018.2884075
DO - 10.1109/TIT.2018.2884075
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AN - SCOPUS:85057827586
SN - 0018-9448
VL - 65
SP - 2673
EP - 2695
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 5
M1 - 8552670
ER -