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A Transformation of Repairing Reed-Solomon Codes from Rack-Aware Storage Model to Homogeneous Storage Model

Yumeng Yang, Han Cai, Xiaohu Tang

TL;DR

This work tackles repairing Reed-Solomon codes in rack-aware storage by introducing a universal repair framework that uses good polynomials to place code symbols into racks, thereby transforming rack-aware repairs into equivalent repairs under the homogeneous storage model. The authors connect residue-coefficient representations to short RS codes so that existing homogeneous-model repair schemes can be leveraged for multi-node failures within and across racks. They extend and reinterpret known rack-aware RS constructions and introduce new codes based on three classes of TB14 good polynomials, providing bandwidth analyses and explicit repair procedures. The framework enables parallelizable repair, clarifies when single-rack repairs can achieve optimal cut-set bounds, and opens the path to multi-rack repair optimality questions in rack-aware setups with practical implications for data-center storage efficiency.

Abstract

In this paper, we address the node repair problem of Reed-Solomon (RS) coded distributed storage systems. Specifically, to overcome the challenges of multiple-node failures of RS codes under the rack-aware storage model, we employ good polynomials to guide the placement of the conventional RS codes into racks and then propose a novel repair framework for the resultant rack-aware RS codes, which can transform its repair to that under the homogeneous storage model. As applications of our repair framework, firstly we present the repair scheme of multiple-node failures for some existing constructions, which only have non-trivial solutions for repairing a single-node failure before. Secondly, we deduce several new constructions of rack-aware RS codes supporting the repair of multiple-node failures within a single rack and across multiple racks respectively.

A Transformation of Repairing Reed-Solomon Codes from Rack-Aware Storage Model to Homogeneous Storage Model

TL;DR

This work tackles repairing Reed-Solomon codes in rack-aware storage by introducing a universal repair framework that uses good polynomials to place code symbols into racks, thereby transforming rack-aware repairs into equivalent repairs under the homogeneous storage model. The authors connect residue-coefficient representations to short RS codes so that existing homogeneous-model repair schemes can be leveraged for multi-node failures within and across racks. They extend and reinterpret known rack-aware RS constructions and introduce new codes based on three classes of TB14 good polynomials, providing bandwidth analyses and explicit repair procedures. The framework enables parallelizable repair, clarifies when single-rack repairs can achieve optimal cut-set bounds, and opens the path to multi-rack repair optimality questions in rack-aware setups with practical implications for data-center storage efficiency.

Abstract

In this paper, we address the node repair problem of Reed-Solomon (RS) coded distributed storage systems. Specifically, to overcome the challenges of multiple-node failures of RS codes under the rack-aware storage model, we employ good polynomials to guide the placement of the conventional RS codes into racks and then propose a novel repair framework for the resultant rack-aware RS codes, which can transform its repair to that under the homogeneous storage model. As applications of our repair framework, firstly we present the repair scheme of multiple-node failures for some existing constructions, which only have non-trivial solutions for repairing a single-node failure before. Secondly, we deduce several new constructions of rack-aware RS codes supporting the repair of multiple-node failures within a single rack and across multiple racks respectively.
Paper Structure (15 sections, 6 theorems, 31 equations, 2 figures, 2 tables)

This paper contains 15 sections, 6 theorems, 31 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and Notations
  4. Repairing Reed-Solomon Codes under the Homogeneous Storage Model
  5. Single Failure
  6. Multiple Failures
  7. Repairing Reed-Solomon Codes under the Rack-Aware Storage Model
  8. A Repair Framework of Rack-Aware Reed-Solomon Codes
  9. Rack-Aware Reed-Solomon Codes via Good Polynomials
  10. Repair of Rack-Aware Reed-Solomon Codes
  11. Some Explicit Rack-Aware Reed-Solomon Codes by means of good polynomials
  12. The Repair of Single-Rack Failure
  13. Interpretation of Known Constructions
  14. New Constructions of Rack-Aware Reed-Solomon Codes
  15. Conclusion

Key Result

Lemma 1

Let $f(x)$ and $h(x)$ respectively be two polynomials of degree $k-1$ and $u$ over ${\mathbb F}_{q^t}$. Set $s=\lceil {k\over u}\rceil$. For any $y\in {\mathbb F}_{q^t}$, denote the residue polynomial $f(x)\bmod\, (h(x)-y)$ by Then, $H_{j}(y)$ is a polynomial of $deg(H_{j}(y))\leq s-1$.

Figures (2)

  • Figure 1: The figure shows the procedure of our repair framework.
  • Figure 2: An example of the single-rack failure.

Theorems & Definitions (19)

  • Definition 1: Reed-Solomon Code
  • Definition 2: Dual Code
  • Definition 3: Rack-Aware Reed-Solomon Code
  • Definition 4: Good Polynomial TB14
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 9 more