A Formula for the I/O Cost of Linear Repair Schemes and Application to Reed-Solomon Codes
Zhongyan Liu, Zhifang Zhang
TL;DR
The paper addresses the I/O cost of linear repair schemes for Reed-Solomon codes in erasure-coded distributed storage, introducing a general formula that ties the I/O cost to the Hamming weight of a related $B$-linear space. This enables tractable lower bounds for full-length RS codes with two or three parities and guides the construction of repair schemes with reduced I/O, while preserving repair bandwidth. A key contribution is a scheme for full-length RS codes using affine $q$-polynomials that achieves an optimal or near-optimal I/O cost in several regimes, and demonstrates that reducing I/O cost can be achieved with limited or zero trade-offs in repair bandwidth. The results have practical impact for implementing RS codes in large-scale storage systems by lowering the data read during node repair and improving overall repair efficiency.
Abstract
Node repair is a crucial problem in erasure-code-based distributed storage systems. An important metric for repair efficiency is the I/O cost which equals the total amount of data accessed at helper nodes to repair a failed node. In this work, a general formula for computing the I/O cost of linear repair schemes is derived from a new perspective, i.e., by investigating the Hamming weight of a related linear space. Applying the formula to Reed-Solomon (RS) codes, we obtain lower bounds on the I/O cost for full-length RS codes with two and three parities. Furthermore, we build linear repair schemes for the RS codes with improved I/O cost. For full-length RS codes with two parities, our scheme meets the lower bound on the I/O cost.