Centralized Multi-Node Repair Regenerating Codes

Abstract

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 e 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 k ≡ 1 mod e, k being the minimum number of nodes needed to reconstruct the entire data. Finally, for k > 2e, e | k and e | d, where d 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 Shah et al.

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

“…Therefore, we transmit 100 symbols in F 1 , which can be normalized to b (n−1)ℓ = 0.4545. Compared with the scheme in [16], which need ℓ = 2.4 × 10 19 and b (n−1)ℓ = 0.25, we provide a tradeoff between ℓ and b. It should be noted that in this example, the RS(12, 8) code needs to be extended from an RS(3, −1) code, which does not exist.…”

Determinant Codes With Helper-Independent Repair for Single and Multiple Failures

“…The next theorem shows that the repair bandwidth for multiple failures can be further reduced in the centralized repair setting [4], [18], by a sequential repair mechanism, and exploiting the repair symbols contributed by already repaired failed nodes which can act as helpers.…”

“…This requires a repair bandwidth from each helper node that scales linearly with the number of failures. There are two types of repair for multiple failures studied in the literature [18]: (i) centralized regenerating codes and (ii) cooperative regenerating codes. In centralized regenerating codes, a single data center is responsible for the repair of all failed nodes.…”

“…Our codes are not restricted only to the extreme points of the trade-off and can operate at any intermediate point on the optimum trade-off. A similar problem is studied in [18], where a class of codes is introduced to operate at the intermediate points of the trade-off, with an improved repair bandwidth for multiple failures. However, this improvement is obtained at the price of degradation of the system's storage capacity as n (the total number of nodes) increases.…”

Code constructions for multi-node exact repair in distributed storage

“…In[31], it was shown that for e 2, when k is a multiple of e,β MBCR =β MBMR ,ᾱ MBCR >ᾱ MBMR ; otherwise, β MBCR >β MBMR .Zorgui M, et al Sci China Inf Sci…”