Repairing Schemes for Tamo-Barg Codes

TL;DR

The paper addresses efficient data repair in rack-aware storage systems when erasures exceed local repair capability. It adapts Tamo-Barg locally repairable codes to a rack-based architecture and proposes two repair schemes that minimize cross-rack bandwidth, proving optimality for at least the single-rack erasure case by extending the cut-set bound to incorporate locality. It also develops a partial-erasure framework and corresponding repair schemes, supplemented by a lower bound on partial-repair bandwidth via a partial information-flow graph. Together, these results advance practical, bandwidth-efficient repair strategies for TB-type LRCs in distributed storage environments, with explicit constructions linked to redundant residue codes and GRS codes. The work also sets groundwork for more general partial repair scenarios and higher-order rack erasure patterns, suggesting avenues for future exploration in robust, locality-aware storage systems.

Abstract

In this paper, the repair problem for erasures beyond locality in locally repairable codes is explored under a practical system setting, where a rack-aware storage system consists of racks, each containing a few parity checks. This is referred to as a rack-aware system with locality. Two repair schemes are devised to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. Additionally, a cut-set bound for locally repairable codes under the rack-aware model with locality is introduced. Using this bound, the second repair scheme is proven to be optimal. Furthermore, the partial-repair problem is considered for locally repairable codes under the rack-aware model with locality, and both repair schemes and bounds are introduced for this scenario.

Figures (7)

• Figure 1: The distributed storage system is organized with a racks-servers structure, where each rack consists of $r+\delta-1$ servers. Within each rack, $\delta-1$ parity checks are included to ensure data integrity. Additionally, to safeguard against rack erasures, $N-K$ rack parity checks (red columns) are stored.
• Figure 2: As a comparison, setting $n=18$, $k=10$, the first figure shows the rack-aware model with one erasure, the second one is corresponding to the $(18,10;6,2)$-RASL for one rack erasure and the last is for the $(18,10;6,2)$-RASL with one partial erasure, where the red nodes refer to the local parity checks.
• Figure 3: Repairing Tamo-Barg codes. The repair problem of the first column is reduced to the repair problem for the corresponding polynomial remainder. Secondly, by Lemma \ref{['lemma_F_a']}, this is equivalent to the repair problem of certain codewords within a Reed-Solomon code, where each of them suffers one erasure. Finally, the repairing of these codewords ensures the recovery of the erased column.
• Figure 4: Partial repairing of Tamo-Barg codes. Herein, we initially reduce the repair problem of failed items in the first column to the repair problem for the coefficients of the corresponding polynomial remainder. Secondly, according to Lemma \ref{['lemma_F_a']}, these coefficients can be recovered by repairing certain codewords within a Reed-Solomon code. Finally, repairing these symbols of the codewords ensures the repair of the desired components.
• Figure 5: An example of the the initial status of the information flow graph, where one possible data collector $DC$ is included.

Theorems & Definitions (38)

• Definition 1
• Definition 2
• Theorem 1: Cut-set bound, dimakis2010networkcadambe2013asymptotic
• Definition 3
• Definition 4
• Lemma 1: gopalan2012localityprakash2012optimal
• Definition 5
• Definition 6
• Remark 1
• Remark 2