748 research outputs found
A Construction of Systematic MDS Codes with Minimum Repair Bandwidth
In a distributed storage system based on erasure coding, an important problem
is the \emph{repair problem}: If a node storing a coded piece fails, in order
to maintain the same level of reliability, we need to create a new encoded
piece and store it at a new node. This paper presents a construction of
systematic -MDS codes for that achieves the minimum repair
bandwidth when repairing from nodes.Comment: Submitted to IEEE Transactions on Information Theory on August 14,
200
Interference Alignment in Regenerating Codes for Distributed Storage: Necessity and Code Constructions
Regenerating codes are a class of recently developed codes for distributed
storage that, like Reed-Solomon codes, permit data recovery from any arbitrary
k of n nodes. However regenerating codes possess in addition, the ability to
repair a failed node by connecting to any arbitrary d nodes and downloading an
amount of data that is typically far less than the size of the data file. This
amount of download is termed the repair bandwidth. Minimum storage regenerating
(MSR) codes are a subclass of regenerating codes that require the least amount
of network storage; every such code is a maximum distance separable (MDS) code.
Further, when a replacement node stores data identical to that in the failed
node, the repair is termed as exact.
The four principal results of the paper are (a) the explicit construction of
a class of MDS codes for d = n-1 >= 2k-1 termed the MISER code, that achieves
the cut-set bound on the repair bandwidth for the exact-repair of systematic
nodes, (b) proof of the necessity of interference alignment in exact-repair MSR
codes, (c) a proof showing the impossibility of constructing linear,
exact-repair MSR codes for d < 2k-3 in the absence of symbol extension, and (d)
the construction, also explicit, of MSR codes for d = k+1. Interference
alignment (IA) is a theme that runs throughout the paper: the MISER code is
built on the principles of IA and IA is also a crucial component to the
non-existence proof for d < 2k-3. To the best of our knowledge, the
constructions presented in this paper are the first, explicit constructions of
regenerating codes that achieve the cut-set bound.Comment: 38 pages, 12 figures, submitted to the IEEE Transactions on
Information Theory;v3 - The title has been modified to better reflect the
contributions of the submission. The paper is extensively revised with
several carefully constructed figures and example
Optimal Rebuilding of Multiple Erasures in MDS Codes
MDS array codes are widely used in storage systems due to their
computationally efficient encoding and decoding procedures. An MDS code with
redundancy nodes can correct any node erasures by accessing all the
remaining information in the surviving nodes. However, in practice,
erasures is a more likely failure event, for . Hence, a natural
question is how much information do we need to access in order to rebuild
storage nodes? We define the rebuilding ratio as the fraction of remaining
information accessed during the rebuilding of erasures. In our previous
work we constructed MDS codes, called zigzag codes, that achieve the optimal
rebuilding ratio of for the rebuilding of any systematic node when ,
however, all the information needs to be accessed for the rebuilding of the
parity node erasure.
The (normalized) repair bandwidth is defined as the fraction of information
transmitted from the remaining nodes during the rebuilding process. For codes
that are not necessarily MDS, Dimakis et al. proposed the regenerating codes
framework where any erasures can be corrected by accessing some of the
remaining information, and any erasure can be rebuilt from some subsets
of surviving nodes with optimal repair bandwidth.
In this work, we study 3 questions on rebuilding of codes: (i) We show a
fundamental trade-off between the storage size of the node and the repair
bandwidth similar to the regenerating codes framework, and show that zigzag
codes achieve the optimal rebuilding ratio of for MDS codes, for any
. (ii) We construct systematic codes that achieve optimal
rebuilding ratio of , for any systematic or parity node erasure. (iii) We
present error correction algorithms for zigzag codes, and in particular
demonstrate how these codes can be corrected beyond their minimum Hamming
distances.Comment: There is an overlap of this work with our two previous submissions:
Zigzag Codes: MDS Array Codes with Optimal Rebuilding; On Codes for Optimal
Rebuilding Access. arXiv admin note: text overlap with arXiv:1112.037
An Alternate Construction of an Access-Optimal Regenerating Code with Optimal Sub-Packetization Level
Given the scale of today's distributed storage systems, the failure of an
individual node is a common phenomenon. Various metrics have been proposed to
measure the efficacy of the repair of a failed node, such as the amount of data
download needed to repair (also known as the repair bandwidth), the amount of
data accessed at the helper nodes, and the number of helper nodes contacted.
Clearly, the amount of data accessed can never be smaller than the repair
bandwidth. In the case of a help-by-transfer code, the amount of data accessed
is equal to the repair bandwidth. It follows that a help-by-transfer code
possessing optimal repair bandwidth is access optimal. The focus of the present
paper is on help-by-transfer codes that employ minimum possible bandwidth to
repair the systematic nodes and are thus access optimal for the repair of a
systematic node.
The zigzag construction by Tamo et al. in which both systematic and parity
nodes are repaired is access optimal. But the sub-packetization level required
is where is the number of parities and is the number of
systematic nodes. To date, the best known achievable sub-packetization level
for access-optimal codes is in a MISER-code-based construction by
Cadambe et al. in which only the systematic nodes are repaired and where the
location of symbols transmitted by a helper node depends only on the failed
node and is the same for all helper nodes. Under this set-up, it turns out that
this sub-packetization level cannot be improved upon. In the present paper, we
present an alternate construction under the same setup, of an access-optimal
code repairing systematic nodes, that is inspired by the zigzag code
construction and that also achieves a sub-packetization level of .Comment: To appear in National Conference on Communications 201
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