One of the primary objectives of a distributed storage system is to reliably store large amounts of source data for long durations using a large number @math of unreliable storage nodes, each with @math bits of storage capacity. Storage nodes fail randomly over time and are replaced with nodes of equal capacity initialized to zeroes, and thus bits are erased at some rate @math . To maintain recoverability of the source data, a repairer continually reads data over a network from nodes at a rate @math , and generates and writes data to nodes based on the read data. The distributed storage source data capacity is the maximum amount of source data that can be reliably stored for long periods of time. We prove the distributed storage source data capacity asymptotically approaches @math as @math and @math grow. This equation expresses a fundamental trade-off between network traffic and storage overhead to reliably store source data.
When s @cite_11 , @cite_1 , @cite_10 are used to repair all objects in a system, the ratio of the amount of data read per node failure to the capacity of a node is approximately @math with respect to any with a periodic timing sequence, which is around a factor of two above the lower bound of approximately @math from Equation .
@cite_10 @cite_1 @cite_11
2102047288 2249486791 2950891324 2765604633
We consider exact repair of failed nodes in maximum distance separable (MDS) code based distributed storage systems. It is well known that an (n, k) MDS code can tolerate failure (erasure) of up to n − k storage disks, when the code is used to store k information elements over n distributed storage disks. The focus of this paper is optimal recovery, in terms of repair bandwidth - the amount of data to be downloaded to repair a failed node - for a single failed node. When a single node fails, it has been previously shown by Dimakis et. al. that the amount of repair bandwidth is at least equation units, when each storage disk stores ℒ units of data. The achievability of this lower bound of equation units, for arbitrary values of (n, k); has been shown previously using asymptotic code constructions based on asymptotic interference alignment. However, the existence of finite codes satisfying this lower bound has been shown only for specific regimes of (n, k) and their existence for arbitrary values of (n, k) remained open. In this paper, we provide the first known construction of a finite code for arbitrary (n, k), which can repair a single failed systematic node by downloading exactly equation units of data. The code that we construct is based on permutation matrices and hence termed the Permutation Code. We consider the problem of designing [n; k] linear codes for distributed storage systems (DSS) that satisfy the (r, t)-Local Repair Property, where any t'(<=t) simultaneously failed nodes can be locally repaired, each with locality r. The parameters n, k, r, t are positive integers such that r<k<n and t <= n-k. We consider the functional repair model and the sequential approach for repairing multiple failed nodes. By functional repair, we mean that the packet stored in each newcomer is not necessarily an exact copy of the lost data but a symbol that keep the (r, t)-local repair property. By the sequential approach, we mean that the t' newcomers are ordered in a proper sequence such that each newcomer can be repaired from the live nodes and the newcomers that are ordered before it. Such codes, which we refer to as (n, k, r, t)-functional locally repairable codes (FLRC), are the most general class of LRCs and contain several subclasses of LRCs reported in the literature. In this paper, we aim to optimize the storage overhead (equivalently, the code rate) of FLRCs. We derive a lower bound on the code length n given t belongs to 2,3 and any possible k, r. For t=2, our bound generalizes the rate bound proved in [14]. For t=3, our bound improves the rate bound proved in [10]. We also give some onstructions of exact LRCs for t belongs to 2,3 whose length n achieves the bound of (n, k, r, t)-FLRC, which proves the tightness of our bounds and also implies that there is no gap between the optimal code length of functional LRCs and exact LRCs for certain sets of parameters. Moreover, our constructions are over the binary field, hence are of interest in practice. MDS (maximum distance separable) array codes are widely used in storage systems due to their computationally efficient encoding and decoding procedures. An MDS code with r redundancy nodes can correct any r erasures by accessing (reading) all the remaining information in both the systematic nodes and the parity (redundancy) nodes. However, in practice, a single erasure is the most likely failure event; hence, a natural question is how much information do we need to access in order to rebuild a single storage node? We define the rebuilding ratio as the fraction of remaining information accessed during the rebuilding of a single erasure. In our previous work we showed that the optimal rebuilding ratio of 1 r is achievable (using our newly constructed array codes) for the rebuilding of any systematic node, however, all the information needs to be accessed for the rebuilding of the parity nodes. Namely, constructing array codes with a rebuilding ratio of 1 r was left as an open problem. In this paper, we solve this open problem and present array codes that achieve the lower bound of 1 r for rebuilding any single systematic or parity node. The repair bandwidth of a code is the minimum amount of data required to repair one or several failed nodes (erasures). For MDS codes, the repair bandwidth is bounded below by the so-called cut-set bound, and codes that meet this bound with equality are said to support optimal repair of one or multiple failed nodes. We consider the problem of repairing multiple failed nodes of Reed-Solomon (RS) codes. In a recent work with I. Tamo (Proc. IEEE FOCS 2017), we gave the first explicit construction of RS codes with optimal repair of any single failed node from any subset of helper nodes. In this paper, we construct explicit RS codes that universally achieve the cut-set bound for the repair of any number of failed nodes from any set of helper nodes. Moreover, the node size of our codes is close to the optimal (smallest possible) node size of codes with such property.