1,464 research outputs found

    Distributed Data Storage with Minimum Storage Regenerating Codes - Exact and Functional Repair are Asymptotically Equally Efficient

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    We consider a set up where a file of size M is stored in n distributed storage nodes, using an (n,k) minimum storage regenerating (MSR) code, i.e., a maximum distance separable (MDS) code that also allows efficient exact-repair of any failed node. The problem of interest in this paper is to minimize the repair bandwidth B for exact regeneration of a single failed node, i.e., the minimum data to be downloaded by a new node to replace the failed node by its exact replica. Previous work has shown that a bandwidth of B=[M(n-1)]/[k(n-k)] is necessary and sufficient for functional (not exact) regeneration. It has also been shown that if k < = max(n/2, 3), then there is no extra cost of exact regeneration over functional regeneration. The practically relevant setting of low-redundancy, i.e., k/n>1/2 remains open for k>3 and it has been shown that there is an extra bandwidth cost for exact repair over functional repair in this case. In this work, we adopt into the distributed storage context an asymptotically optimal interference alignment scheme previously proposed by Cadambe and Jafar for large wireless interference networks. With this scheme we solve the problem of repair bandwidth minimization for (n,k) exact-MSR codes for all (n,k) values including the previously open case of k > \max(n/2,3). Our main result is that, for any (n,k), and sufficiently large file sizes, there is no extra cost of exact regeneration over functional regeneration in terms of the repair bandwidth per bit of regenerated data. More precisely, we show that in the limit as M approaches infinity, the ratio B/M = (n-1)/(k(n-k))$

    On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage

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    The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. In this paper, we address bandwidth-optimal (n,k,d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to arbitrary d survivor nodes, where k<=d<=n-1. We show the existence of Exact-Repair MDS codes that achieve minimum repair bandwidth (matching the cutset lower bound) for arbitrary admissible (n,k,d), i.e., k<n and k<=d<=n-1. Our approach is based on interference alignment techniques and uses vector linear codes which allow to split symbols into arbitrarily small subsymbols.Comment: 20 pages, 6 figure

    Exact Regeneration Codes for Distributed Storage Repair Using Interference Alignment

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    The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. On one end of this spectrum of Regenerating Codes are Minimum Storage Regenerating (MSR) codes that can match the minimum storage cost of MDS codes while also significantly reducing repair bandwidth. In this paper, we describe Exact-MSR codes which allow for any failed nodes (whether they are systematic or parity nodes) to be regenerated exactly rather than only functionally or information-equivalently. We show that Exact-MSR codes come with no loss of optimality with respect to random-network-coding based MSR codes (matching the cutset-based lower bound on repair bandwidth) for the cases of: (a) k/n <= 1/2; and (b) k <= 3. Our constructive approach is based on interference alignment techniques, and, unlike the previous class of random-network-coding based approaches, we provide explicit and deterministic coding schemes that require a finite-field size of at most 2(n-k).Comment: to be submitted to IEEE Transactions on Information Theor

    Interference Alignment in Regenerating Codes for Distributed Storage: Necessity and Code Constructions

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    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

    Exact Optimized-cost Repair in Multi-hop Distributed Storage Networks

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    The problem of exact repair of a failed node in multi-hop networked distributed storage systems is considered. Contrary to the most of the current studies which model the repair process by the direct links from surviving nodes to the new node, the repair is modeled by considering the multi-hop network structure, and taking into account that there might not exist direct links from all the surviving nodes to the new node. In the repair problem of these systems, surviving nodes may cooperate to transmit the repair traffic to the new node. In this setting, we define the total number of packets transmitted between nodes as repair-cost. A lower bound of the repaircost can thus be found by cut-set bound analysis. In this paper, we show that the lower bound of the repair-cost is achievable for the exact repair of MDS codes in tandem and grid networks, thus resulting in the minimum-cost exact MDS codes. Further, two suboptimal (achievable) bounds for the large scale grid networks are proposed.Comment: (To appear in ICC 2014

    A Construction of Systematic MDS Codes with Minimum Repair Bandwidth

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    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 (n,k)(n,k)-MDS codes for 2k≤n2k\le n that achieves the minimum repair bandwidth when repairing from k+1k+1 nodes.Comment: Submitted to IEEE Transactions on Information Theory on August 14, 200
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