You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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 functional repair, and propose a general family of erasure codes called rack-aware regenerating codes (RRC) , which achieve the optimal trade-off. We further propose exact repair constructions of RRC that have minimum storage redundancy and minimum cross-rack repair bandwidth, respectively. We show that (i) the minimum storage redundancy constructions support a wide range of parameters and have cross-rack repair bandwidth that is strictly less than that of the classical minimum storage regenerating codes in most cases, and (ii) the minimum cross-rack repair bandwidth constructions support all the parameters and have less cross-rack repair bandwidth than that of the minimum bandwidth regenerating codes for almost all of the parameters.

Cited Abstract(s):
[1]: We study a generalization of the setting of regenerating codes, motivated by applications to storage systems consisting of clusters of storage nodes. There are @math clusters in total, with @math nodes per cluster. A data file is coded and stored across the @math nodes, with each node storing @math symbols. For availability of data, we require that the file be retrievable by downloading the entire content from any subset of @math clusters. Nodes represent entities that can fail. We distinguish between intra-cluster and inter-cluster bandwidth (BW) costs during node repair. Node-repair in a cluster is accomplished by downloading @math symbols each from any set of @math other clusters, dubbed remote helper clusters, and also up to @math symbols each from any set of @math surviving nodes, dubbed local helper nodes, in the host cluster. We first identify the optimal trade-off between storage-overhead and inter-cluster repair-bandwidth under functional repair, and also present optimal exact-repair code constructions for a class of parameters. The new trade-off is strictly better than what is achievable via space-sharing existing coding solutions, whenever @math . We then obtain sharp lower bounds on the necessary intra-cluster repair BW to achieve optimal trade-off. Under functional repair, random linear network codes (RLNCs) simultaneously optimize usage of both inter- and intra-cluster repair BW; simulation results based on RLNCs suggest optimality of the bounds on intra-cluster repair-bandwidth. Our bounds reveal the interesting fact that, while it is beneficial to increase the number of local helper nodes @math in order to improve the storage-vs-inter-cluster-repair-BW trade-off, increasing @math not only increases intra-cluster BW in the host-cluster, but also increases the intra-cluster BW in the remote helper clusters. We also analyze resilience of the clustered storage system against passive eavesdropping by providing file-size bounds and optimal code constructions.
The closest related work to ours is by Prakash [1] . In their model, a file needs to be retrieved from a certain number of racks, and hence @math must be a multiple of number of nodes in each rack. On the other hand, our model allows a file to be retrieved from any @math nodes. Therefore, our RRC can tolerate more failure patterns than the codes in [1] . We show that the trade-off curve of RRC coincides with the optimal trade-off curve in [1] when @math is a multiple of number of nodes in each rack, yet our exact repair constructions for MSRR codes and MBRR codes can support much more parameters than that of the minimum storage codes and the minimum bandwidth codes in [1] , respectively (see for details). More importantly, the cross-rack repair bandwidth of our MSRR codes with additional parameters is strictly less than that of MSRR codes with the nearest @math that is a multiple of number of nodes in each rack (see the remark in ). Table compares our RRC with the other related work for erasure-coded data centers.