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Network Coding for Joint Storage and Transmission with Minimum Cost
 In ISIT
, 2006
"... Abstract — Network coding provides elegant solutions to many data transmission problems. The usage of coding for distributed data storage has also been explored. In this work, we study a joint storage and transmission problem, where a source transmits a file to storage nodes whenever the file is upd ..."
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Abstract — Network coding provides elegant solutions to many data transmission problems. The usage of coding for distributed data storage has also been explored. In this work, we study a joint storage and transmission problem, where a source transmits a file to storage nodes whenever the file is updated, and clients read the file by retrieving data from the storage nodes. The cost includes the transmission cost for file update and file read, as well as the storage cost. We show that such a problem can be transformed into a pure flow problem and is solvable in polynomial time using linear programming. Coding is often necessary for obtaining the optimal solution with the minimum cost. However, we prove that for networks of generalized tree structures, where adjacent nodes can have asymmetric links between them, file splitting — instead of coding — is sufficient for achieving optimality. In particular, if there is no constraint on the numbers of bits that can be stored in storage nodes, there exists an optimal solution that always transmits and stores the file as a whole. The proof is accompanied by an algorithm that optimally assigns file segments to storage nodes. I.
Interleaving schemes on circulant graphs with two offsets
 IEEE Transactions on Information Theory
"... Interleaving is used for errorcorrecting on a bursty noisy channel. Given a graph G describing the topology of the channel, we label the vertices of G so that each labelset is sufficiently sparse. The interleaving scheme corrects for any error burst of size at most t; it is a labeling where the di ..."
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Interleaving is used for errorcorrecting on a bursty noisy channel. Given a graph G describing the topology of the channel, we label the vertices of G so that each labelset is sufficiently sparse. The interleaving scheme corrects for any error burst of size at most t; it is a labeling where the distance between any two vertices in the same labelset is at least t. We consider interleaving schemes on infinite circulant graphs with two offsets 1 and d. In such graph the vertices are integers; edge ij exists if and only if i − j  ∈ {1, d}. Our goal is to minimize the number of labels used. Our constructions are covers of the graph by the minimal number of translates of some labelset S. We focus on minimizing the index of S, which is the inverse of its density rounded up. We establish lower bounds and prove that our constructions are optimal or almost optimal, both for the index of S and for the number of labels. Keywords: Error bursts, errorcorrecting codes, interleaving schemes, circulant graphs.
Optimal interleaving on tori
 IN PROC. IEEE INT. SYMP. INFORMATION THEORY (ISIT2004)
, 2004
"... This paper studies tinterleaving on twodimensional tori, which is defined by the property that every connected subgraph of order t in the torus is labelled by t distinct integers. This is the first time that the tinterleaving problem is solved for graphs of modular structures. tinterleaving on t ..."
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This paper studies tinterleaving on twodimensional tori, which is defined by the property that every connected subgraph of order t in the torus is labelled by t distinct integers. This is the first time that the tinterleaving problem is solved for graphs of modular structures. tinterleaving on tori has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. We say that a torus can be perfectly tinterleaved if its tinterleaving number — the minimum number of distinct integers needed to tinterleave the torus — meets the spherepacking lower bound. We prove the necessary and sufficient conditions for tori that can be perfectly tinterleaved, and present efficient perfect tinterleaving constructions. The most important contribution of this paper is to prove that when a torus is large enough in both dimensions, its tinterleaving number is at most one more than the spherepacking lower bound, and to present an optimal and efficient tinterleaving scheme for such tori. Then we prove bounds for the tinterleaving numbers of the remaining cases, completing a general characterization of the tinterleaving problem on 2dimensional tori.