The paper is devoted to the problem of erasure coding in distributed storage. We consider a model of storage that assumes that nodes are organized into equally sized groups, called racks, that within each group the nodes can communicate freely without taxing the system bandwidth, and that the only information transmission that counts is the one between the racks. This assumption implies that the nodes within each of the racks can collaborate before providing information to the failed node. The main emphasis of the paper is on code construction for this storage model. We present an explicit family of MDS array codes that support recovery of a single failed node from any number of helper racks using the minimum possible amount of inter-rack communication (such codes are said to provide optimal repair). The codes are constructed over finite fields of size comparable to the code length.We also derive a bound on the number of symbols accessed at helper nodes for the purposes of repair, and construct a code family that approaches this bound, while still maintaining the optimal repair property.Finally, we present a construction of scalar Reed-Solomon codes that support optimal repair for the rackoriented storage model.The problems of centralized and cooperative repair have been addressed in multiple recent papers, and there are explicit constructions of optimal-repair regenerating codes that cover the entire range of admissible parameters, require small-size ground alphabet compared to the length n of the encoding block, and attain the smallest possible repair bandwidth [16], [23], [28], [19], [27], [30], [13] (more references are given in a recent survey [2]). The availability of optimal constructions has motivated a shift of attention toward studying data recovery not only under communication, but also connectivity constraints, in other words, storage models in which communication cost between nodes differs depending on their location in the storage cluster. One of the simple extensions from the basic setting of homogeneous storage suggests that the nodes are joined into several groups (clusters), and repair of a node can be based on information from both the nodes within its own group and from nodes in the other groups. This permits to differentiate between communication within the cluster and the inter-cluster downloads, and the natural assumption is that the former is easier (contributes less to the repair bandwidth) than the latter.Erasure coding for clustered architectures was introduced several years ago and affords several variations. One of the first questions analyzed for heterogeneous storage models was related to repair under the condition that the system contains a group of nodes, downloading information from which contributes more to the repair bandwidth than downloading the same amount of information from the other nodes [1]. Later works [6], [14] observed that a more realistic version of non-homogeneous storage should assume that the cost of downloading information depends on the relative location of th...
In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x 1 , . . . , x n ) ∈ {0, 1} n bit by bit via a channel limited to at most pn corruptions. The channel is "online" in the sense that at the ith step of communication the channel decides whether to corrupt the ith bit or not based on its view so far, i.e., its decision depends only on the transmitted bits (x 1 , . . . , x i ). This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has a full knowledge on the sent codeword x.In this work we study the capacity of binary online channels for two corruption models: the bit-flip model in which the channel may flip at most pn of the bits of the transmitted codeword, and the erasure model in which the channel may erase at most pn bits of the transmitted codeword. Specifically, for both error models we give a full characterization of the capacity as a function of p.The online channel (in both the bit-flip and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we present and analyze a coding scheme that improves on the previously suggested lower bounds and matches the previously suggested upper bounds thus implying a tight characterization.
In the q-ary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x 1 , . . . , x n ) ∈ {0, 1, . . . , q − 1} n symbol by symbol via a channel limited to at most pn errors and/or p n erasures. The channel is "online" in the sense that at the ith step of communication the channel decides whether to corrupt the ith symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols (x 1 , . . . , x i ). This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has a full knowledge on the sent codeword x.In this work we study the capacity of q-ary online channels for a combined corruption model, in which the channel may impose at most pn errors and at most p n erasures on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we give a full characterization of the capacity as a function of q, p, and p .where α q (p) = 1 − 2q q−1 (p −p) − q q−1 p .In fact, as direct by-products of the analysis of our coding scheme, we can show that even if Calvin has "small" lookahead, the capacity is essentially unchanged. More precisely, if for any constant > 0, Calvin decides whether to tamper with the i-th symbol of the codeword based only on the symbols (x 1 , x 2 , · · · , x j ), where j = min{n, i+n }, then the capacity of the corresponding "n -lookahead is at most f ( ) less than the corresponding C we show in Theorems 1.1 above (for some continuous f ). We provide a rough argument in support of this claim in the Remark at the end of Section 3. Previous WorkWe start by briefly summarizing the state-of-the-art for erasure and error adversarial channels, for both omniscient and oblivious adversaries. The optimal rate of communication over binary omniscient adversarial channels (for both erasure and error) are long standing open problems in coding theory. The best known lower bounds for the problems derive from the Gilbert-Varshamov codes (the GV bound) [1,2], and the tightest upper bounds (the MRRW bounds) from the work by McEliece et al. [3].The literature on Arbitrarily Varying Channels (AVCs, e.g., [10]) implies that the capacity of the binary oblivious adversarial error channel is 1 − H(p), and that of oblivious adversarial erasure channels is 1 − p; these match the well-known capacities of the corresponding "random noise" channels with bits flipped or erased Bernoulli(p), but are attainable even for noise patterns that can be chosen (up to an overall constraint of a p-fraction corruptions) by an adversary with full knowledge of the codebook, but no knowledge of the actually transmitted codeword. 1 An alternate proof of the capacity of the binary oblivious bit-flip channel was presented in [11] by Langberg, and a computationally efficient scheme achieving this rate was presented in [12] by Guruswami and Smith.We now turn to the causal...
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