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]: Distributed storage systems provide large-scale re- liable data storage services by spreading redundancy across a large group of storage nodes. In such big systems, node failures take place on a regular basis. When a node fails or leaves the system, to maintain the same level of redundancy, it is expected to regenerate the redundant data at a replacement node as soon as possible. Previous studies aim to minimize the network traffic in the regeneration process, but in practical networks, where link capacities vary in a wide range, minimizing network traffic does not always mean minimizing regeneration time. Considering the heterogeneous link capacities, proposed a tree-structured regeneration scheme, called RCTREE, to bypass the low-capacitated link encountered in direct transmissions. However, we find that RCTREE may rapidly lose data integrity after several regenerations. In this paper, we reconsider the problem of minimizing regeneration time in networks with heterogeneous link capacities. We derive the minimum amount of data to be transmitted through each link to preserve data integrity. We prove that building an optimal regeneration tree is NP-complete and propose a heuristic algorithm for a near-optimal solution. We further introduce a flexible regeneration scheme, which allows providers to generate different amount of coded data. Simulation results show that the flexible tree-structured regeneration scheme can reduce the regeneration time significantly.

[2]: In the distributed storage setting introduced by , B units of data are stored across n nodes in the network in such a way that the data can be recovered by connecting to any k nodes. Additionally one can repair a failed node by connecting to any d nodes while downloading at most β units of data from each node. In this paper, we introduce a flexible framework in which the data can be recovered by connecting to any number of nodes as long as the total amount of data downloaded is at least B. Similarly, regeneration of a failed node is possible if the new node connects to the network using links whose individual capacity is bounded above by β max and whose sum capacity equals or exceeds a predetermined parameter γ. In this flexible setting, we obtain the cut-set lower bound on the repair bandwidth along with a constructive proof for the existence of codes meeting this bound for all values of the parameters. An explicit code construction is provided which is optimal in certain parameter regimes.

[3]: Distributed storage systems provide large-scale reliable data storage by storing a certain degree of redundancy in a decentralized fashion on a group of storage nodes. To recover from data losses due to the instability of these nodes, whenever a node leaves the system, additional redundancy should be regenerated to compensate such losses. In this context, the general objective is to minimize the volume of actual network traffic caused by such regenerations. A class of codes, called regenerating codes, has been proposed to achieve an optimal trade-off curve between the amount of storage space required for storing redundancy and the network traffic during the regeneration. In this paper, we jointly consider the choices of regenerating codes and network topologies. We propose a new design, referred to as RCTREE, that combines the advantage of regenerating codes with a tree-structured regeneration topology. Our focus is the efficient utilization of network links, in addition to the reduction of the regeneration traffic. With the extensive analysis and quantitative evaluations, we show that RCTREE is able to achieve a both fast and stable regeneration, even with departures of storage nodes during the regeneration.

[4]: The capacity of heterogeneous distributed storage systems under repair dynamics is studied. Examples of these systems include peer-to-peer storage clouds, wireless, and Internet caching systems. Nodes in a heterogeneous system can have different storage capacities and different repair bandwidths. Lower and upper bounds on the system capacity are given. These bounds depend on either the average resources per node, or on a detailed knowledge of the node characteristics. Moreover, the case in which nodes may be compromised by an adversary (passive or active) is addressed and bounds on the secure capacity of the system are derived. One implication of these new results is that symmetric repair maximizes the capacity of a homogeneous system, which justifies the model widely used in the literature.

[5]: Distributed storage systems are mainly justified due to the limited amount of storage capacity and improving the reliability through distributing data over multiple storage nodes. On the other hand, it may happen the data is stored in unreliable nodes, while it is desirable the end user to have a reliable access to the stored data. So, in an event that a node is damaged, to prevent the system reliability to regress, it is necessary to regenerate a new node with the same amount of stored data as the damaged node to retain the number of storage nodes, thereby having the previous reliability. This requires the new node to connect to some of existing nodes and downloads the required information, thereby occupying some bandwidth, called the repair bandwidth. On the other hand, it is more likely the cost of downloading varies across different nodes. This paper aims at investigating the theoretical cost-bandwidth tradeoff, and more importantly, it is demonstrated that any point on this curve can be achieved through the use of the so called generalized regenerating codes which is an enhancement of the regenerating codes introduced by [1].
Flexible RC [2] is designed for heterogeneous storage systems that can achieve the lower bound of repair bandwidth. Combined with tree-structured regeneration topology, it is shown that RC can further save the network bandwidth [3] [1] . Some studies [5] [4] focus on the capacity bound for a heterogeneous model. However, all the studies do not distinguish the costs between intra-rack and cross-rack communications in data centers.