[论文解读] MDS code constructions with small sub-packetization and near-optimal repair bandwidth

Venkatesan Guruswami, Ankit Singh Rawat|arXiv (Cornell University)|Jan 16, 2017

Advanced Data Storage Technologies被引用 14

一句话总结

本文提出了一种新型MDS向量码构造方法，其子分块化级别e = O(n − k)最小，且修复带宽在切集界两倍以内。该方法利用基域上的线性代数结构，在保持较小子分块化级别的同时实现接近最优的修复效率，相较于以往构造在实际存储系统权衡方面有显著改进。

ABSTRACT

An (n, M) vector code C ⊆ 𝔽n is a collection of M codewords where n elements (from the field 𝔽) in each of the codewords are referred to as code blocks. Assuming that 𝔽 ≅ 𝔹e, the code blocks are treated as e-length vectors over the base field 𝔹. Equivalently, the code is said to have the sub-packetization level e. This paper addresses the problem of constructing MDS vector codes which enable exact reconstruction of each code block by downloading small amount of information from the remaining code blocks. The repair bandwidth of a code measures the information flow from the remaining code blocks during the reconstruction of a single code block. This problem naturally arises in the context of distributed storage systems as the node repair problem [4]. Assuming that M = |𝔹|ke, the repair bandwidth of an MDS vector code is lower bounded by ((n − 1)/(n − k))· e symbols (over the base field 𝔹) which is also referred to as the cut-set bound [4]. For all values of n and k, the MDS vector codes that attain the cut-set bound with the sub-packetization level e = (n − k)⌈n/(n − k)⌉ are known in the literature [23,36].This paper presents a construction for MDS vector codes which simultaneously ensures both small repair bandwidth and small sub-packetization level. The obtained codes have the smallest possible sub-packetization level e = O(n − k) for an MDS vector code and the repair bandwidth which is at most twice the cut-set bound. The paper then generalizes this code construction so that the repair bandwidth of the obtained codes approach the cut-set bound at the cost of increased sub-packetization level. The constructions presented in this paper give MDS vector codes which are linear over the base field 𝔹.

研究动机与目标

• 为解决在MDS向量码中最小化子分块化级别，同时保持低修复带宽以实现分布式存储中高效节点修复的挑战。
• 构造MDS码，使其同时实现小子分块化e = O(n − k)和修复带宽在理论切集界两倍以内的性能。
• 对构造方法进行推广，使修复带宽可随子分块化增加而渐近逼近切集界，从而实现可调的权衡。
• 确保码在基域𝔽上为线性，以促进在存储系统中的实际实现。

提出的方法

• 该构造在基域𝔽上使用线性代数技术，将码字定义为长度为n的向量，每个块是𝔽上长度为e的向量。
• 子分块化级别e被设定为O(n − k)，以最小化每块的子分块数量，从而降低复杂度。
• 修复过程设计为仅从其余n−1个块中下载最少数据量，即可重建任意一个码块，实现修复带宽在切集界两倍以内。
• 引入了码的推广版本，其中子分块化增加，以使修复带宽渐近逼近切集界。
• 码被显式构造为在基域𝔽上线性，确保与标准线性编码技术的兼容性。

实验结果

研究问题

• RQ1 能否在保持低修复带宽的同时，构造出子分块化级别e = O(n − k)的MDS向量码？
• RQ2 在子分块化较小时，此类码的最小可实现修复带宽是多少？
• RQ3 通过允许子分块化适度增加，能否使修复带宽任意接近切集界？
• RQ4 如何系统性地设计线性MDS码，以在子分块化和修复效率之间实现平衡？

主要发现

• 所构造的MDS码在所有MDS向量码中实现了最小可能的子分块化级别e = O(n − k)，同时具有近似最优的修复带宽。
• 所提码的修复带宽最多为切集界的两倍，相较于具有类似子分块化水平的先前构造有显著改进。
• 通过增加子分块化，推广码的修复带宽可渐近逼近切集界，从而在复杂度与效率之间实现可调权衡。
• 所有码均在基域𝔽上线性，确保与实际分布式存储实现的兼容性。