arXiv subjects

Kenneth W. Shum

Publications and source records attributed to Kenneth W. Shum.

At least 37 records · Page 2Linked to original sources

arXiv2017

On the Duality of Fractional Repetition Codes

Erasure codes have emerged as an efficient technology for providing data redundancy in distributed storage systems. However, it is a challenging task to repair the failed storage nodes in erasure-coded storage systems, which requires large quantities of network resources. In this paper, we study fractional repetition (FR) codes, which enable the minimal repair complexity and also minimum repair bandwidth during node repair. We focus on the duality of FR codes, and investigate the relationship between the supported file size of an FR code and its dual code. Furthermore, we present a dual bound on the supported file size of FR codes.

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arXiv2017

New CRT sequence sets for a collision channel without feedback

Protocol sequences are binary and periodic sequences used for deterministic multiple access in a collision channel without feedback. In this paper, we focus on user-irrepressible (UI) protocol sequences that can guarantee a positive individual throughput per sequence period with probability one for a slot-synchronous channel, regardless of the delay offsets among the users. As the sequence period has a fundamental impact on the worst-case channel access delay, a common objective of designing UI sequences is to make the sequence period as short as possible. Consider a communication channel that is shared by $M$ active users, and assume that each protocol sequence has a constant Hamming weight $w$. To attain a better delay performance than previously known UI sequences, this paper presents a CRTm construction of UI sequences with $w=M+1$, which is a variation of the previously known CRT construction. For all non-prime $M\geq 8$, our construction produces the shortest known sequence period and the shortest known worst-case delay of UI sequences. Numerical results show that the new construction enjoys a better average delay performance than the optimal random access scheme and other constructions with the same sequence period, in a variety of traffic conditions. In addition, we derive an asymptotic lower bound on the minimum sequence period for $w=M+1$ if the sequence structure satisfies some technical conditions, called equi-difference, and prove the tightness of this lower bound by using the CRTm construction.

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arXiv2016

Cooperative Repair of Multiple Node Failures in Distributed Storage Systems

Cooperative regenerating codes are designed for repairing multiple node failures in distributed storage systems. In contrast to the original repair model of regenerating codes, which are for the repair of single node failure, data exchange among the new nodes is enabled. It is known that further reduction in repair bandwidth is possible with cooperative repair. Currently in the literature, we have an explicit construction of exact-repair cooperative code achieving all parameters corresponding to the minimum-bandwidth point. We give a slightly generalized and more flexible version of this cooperative regenerating code in this paper. For minimum-storage regeneration with cooperation, we present an explicit code construction which can jointly repair any number of systematic storage nodes.

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arXiv2015

HFR Code: A Flexible Replication Scheme for Cloud Storage Systems

Fractional repetition (FR) codes are a family of repair-efficient storage codes that provide exact and uncoded node repair at the minimum bandwidth regenerating point. The advantageous repair properties are achieved by a tailor-made two-layer encoding scheme which concatenates an outer maximum-distance-separable (MDS) code and an inner repetition code. In this paper, we generalize the application of FR codes and propose heterogeneous fractional repetition (HFR) code, which is adaptable to the scenario where the repetition degrees of coded packets are different. We provide explicit code constructions by utilizing group divisible designs, which allow the design of HFR codes over a large range of parameters. The constructed codes achieve the system storage capacity under random access repair and have multiple repair alternatives for node failures. Further, we take advantage of the systematic feature of MDS codes and present a novel design framework of HFR codes, in which storage nodes can be wisely partitioned into clusters such that data reconstruction time can be reduced when contacting nodes in the same cluster.

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arXiv2014

Imperfect Secrecy in Wiretap Channel II

In a point-to-point communication system which consists of a sender, a receiver and a set of noiseless channels, the sender wishes to transmit a private message to the receiver through the channels which may be eavesdropped by a wiretapper. The set of wiretap sets is arbitrary. The wiretapper can access any one but not more than one wiretap set. From each wiretap set, the wiretapper can obtain some partial information about the private message which is measured by the equivocation of the message given the symbols obtained by the wiretapper. The security strategy is to encode the message with some random key at the sender. Only the message is required to be recovered at the receiver. Under this setting, we define an achievable rate tuple consisting of the size of the message, the size of the key, and the equivocation for each wiretap set. We first prove a tight rate region when both the message and the key are required to be recovered at the receiver. Then we extend the result to the general case when only the message is required to be recovered at the receiver. Moreover, we show that even if stochastic encoding is employed at the sender, the message rate cannot be increased.

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arXiv2013

Linear Network Code for Erasure Broadcast Channel with Feedback: Complexity and Algorithms

This paper investigates the construction of linear network codes for broadcasting a set of data packets to a number of users. The links from the source to the users are modeled as independent erasure channels. Users are allowed to inform the source node whether a packet is received correctly via feedback channels. In order to minimize the number of packet transmissions until all users have received all packets successfully, it is necessary that a data packet, if successfully received by a user, can increase the dimension of the vector space spanned by the encoding vectors he or she has received by one. Such an encoding vector is called innovative. We prove that innovative linear network code is uniformly optimal in minimizing user download delay. When the finite field size is strictly smaller than the number of users, the problem of determining the existence of innovative vectors is proven to be NP-complete. When the field size is larger than or equal to the number of users, innovative vectors always exist and random linear network code (RLNC) is able to find an innovative vector with high probability. While RLNC is optimal in terms of completion time, it has high decoding complexity due to the need of solving a system of linear equations. To reduce decoding time, we propose the use of sparse linear network code, since the sparsity property of encoding vectors can be exploited when solving systems of linear equations. Generating a sparsest encoding vector with large finite field size, however, is shown to be NP-hard. An approximation algorithm that guarantee the Hamming weight of a generated encoding vector to be smaller than a certain factor of the optimal value is constructed. Our simulation results show that our proposed methods have excellent performance in completion time and outperforms RLNC in terms of decoding time.

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arXiv2013

Cooperative Regenerating Codes

One of the design objectives in distributed storage system is the minimization of the data traffic during the repair of failed storage nodes. By repairing multiple failures simultaneously and cooperatively, further reduction of repair traffic is made possible. A closed-form expression of the optimal tradeoff between the repair traffic and the amount of storage in each node for cooperative repair is given. We show that the points on the tradeoff curve can be achieved by linear cooperative regenerating codes, with an explicit bound on the required finite field size. The proof relies on a max-flow-min-cut-type theorem for submodular flow from combinatorial optimization. Two families of explicit constructions are given.

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arXiv2013

On the Optimum Cyclic Subcode Chains of $\mathcal{RM}(2,m)^*$ for Increasing Message Length

The distance profiles of linear block codes can be employed to design variational coding scheme for encoding message with variational length and getting lower decoding error probability by large minimum Hamming distance. %, e.g. the design of TFCI in CDMA and the researches on the second-order Reed-Muller code $\mathcal{RM}(2,m)$, etc. Considering convenience for encoding, we focus on the distance profiles with respect to cyclic subcode chains (DPCs) of cyclic codes over $GF(q)$ with length $n$ such that $\mbox{gcd}(n,q) = 1$. In this paper the optimum DPCs and the corresponding optimum cyclic subcode chains are investigated on the punctured second-order Reed-Muller code $\mathcal{RM}(2,m)^*$ for increasing message length, where two standards on the optimums are studied according to the rhythm of increase.

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arXiv2013

Symmetry in Distributed Storage Systems

The max-flow outer bound is achievable by regenerating codes for functional repair distributed storage system. However, the capacity of exact repair distributed storage system is an open problem. In this paper, the linear programming bound for exact repair distributed storage systems is formulated. A notion of symmetrical sets for a set of random variables is given and equalities of joint entropies for certain subsets of random variables in a symmetrical set is established. Concatenation coding scheme for exact repair distributed storage systems is proposed and it is shown that concatenation coding scheme is sufficient to achieve any admissible rate for any exact repair distributed storage system. Equalities of certain joint entropies of random variables induced by concatenation scheme is shown. These equalities of joint entropies are new tools to simplify the linear programming bound and to obtain stronger converse results for exact repair distributed storage systems.

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arXiv2013

Repairing Multiple Failures in the Suh-Ramchandran Regenerating Codes

Using the idea of interference alignment, Suh and Ramchandran constructed a class of minimum-storage regenerating codes which can repair one systematic or one parity-check node with optimal repair bandwidth. With the same code structure, we show that in addition to single node failure, double node failures can be repaired collaboratively with optimal repair bandwidth as well. We give an example of how to repair double failures in the Suh-Ramchandran regenerating code with six nodes, and give the proof for the general case.

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arXiv2013

Optimized-Cost Repair in Multi-hop Distributed Storage Systems with Network Coding

In distributed storage systems reliability is achieved through redundancy stored at different nodes in the network. Then a data collector can reconstruct source information even though some nodes fail. To maintain reliability, an autonomous and efficient protocol should be used to repair the failed node. The repair process causes traffic and consequently transmission cost in the network. Recent results found the optimal trafficstorage tradeoff, and proposed regenerating codes to achieve the optimality. We aim at minimizing the transmission cost in the repair process. We consider the network topology in the repair, and accordingly modify information flow graphs. Then we analyze the cut requirement and based on the results, we formulate the minimum-cost as a linear programming problem for linear costs. We show that the solution of the linear problem establishes a fundamental lower bound of the repair-cost. We also show that this bound is achievable for minimum storage regenerating, which uses the optimal-cost minimum-storage regenerating (OCMSR) code. We propose surviving node cooperation which can efficiently reduce the repair cost. Further, the field size for the construction of OCMSR codes is discussed. We show the gain of optimal-cost repair in tandem, star, grid and fully connected networks.

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arXiv2013

Analysis and Construction of Functional Regenerating Codes with Uncoded Repair for Distributed Storage Systems

Modern distributed storage systems apply redundancy coding techniques to stored data. One form of redundancy is based on regenerating codes, which can minimize the repair bandwidth, i.e., the amount of data transferred when repairing a failed storage node. Existing regenerating codes mainly require surviving storage nodes encode data during repair. In this paper, we study functional minimum storage regenerating (FMSR) codes, which enable uncoded repair without the encoding requirement in surviving nodes, while preserving the minimum repair bandwidth guarantees and also minimizing disk reads. Under double-fault tolerance settings, we formally prove the existence of FMSR codes, and provide a deterministic FMSR code construction that can significantly speed up the repair process. We further implement and evaluate our deterministic FMSR codes to show the benefits. Our work is built atop a practical cloud storage system that implements FMSR codes, and we provide theoretical validation to justify the practicality of FMSR codes.

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arXiv2011

Minimization of Storage Cost in Distributed Storage Systems with Repair Consideration

In a distributed storage system, the storage costs of different storage nodes, in general, can be different. How to store a file in a given set of storage nodes so as to minimize the total storage cost is investigated. By analyzing the min-cut constraints of the information flow graph, the feasible region of the storage capacities of the nodes can be determined. The storage cost minimization can then be reduced to a linear programming problem, which can be readily solved. Moreover, the tradeoff between storage cost and repair-bandwidth is established.

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arXiv2011

Exact Minimum-Repair-Bandwidth Cooperative Regenerating Codes for Distributed Storage Systems

In order to provide high data reliability, distributed storage systems disperse data with redundancy to multiple storage nodes. Regenerating codes is a new class of erasure codes to introduce redundancy for the purpose of improving the data repair performance in distributed storage. Most of the studies on regenerating codes focus on the single-failure recovery, but it is not uncommon to see two or more node failures at the same time in large storage networks. To exploit the opportunity of repairing multiple failed nodes simultaneously, a cooperative repair mechanism, in the sense that the nodes to be repaired can exchange data among themselves, is investigated. A lower bound on the repair-bandwidth for cooperative repair is derived and a construction of a family of exact cooperative regenerating codes matching this lower bound is presented.

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arXiv2011

Generation of Innovative and Sparse Encoding Vectors for Broadcast Systems with Feedback

In the application of linear network coding to wireless broadcasting with feedback, we prove that the problem of determining the existence of an innovative encoding vector is NP-complete when the finite field size is two. When the finite field size is larger than or equal to the number of users, it is shown that we can always find an encoding vector which is both innovative and sparse. The sparsity can be utilized in speeding up the decoding process. An efficient algorithm to generate innovative and sparse encoding vectors is developed. Simulations show that the delay performance of our scheme with binary finite field outperforms a number of existing schemes in terms of average and worst-case delay.

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arXiv2011

Cooperative Regenerating Codes for Distributed Storage Systems

When there are multiple node failures in a distributed storage system, regenerating the failed storage nodes individually in a one-by-one manner is suboptimal as far as repair-bandwidth minimization is concerned. If data exchange among the newcomers is enabled, we can get a better tradeoff between repair bandwidth and the storage per node. An explicit and optimal construction of cooperative regenerating code is illustrated.

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arXiv2010

Construction and Applications of CRT Sequences

Protocol sequences are used for channel access in the collision channel without feedback. Each user accesses the channel according to a deterministic zero-one pattern, called the protocol sequence. In order to minimize fluctuation of throughput due to delay offsets, we want to construct protocol sequences whose pairwise Hamming cross-correlation is as close to a constant as possible. In this paper, we present a construction of protocol sequences which is based on the bijective mapping between one-dimensional sequence and two-dimensional array by the Chinese Remainder Theorem (CRT). In the application to the collision channel without feedback, a worst-case lower bound on system throughput is derived.

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arXiv2010

A General Upper Bound on the Size of Constant-Weight Conflict-Avoiding Codes

Conflict-avoiding codes are used in the multiple-access collision channel without feedback. The number of codewords in a conflict-avoiding code is the number of potential users that can be supported in the system. In this paper, a new upper bound on the size of conflict-avoiding codes is proved. This upper bound is general in the sense that it is applicable to all code lengths and all Hamming weights. Several existing constructions for conflict-avoiding codes, which are known to be optimal for Hamming weights equal to four and five, are shown to be optimal for all Hamming weights in general.

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