Reducing The Sub-packetization Level of Optimal-Access Cooperative MSR Codes
Yaqian Zhang, Jingke Xu
TL;DR
The paper tackles reducing the sub-packetization $\ell$ of optimal-access cooperative MSR codes for two erasures in distributed storage. It introduces two MDS array-code building blocks, $\mathcal{C}_{\mathrm{I}}$ and $\mathcal{C}_{\mathrm{II}}$, and builds a larger code $\mathcal{C}$ by stacking these blocks across $m$ dimensions, yielding $\ell=r^{m}$ with $m=\binom{n}{2}-\lfloor n/r\rfloor(\binom{r}{2}-1)$. The authors prove that the resulting code is MDS and achieves optimal-access repair for two erasures, while covering all $\binom{n}{2}$ erasure patterns via a carefully designed parity-check structure and dimension-stacking. This yields a concrete sub-packetization reduction by a factor of $1/r^{\lfloor n/r\rfloor(\binom{r}{2}-1)}$ compared to the previous $\ell=r^{\binom{n}{2}}$, reducing disk I/O and repair complexity in practice. The work opens avenues for further reducing sub-packetization and extending the approach to more general $h>2$ erasures and $d$-dependent settings, potentially enabling robust, low-IO cooperative MSR codes in large-scale storage systems.
Abstract
Cooperative MSR codes are a kind of storage codes which enable optimal-bandwidth repair of any $h\geq2$ node erasures in a cooperative way, while retaining the minimum storage as an $[n,k]$ MDS code. Each code coordinate (node) is assumed to store an array of $\ell$ symbols, where $\ell$ is termed as sub-packetization. Large sub-packetization tends to induce high complexity, large input/output in practice. To address the disk IO capability, a cooperative MSR code is said to have optimal-access property, if during node repair, the amount of data accessed at each helper node meets a theoretical lower bound. In this paper, we focus on reducing the sub-packetization of optimal-access cooperative MSR codes with two erasures. At first, we design two crucial MDS array codes for repairing a specific repair pattern of two erasures with optimal access. Then, using the two codes as building blocks and by stacking up of the two codes for several times, we obtain an optimal-access cooperative MSR code with two erasures. The derived code has sub-packetization $\ell=r^{\binom{n}{2}-\lfloor\frac{n}{r}\rfloor(\binom{r}{2}-1)}$ where $r=n-k$, and it reduces $\ell$ by a fraction of $1/r^{\lfloor\frac{n}{r}\rfloor(\binom{r}{2}-1)}$ compared with the state of the art ($\ell=r^{\binom{n}{2}}$).