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 paper, we focus on the problem of repairing multiple failures in a centralized way, which can be desirable in many data storage configurations; furthermore, we show that a significant repair traffic reduction is possible. First, the fundamental trade-off 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 solution to an integer optimization problem, for which a closed-form expression is derived. Expressions of the extreme points, namely the minimum storage multi-node repair (MSMR) and minimum bandwidth multi-node repair (MBMR) points, are obtained. Second, we describe a general framework for converting single erasure minimum storage regenerating codes to MSMR codes. The repair strategy for $e$ failures is similar to that for a single failure; however, certain extra requirements need to be satisfied by the repairing functions for a single failure. For illustration, the framework is applied to product-matrix codes and interference alignment codes. Furthermore, we prove that the functional MBMR point is not achievable for linear exact-repair codes. We also show that the exact-repair minimum bandwidth cooperative repair codes achieve an interior point, that lies near the MBMR point, when $k \equiv 1 \mod e$ , $k$ being the minimum number of nodes needed to reconstruct the entire data. Finally, for $k> 2e, e\mid k$ , and $e \mid d$ , where $d$ is the number of helper nodes during repair, we show that the functional repair trade-off is not achievable under exact repair, except for maybe a small portion near the MSMR point, which parallels the results for single-erasure repair by Shah et al.

Centralized Multi-Node Repair Regenerating Codes

We propose a delayed parity generation technique for maximum distance separable (MDS) storage codes, for two possible applications: the first is to improve the write-speed during data intake where only a subset of the parities are initially produced and stored into the system, and the rest can be produced from the stored data during a later time of lower system load; the second is to provide better adaptivity, where a lower number of parities can be chosen initially in a storage system, and more parities can be produced when the existing ones are not sufficient to guarantee the needed reliability or performance. In both applications, it is important to reduce the data access as much as possible during the delayed parity generation procedure. For this purpose, we first identify the fundamental limit for delayed parity generation through a connection to the well-known multicast network coding problem, then provide an explicit and low-complexity code transformation that is applicable on any MDS codes to obtain optimal codes. The problem we consider is closely related to the regenerating code problem, however the proposed codes are much simpler and have a much smaller subpacketization factor than regenerating codes, and thus our result in fact shows that blindly adopting regenerating codes in these two settings is unnecessary and wasteful.

Delayed Parity Generation in MDS Storage Codes

In order to guarantee data reliability, erasure codes have been used in distributed storage systems. Nevertheless, this mechanism suffers from the repair problem that excess data are needed to repair a single failure, causing both high bandwidth consuming for the network and heavy computing load on the replacement node. To reduce repair traffic, researchers pointed out the tradeoff between storage and repair traffic and proposed regenerating codes by combining network coding. However, the combination only focuses on the storage terminal and the construction of the codes is quite complicated. Therefore, this paper further combines network coding with network structure and proposes a repair tree model based on general erasure codes to simplify the repair procedure. By decomposing repair computing and distributing it among the tree nodes, our model can mitigate the computing tension. The performance of repair tree is analyzed and evaluated by preliminary emulation. The result shows it can make about three times faster computing than conventional measure and the repair throughput is doubled if there are network bottlenecks. For proper topology, it can significantly reduce the repair traffic. We present algorithms to generate trees across the network topology. At last, we present the idea of extending repair tree to repair multiple failures.

Repair Tree: Fast Repair for Single Failure in Erasure-Coded Distributed Storage Systems

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 solution to an integer optimization problem, for which a closed-form expression is derived. Expressions of the extreme points, namely the minimum storage multi-node repair (MSMR) and minimum bandwidth multi-node repair (MBMR) points, are obtained. Second, we describe a general framework for converting single erasure minimum storage regenerating codes to MSMR codes. The repair strategy for $e$ failures is similar to that for single failure, however certain extra requirements need to be satisfied by the repairing functions for single failure. For illustration, the framework is applied to product-matrix codes and interference alignment codes. Furthermore, we prove that the functional MBMR point is not achievable for linear exact repair codes. We also show that exact-repair minimum bandwidth cooperative repair (MBCR) codes achieve an interior point, that lies near the MBMR point, when $k \equiv 1 \mod e$, $k$ being the minimum number of nodes needed to reconstruct the entire data. Finally, for $k> 2e, e\mid k$ and $e \mid d$, where $d$ is the number of helper nodes during repair, we show that the functional repair tradeoff is not achievable under exact repair, except for maybe a small portion near the MSMR point, which parallels the results for single erasure repair by Shah et al.

A method is provided that is implemented by a network device to simplify a topology graph of a network to generate a multicast distribution tree, the method to reduce the complexity of the topology graph while enabling a creation of the multicast distribution tree such that the computational complexity of generating the multicast distribution tree is reduced, the method including computing a shortest path to all nodes of the topology graph rooted at a source node S, determining a metric for each adjacency on each shortest path of the topology graph for the multicast group G, construct an (S, G) graph with only source node S, leaves and candidate replication points, and prune the (S, G) graph using a set of pruning processes to fully resolve the multicast distribution tree, where full resolution can be determined, and the first set of pruning processes if successful are known to produce a minimum cost tree.

Graph construction for computed spring multicast

The present invention relates to IP network communication technology. It provides a kind of IP multicast data layer distribution method and system. The method includes: first code original multicast data into multiple sub-flow, separately specify different shunt sub-source for every sub-flow, distribute sub-group address, create sub-group state, etc, form shunt structure. Then the sink send join news to each shunt sub-source, join each shunt sub-source sub-group and become the leaf node of each sub-tree. And then each shunt sub-source determine the intermediate node according to the port receiving join news and construct multicast distribution-tree. If the overlapping portion between the multicast distribution trees produce congestion, in the congestion nodes network coding is used to eliminate congestion and construct coding sub-tree according to the way of building multicast distribution tree from the below of the coding nodes.

Ip multicast layered distribution method and system

Self-adaptive network transmission control method and system based on TCP (Transmission Control Protocol) protocol

To reduce multiple-failure repair traffic in erasure coded storage systems, Patrick Lee et al. introduce concurrent framework-based minimal-storage regenerating codes (RGCs). The approach is simpler and more practical than the cooperative mechanism in non-fully distributed environment. This study unifies such class of codes as concurrent RGC and further studies the characteristics by analysing the cut-based information flow graph. The authors present a general storage–bandwidth tradeoff and give closed-form expressions for the points on the curve including the minimal-bandwidth point. They show that the concurrent RGC can be constructed by reforming the existing single-node RGC or multiple-node cooperative RGC. Moreover, a connection to strong-maximum distance separable is also analysed.

Concurrent regenerating codes

The present invention is applied to the technical field of IP network communications. Provided are a method and system for layered distribution of IP multicast data. The method comprises: coding a raw multicast data flow into a plurality of subflows first, then designating a shunt sub-information source for each of the subflows, allocating subgroup addresses and establishing subgroup states and so on to form a shunt structure, and then an information sink sending a Join message to each of the shunt sub-information sources and joining each of the shunt sub-information sources to become the leaf nodes of each subtree, each node determining a middle node of each subtree according to the port which receives the Join message to form a multicast distribution tree, and if congestion occurs in the overlaps among the multicast distribution trees, using network coding to eliminate the congestion at the congestion node, and constructing a coding subtree below a coding node according to a method for constructing a shunt subtree. The present invention can provide higher bandwidth and better stability than traditional multicast tree structures and can provide differentiated services for heterogeneous information sinks.

Huayu Zhang

Papers

Ip multicast layered distribution method and system

Repair Tree: Fast Repair for Single Failure in Erasure-Coded Distributed Storage Systems

Self-adaptive network transmission control method and system based on TCP (Transmission Control Protocol) protocol

Concurrent regenerating codes

Method and system for layered distribution of IP multicast data