Trace Repair Never Loses to Classical Repair: Exact and Explicit Helper Nodes Selection

TL;DR

This paper tackles the problem of bandwidth-efficient single-erasure repair for Reed-Solomon codes in distributed storage by unifying the Guruswami–Wootters trace-repair framework with Lin's zero-forcing and Liu et al.'s subspace dependencies. It introduces Trace-Repair Compatible polynomials and establishes an exact dimension relationship with the subspace $\mathcal{W}_{k,\mathcal{S}}$, enabling an explicit optimization (Algorithm O) over the exclusion set $\mathcal{S}$ and cyclotomic cosets. The authors provide an explicit construction of helper nodes and prove a universal bandwidth bound of $k\log|\mathbb{F}|$ bits for trace-repair applicable scenarios, improving over classical repair for all $k\le n-q^{t-1}$. Their contributions include precise dimension formulas, an optimization framework, and practical procedures that outperform state-of-the-art schemes across several finite-field regimes. Overall, the work delivers both tight theoretical guarantees and implementable repair protocols for RS codes in distributed storage.

Abstract

Repairing Reed-Solomon codes with low bandwidth is a central challenge in distributed storage. Following the trace-repair framework of Guruswami and Wootters (2017), recent works by Lin (2023) and Liu-Wan-Xing (2024) provided significant improvements in bandwidth using two distinct ideas. Lin constructed a trace-repair scheme that requires no contribution from a set of predetermined nodes $\mathscr{S}$, while Liu-Wan-Xing identified linear dependencies among the downloaded traces, relating the number of dependent traces to the dimension of a subspace $\mathscr{W}_k$. In this work, we fully utilize and unify these ideas. We compute the exact dimension of $\mathscr{W}_{k,\mathscr{S}}$ (a generalization of $\mathscr{W}_k$). We identify the trade-off between the set size $|\mathscr{S}|$ and the dimension $\dim(\mathscr{W}_{k,\mathscr{S}})$. We provide an algorithm to find the combination that results in the lowest bandwidth. Furthermore, we provide an explicit choice of the helper nodes for the repair. Finally, we prove that our optimized scheme never loses to the classical repair scheme, establishing a bandwidth guarantee of at most $k\log|\mathbb{F}|$ bits for all dimension $k$ and field $\mathbb{F}$, whenever the trace repair is applicable.

Figures (3)

• Figure 1: The plots illustrate the repair bandwidth (in base field symbols) required for varying $k$. This shows empirically that our scheme never requires more bandwidth than the classical scheme (gray circles) or the schemes by Lin Lin2023 and Liu et al. Liu2024.
• Figure 2: Visual comparison of the repair bandwidth as the pruning progresses for various $k$ values with $q = 4$ and $t = 5$. The horizontal axis represents the number of cosets pruned from $\Xi_k^*$. The vertical axis represents the resulting repair bandwidth, while the red dot represents the minimum repair bandwidth. We observe that the graph is shifted as $k$ increases.
• Figure 3: Visual comparison of the repair bandwidth as the pruning progresses for a fixed $k = 10$ and $\mathbb{F} = {\mathrm{GF}}(256)$ but varying $t$. The horizontal axis represents the number of cosets pruned from $\Xi_k^*$. The vertical axis represents the resulting repair bandwidth, while the red dot represents the minimum repair bandwidth. We observe that the graph looks more convex-like as $|\mathbb{B}|$ increases.

Theorems & Definitions (26)

• Theorem 1
• Definition 1
• Theorem 2: Dau and Milenkovic Dau2017
• Theorem 3: Liu et al. Liu2024
• Definition 2: Trace-Repair Compatible Polynomial
• Theorem 4
• proof
• proof : Proof of Theorem \ref{['thm:main_theorem']}
• Remark 3
• Definition 4