Repair Locality with Multiple Erasure Tolerance (1306.4774v1)

Abstract: In distributed storage systems, erasure codes with locality $r$ is preferred because a coordinate can be recovered by accessing at most $r$ other coordinates which in turn greatly reduces the disk I/O complexity for small $r$. However, the local repair may be ineffective when some of the $r$ coordinates accessed for recovery are also erased. To overcome this problem, we propose the $(r,\delta)_c$-locality providing $\delta -1$ local repair options for a coordinate. Consequently, the repair locality $r$ can tolerate $\delta-1$ erasures in total. We derive an upper bound on the minimum distance $d$ for any linear $[n,k]$ code with information $(r,\delta)_c$-locality. For general parameters, we prove existence of the codes that attain this bound when $n\geq k(r(\delta-1)+1)$, implying tightness of this bound. Although the locality $(r,\delta)$ defined by Prakash et al provides the same level of locality and local repair tolerance as our definition, codes with $(r,\delta)_c$-locality are proved to have more advantage in the minimum distance. In particular, we construct a class of codes with all symbol $(r,\delta)_c$-locality where the gain in minimum distance is $\Omega(\sqrt{r})$ and the information rate is close to 1.

• The paper introduces a new class of erasure codes called $(r,\delta)_c$-locality codes, designed to offer $\delta-1$ local repair options for enhanced robustness in distributed storage.
• It derives a new theoretical upper bound on the minimum distance for these codes, demonstrating they can achieve significantly higher minimum distance than traditional $(r,\delta)$ codes.
• The study presents construction methods for optimal codes achieving this bound, highlighting their practical implication for fortified data resilience and improved repair efficiency.

An Insight into Repair Locality with Multiple Erasure Tolerance

The paper presented explores the optimization of distributed storage systems via erasure codes that enhance repair locality, a pivotal factor in reducing disk I/O complexity and bandwidth during data recovery processes. This paper introduces the concept of $(r,\delta)_c$-locality codes, which innovate by providing $\delta-1$ local repair options, significantly enhancing the system's robustness against failures.

Theoretical Contributions

The authors develop a comprehensive theoretical framework to assess the performance of $(r,\delta)_c$-locality codes. A major contribution is the derivation of an upper bound on the minimum distance $d$ for a linear $[n,k]$ code under this new locality definition. The bound is formulated as $d \le n - k + 1 - \mu$, where $$\mu = \lceil \frac{(k-1)(\delta-1)+1}{(r-1)(\delta-1)+1} \rceil -1$$. This establishes a direct relationship between the system's redundancy (reflected in $n$ and $k$) and its resilience (related to $d$ and $\delta$).

Results and Implications

A striking conclusion from the paper is that $(r,\delta)_c$-locality codes can outperform traditionally defined locality $(r,\delta)$ codes in terms of minimum distance. This is achieved through strategic construction methods, such as leveraging projective planes and appropriate generator matrices, allowing these codes to achieve a higher minimum distance without increasing code length. Notably, the gain in minimum distance is significant, denoted as $\Omega(\sqrt{r})$, while maintaining an information rate close to unity.

The paper provides extensive results showcasing the efficacy of codes with $(r,\delta)_c$-locality. It demonstrates the existence of a class of optimal codes attaining the proposed theoretical bound and further explores specific constructions like 'square codes', which highlight superior minimum distance at high information rates.

Practical and Theoretical Implications

Practically, the enhancement in erasure tolerance directly translates to fortified data resilience in distributed storage systems, especially prevalent in large-scale systems where node failures are common. The improvements in repair processes security and efficiency are critical in sustaining system reliability and performance.

Theoretically, the results presented in this paper extend the applicability of linear coding theories by integrating combinatorial perspectives into the tolerance evaluations. It invites further exploration into code constructions that better adapt to varying system constraints and potential failures.

Future Directions

The paper opens avenues for future research in the development and refinement of code structures, emphasizing adaptability in diverse failure scenarios. Exploring diverse combinatorial structures for constructing generator matrices and further reducing repair complexity are prospective research avenues. Additionally, integrating machine learning techniques for predicting failure patterns and dynamically adjusting code parameters could enhance the adaptability of these systems.

In sum, this paper enriches the field of distributed storage systems by proposing an advanced erasure correction methodology that aligns with contemporary demands for efficiency, resilience, and scalability.