In a distributed storage system, recovering from multiple failures is a critical and frequent task that is crucial for maintaining the system's reliability and fault-tolerance. In this work, we focus on the problem of repairing multiple failures in a centralized way, which can be desirable in many data storage configurations, and we show that a significant repair traffic reduction is possible. First, the fundamental tradeoff between the repair bandwidth and the storage size for functional repair is established. Using a graph-theoretic formulation, the optimal tradeoff is identified as the solution to an integer optimization problem, for which a closed-form expression is derived. Expressions of the extreme points, namely the minimum storage multi-node repair (MSMR) and minimum bandwidth multi-node repair (MBMR) points, are obtained. Second, we describe a general framework for converting single erasure minimum storage regenerating codes to MSMR codes. The repair strategy for @math failures is similar to that for single failure, however certain extra requirements need to be satisfied by the repairing functions for single failure. For illustration, the framework is applied to product-matrix codes and interference alignment codes. Furthermore, we prove that the functional MBMR point is not achievable for linear exact repair codes. We also show that exact-repair minimum bandwidth cooperative repair (MBCR) codes achieve an interior point, that lies near the MBMR point, when @math , @math being the minimum number of nodes needed to reconstruct the entire data. Finally, for @math and @math , where @math is the number of helper nodes during repair, we show that the functional repair tradeoff is not achievable under exact repair, except for maybe a small portion near the MSMR point, which parallels the results for single erasure repair by
Cooperative regenerating codes (also known as coordinated regenerating codes) have been studied to address the repair of multiple erasures @cite_12 @cite_23 in a distributed manner. In this framework, each replacement node downloads information from @math helpers in the first stage. Then, the replacement nodes exchange information between themselves before regenerating the lost nodes. Cooperative regenerating codes that achieve the extreme points on the cooperative tradeoff have been developed; namely, minimum storage cooperative regenerating (MSCR) codes @cite_29 @cite_23 @cite_11 and minimum bandwidth cooperative regeneration (MBCR) codes @cite_48 .
@cite_48 @cite_29 @cite_23 @cite_12 @cite_11
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We give an explicit construction of exact cooperative regenerating codes at the MBCR (minimum bandwidth cooperative regeneration) point. Before the paper, the only known explicit MBCR codes are given with parameters n = d + r and d = k, while our construction applies to all possible values of n, k, d, r. The code has a brief expression in the polynomial form and the data reconstruction is accomplished by bivariate polynomial interpolation. It is a scalar code and operates over a finite field of size q ≥ n. Besides, we establish several subspace properties for linear exact MBCR codes. Based on these properties we prove that linear exact MBCR codes cannot achieve repair-by-transfer. In order to provide high data reliability, distributed storage systems disperse data with redundancy to multiple storage nodes. Regenerating codes is a new class of erasure codes to introduce redundancy for the purpose of improving the data repair performance in distributed storage. Most of the studies on regenerating codes focus on the single-failure recovery, but it is not uncommon to see two or more node failures at the same time in large storage networks. To exploit the opportunity of repairing multiple failed nodes simultaneously, a cooperative repair mechanism, in the sense that the nodes to be repaired can exchange data among themselves, is investigated. A lower bound on the repair-bandwidth for cooperative repair is derived and a construction of a family of exact cooperative regenerating codes matching this lower bound is presented. 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. We show that cooperative model is stronger than the centralized one, in the sense that MDS codes with optimal repair bandwidth under the former model have the same property under the latter one. In this paper we present explicit constructions of MDS codes with optimal cooperative repair for all possible parameters. More precisely, given any @math such that @math we construct explicit @math MDS codes over the field @math of size @math that can optimally repair any @math erasures from any @math 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 shows that two rounds of communication in cooperative repair always suffice, and that the proposed codes achieve the optimal repair bandwidth using the smallest possible number of rounds. Erasure coding techniques are used to increase the reliability of distributed storage systems while minimizing storage overhead. Also of interest is minimization of the bandwidth required to repair the system following a node failure. In a recent paper, characterize the tradeoff between the repair bandwidth and the amount of data stored per node. They also prove the existence of regenerating codes that achieve this tradeoff. In this paper, we introduce Exact Regenerating Codes, which are regenerating codes possessing the additional property of being able to duplicate the data stored at a failed node. Such codes require low processing and communication overheads, making the system practical and easy to maintain. Explicit construction of exact regenerating codes is provided for the minimum bandwidth point on the storage-repair bandwidth tradeoff, relevant to distributed-mail-server applications. A subspace based approach is provided and shown to yield necessary and sufficient conditions on a linear code to possess the exact regeneration property as well as prove the uniqueness of our construction. Also included in the paper, is an explicit construction of regenerating codes for the minimum storage point for parameters relevant to storage in peer-to-peer systems. This construction supports a variable number of nodes and can handle multiple, simultaneous node failures. All constructions given in the paper are of low complexity, requiring low field size in particular.