In large data centers, storage nodes are organized in racks, and the cross-rack transmission dominates the bandwidth cost. For the repair of single node failures, codes achieving the tradeoff between the storage redundancy and cross-rack repair bandwidth are called rack-aware regenerating codes (RRCs). In this work, we give the first explicit construction of RRCs with the minimum repair bandwidth (i.e., the cross-rack bandwidth equals the storage size per node). Our construction applies to all admissible parameters and has the lowest sub-packetization level. Moreover, the underlying finite fields are of size comparable to the number of storage nodes, which makes our codes more implementable in practice. Finally, for the convenience of practical use, we also establish a transformation to convert our codes into systematic codes.
In distributed storage systems, cooperative regenerating codes tradeoff storage for repair bandwidth in the case of multiple node failures. In rack-aware distributed storage systems, there is no cost associated with transferring symbols within a rack. Hence, the repair bandwidth will only take into account cross-rack transfer. Rack-aware regenerating codes for the case of single node failures have been studied and their repair bandwidth tradeoff characterized. In this paper, we consider the framework of rack-aware cooperative regenerating codes for the case of multiple node failures where the …
Erasure coding is widely used for massive storage in data centers to achieve high fault tolerance and low storage redundancy. Since the cross-rack communication cost is often high, it is critical to design erasure codes that minimize the cross-rack repair bandwidth during failure repair. In this paper, we analyze the optimal trade-off between storage redundancy and cross-rack repair bandwidth specifically for data centers, subject to the condition that the original data can be reconstructed from a sufficient number of any non-failed nodes. We characterize the optimal trade-off curve under func…
We consider the rack-aware storage system where $n=\bar{n}u$ nodes are organized in $\bar{n}$ racks each containing $u$ nodes, and any $k=\bar{k}u+u_0~(0\leq u_0<u)$ nodes can retrieve the original data file. More importantly, the cross-rack communication cost is much more expensive than the intra-rack communication cost, so that the latter is usually neglected in the system bandwidth. The MSRR (minimum storage rack-aware regenerating) code is an important variation of regenerating codes that achieves the optimal repair bandwidth for single node failures in the rack-aware model. However, expli…
We consider the rack-aware storage system where \(n\) nodes are organized in \(\bar{n}\) racks each containing \(u\) nodes, and any \(k\) nodes can retrieve the stored file. Moreover, any single node erasure can be recovered by downloading data from \(\bar{d}\) helper racks as well as the remaining \(u\!-\!1\) nodes in the same rack. Previous work mostly focuses on minimizing the cross-rack repair bandwidth under the condition \(\bar{d}\geq \bar{k}\), where \(\bar{k}=\lfloor\frac{k}{u}\rfloor\). However, \(\bar{d}\geq \bar{k}\) is not an intrinsic condition for the rack-aware storage model. In…
In a modern distributed storage system, storage nodes are organized in racks, and the cross-rack communication dominates the system bandwidth. In We study the rack-aware storage system where all storage nodes are organized in racks and within each rack the nodes can communicate freely without taxing the system bandwidth. Rack-aware regenerating codes (RRCs) were proposed for minimizing the repair bandwidth for single erasures. In the initial setting of RRCs, the repair of a single node requires the participation of all the remaining nodes in the rack containing the failed node as well as a lar…
In modern practical data centers, storage nodes are usually organized into equally sized groups, which is called racks. The cost of cross-rack communication is much more expensive compared with the intra-rack communication cost. The codes for this system are called rack-aware regenerating codes. Similar to standard minimum storage regenerating (MSR) codes, it is a challenging task to construct minimum storage rack-aware regenerating (MSRR) codes achieving the cut-set bound. The known constructions of MSRR codes achieving the cut-set bound give codes with alphabet size $q$ exponential in the co…
The minimum storage rack-aware regenerating (MSRR) code is a variation of regenerating codes that achieves the optimal repair bandwidth for a single node failure in the rack-aware model. The authors in~\cite{Chen-Barg2019} and~\cite{Zhou-Zhang2021} provided explicit constructions of MSRR codes for all parameters to repair a single failed node. This paper generalizes the results in~\cite{Chen-Barg2019} to the case of multiple node failures. We propose a class of MDS array codes and scalar Reed-Solomon (RS) codes, and show that these codes have optimal repair bandwidth and error resilient capabi…
The Storage-Repair-Bandwidth Trade-off of Exact Repair Linear Regenerating Codes for the Case $d = k = n-1$
本文研究了精确修复线性再生码的设定问题。在这种设置下，我们推导出一个新的存储-修复-带宽权衡的情况下，当$d = k = n -1$，其中$(n, k, d)$是参数的再生代码，其通常的含义。结合Tian et. [1]，我们证明了当$d = k = n -1$ .通过分析线性再生码的对偶码，导出了新的外界。
A code construction and repair scheme for optimal functional regeneration of multiple node failures is presented, which is based on stitching together short MDS codes on carefully chosen sets of points lying on a linearized polynomial. The nodes are connected wirelessly, hence all transmissions by helper nodes during a repair round are available to all the nodes being repaired. The scheme is simple and practical because of low subpacketization, low I/O cost and low computational cost. Achievability of the minimum-bandwidth regenerating (MBR) point, as well as an interior point, on the optimal …
A private information retrieval (PIR) scheme allows a user to retrieve a file from a database without revealing any information on the file being requested. As of now, PIR schemes have been proposed for several kinds of storage systems, including replicated and MDS-coded data. In this paper, the problem of constructing a PIR scheme on regenerating codes is considered. A regenerating code is a storage code whose codewords are distributed among $n$ nodes, enabling efficient storage of files, as well as low-bandwidth retrieval of files and repair of nodes. In this work, a PIR scheme on regenerati…
Decoding and repair schemes are proposed for shift-exclusive-or (shift-XOR) product-matrix (PM) regenerating codes, which outperform the existing schemes in terms of both communication and computation costs. In particular, for the shift-XOR minimum bandwidth regenerating (MBR) codes, our decoding and repair schemes have the optimal transmission bandwidth and can be implemented in-place without extra storage space for intermediate XOR results. Technically, our schemes involve an in-place algorithm for solving a system of shift-XOR equations, called \emph{shift-XOR elimination}, which does not h…
In this paper, a new repair scheme for a modified construction of MDS codes is studied. The obtained repair scheme has optimal bandwidth for multiple failed nodes under the cooperative repair model. In addition, the repair scheme has relatively low access property, where the number of data accessed is less than two times the optimal value.
Cooperative regenerating codes are regenerating codes designed to tradeoff storage for repair bandwidth in case of multiple node failures. Minimum storage cooperative regenerating (MSCR) codes are a class of cooperative regenerating codes which achieve the minimum storage point of the tradeoff. Recently, these codes have been constructed for all possible parameters $(n,k,d,h)$, where $h$ erasures are repaired by contacting any $d$ surviving nodes. However, these constructions have very large sub-packetization. $ε$-MSR codes are a class of codes introduced to tradeoff subpacketization level for…
In coding for distributed storage systems, efficient data reconstruction and repair through accessing a predefined number of arbitrarily chosen storage nodes is guaranteed by regenerating codes. Traditionally, code parameters, specially the number of helper nodes participating in a repair process, are predetermined. However, depending on the state of the system and network traffic, it is desirable to adapt such parameters accordingly in order to minimize the cost of repair. In this work a class of regenerating codes with minimum storage is introduced that can simultaneously operate at the opti…
The problem of exact-repair regenerating codes against eavesdropping attack is studied. The eavesdropping model we consider is that the eavesdropper has the capability to observe the data involved in the repair of a subset of $\ell$ nodes. An $(n,k,d,\ell)$ secure exact-repair regenerating code is an $(n,k,d)$ exact-repair regenerating code that is secure under this eavesdropping model. It has been shown that for some parameters $(n,k,d,\ell)$, the associated optimal storage-bandwidth tradeoff curve, which has one corner point, can be determined. The focus of this paper is on characterizing su…
We continue our study of regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this problem, the failed node downloads the information stored at a subset of vertices of the graph for the purpose of recovering the lost data. This information is moved across the network, and the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. This problem was formulated in our recent work (IEEE IT Transactions, May 2022) where we showed that processing of the information at the intermediate nodes …
In a distributed storage system, recovering from multiple failures is a critical and frequent task that is crucial for maintaining the system's reliability and fault-tolerance. In this work, we focus on the problem of repairing multiple failures in a centralized way, which can be desirable in many data storage configurations, and we show that a significant repair traffic reduction is possible. First, the fundamental tradeoff between the repair bandwidth and the storage size for functional repair is established. Using a graph-theoretic formulation, the optimal tradeoff is identified as the solu…
