Repairing Reed-Solomon Codes

TL;DR

This paper investigates the exact repair problem for Reed-Solomon codes in distributed storage, demonstrating their optimality in certain regimes and providing new repair schemes with reduced bandwidth.

Contribution

It characterizes MDS codes with linear repair schemes and introduces improved repair schemes for RS codes, including practical applications in large-scale systems.

Findings

Reed-Solomon codes are optimal regenerating codes in some parameter regimes.

Proposed repair schemes reduce bandwidth compared to naive algorithms.

Applied methods to a real-world (14,10)-RS code used in Facebook Hadoop cluster.

Abstract

We study the performance of Reed-Solomon (RS) codes for the \em exact repair problem \em in distributed storage. Our main result is that, in some parameter regimes, Reed-Solomon codes are optimal regenerating codes, among MDS codes with linear repair schemes. Moreover, we give a characterization of MDS codes with linear repair schemes which holds in any parameter regime, and which can be used to give non-trivial repair schemes for RS codes in other settings. More precisely, we show that for $k$-dimensional RS codes whose evaluation points are a finite field of size $n$, there are exact repair schemes with bandwidth $(n-1)\log((n-1)/(n-k))$ bits, and that this is optimal for any MDS code with a linear repair scheme. In contrast, the naive (commonly implemented) repair algorithm for this RS code has bandwidth $k\log(n)$ bits. When the entire field is used as evaluation points, the number of nodes $n$ is much larger than the number of bits per node (which is $O(\log(n))$), and so this result holds only when the degree of sub-packetization is small. However, our method applies in any parameter regime, and to illustrate this for high levels of sub-packetization we give an improved repair scheme for a specific (14,10)-RS code used in the Facebook Hadoop Analytics cluster.