Access Versus Bandwidth in Codes for Storage

Abstract

Maximum distance separable (MDS) codes are widely used in storage systems to protect against disk (node) failures. A node is said to have capacity l over some field F, if it can store that amount of symbols of the field. An (n, k, l) MDS code uses n nodes of capacity l to store k information nodes. The MDS property guarantees the resiliency to any n − k node failures. An optimal bandwidth (resp. optimal access) MDS code communicates (resp. accesses) the minimum amount of data during the repair process of a single failed node. It was shown that this amount equals a fraction of 1/(n − k) of data stored in each node. In previous optimal bandwidth constructions, l scaled polynomially with k in codes with asymptotic rate < 1. Moreover, in constructions with a constant number of parities, i.e. rate approaches 1, l is scaled exponentially w.r.t. k. In this paper, we focus on the later case of constant number of parities n − k = r, and ask the following question: Given the capacity of a node l what is the largest number of information disks k in an optimal bandwidth (resp. access) (k + r, k, l) MDS code. We give an upper bound for the general case, and two tight bounds in the special cases of two important families of codes. Moreover, the bounds show that in some cases optimal-bandwidth code has larger k than optimal-access code, and therefore these two measures are not equivalent.

MDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth

“…Some converse results on the sub-packetization level that is necessary for an MDS code to attain the cut-set bound are presented in [9,29]. For d = n − 1, Goparaju et al [9] show that an exact-repairable MDS code that downloads the same number of symbols from each of the contacted code blocks and employs linear repair schemes satisfies the following bound on its sub-packetization level.…”

“…On the other hand, Tamo et al [29] obtain the following lower bound on the sub-packetization level of an MDS vector code which enables exact repair using repair-bytransfer schemes.…”

A Framework of Constructions of Minimal Storage Regenerating Codes With the Optimal Access/Update Property

“…Secondly, as shown in [19], such a (k + 2, k) MSR code can be transformed to a (k + 1, k − 1) MSR code in canonical form. Thus we only consider MSR codes in canonical form since the difference between the numbers of their nodes k + 2 and k + 1 is negligible.…”

“…A systematic node is said to have the optimal access property if the computation within the surviving nodes is not required during the repair procedure [20]. For some applications such as data centers, the access to information is more costly than the transmission, which may cause a bottleneck if the amount of the former is larger than that of latter [19]. Hence, an MSR code with more systematic nodes possessing the optimal access property is preferred.…”

“…The number of distinct such matrices satisfying R2 and R3 appears to be greatly limited. In [18], [19], it is shown that the largest number of systematic nodes of an MSR code with the optimal access property (resp. both the optimal access property and the optimal update property) is 2 log 2 α (resp.…”