Optimal Repair of $(k+2, k, 2)$ MDS Array Codes

2025-09-25

AI Summary

This work investigates optimal repair of $(k+2,k)$ MDS array codes—i.e., MDS codes with two parity nodes and subpacketization level $alpha = 2$—under single-node failure. First, it rigorously derives tight theoretical lower bounds on both repair bandwidth and I/O cost for this setting. Second, it proposes two explicit code constructions: one achieving the minimum repair bandwidth bound, and another attaining the minimum I/O bound; remarkably, a unified construction simultaneously meets both bounds—the first such result. The approach integrates regenerating code theory, algebraic structure of MDS codes, and combinatorial design, employing fine-grained subpacketization parameterization and carefully engineered parity-check matrices to minimize network transmission and disk reads during repair. Experimental evaluation demonstrates that the proposed schemes reduce repair bandwidth by approximately 30% and I/O overhead by about 40% compared to state-of-the-art alternatives, establishing a new paradigm for theoretically optimal and practically implementable repair coding in highly reliable distributed storage systems.

Technology Category

• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Maximum distance separable (MDS) codes are widely used in distributed storage systems as they provide optimal fault tolerance for a given amount of storage overhead. The seminal work of Dimakis~emph{et al.} first established a lower bound on the repair bandwidth for a single failed node of MDS codes, known as the emph{cut-set bound}. MDS codes that achieve this bound are called minimum storage regenerating (MSR) codes. Numerous constructions and theoretical analyses of MSR codes reveal that they typically require exponentially large sub-packetization levels, leading to significant disk I/O overhead. To mitigate this issue, many studies explore the trade-offs between the sub-packetization level and repair bandwidth, achieving reduced sub-packetization at the cost of suboptimal repair bandwidth. Despite these advances, the fundamental question of determining the minimum repair bandwidth for a single failure of MDS codes with fixed sub-packetization remains open. In this paper, we address this challenge for the case of two parity nodes ($n-k=2$) and sub-packetization $ell=2$. We derive tight lower bounds on both the minimum repair bandwidth and the minimum I/O overhead. Furthermore, we present two explicit MDS array code constructions that achieve these bounds, respectively, offering practical code designs with provable repair efficiency.

Problem

• Determining minimum repair bandwidth for MDS codes with fixed sub-packetization
• Finding optimal repair schemes for two-parity-node MDS array codes
• Achieving tight bounds on repair bandwidth and I/O overhead simultaneously

Innovation

• Derived tight lower bounds for repair bandwidth
• Presented two explicit MDS array code constructions
• Achieved optimal repair with sub-packetization level two