On Convolutional Network Coding

Li, S.-Y.R.; Yeung, Raymond W.

doi:10.1109/isit.2006.261653

Cited by 48 publications

(23 citation statements)

References 6 publications

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“…Then a polynomial time deterministic algorithm was obtained to construct optimal linear coding schemes [23] for multicasting in directed acyclic graphs. Convolution codes were used [11,28] to extend the algorithm in [23] for multicasting in general directed graphs. Subsequently, a simple polynomial time randomized algorithm was obtained for constructing optimal linear coding schemes for multicasting [18], and our algorithm is based on this approach.…”

Section: Related Workmentioning

confidence: 99%

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Graph Connectivities, Network Coding, and Expander Graphs

Cheung

Lau

Leung

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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Abstract. We present a new algebraic formulation for computing edge connectivities in a directed graph, using ideas developed in network coding. This reduces the problem of computing edge connectivities to solving systems of linear equations, thus allowing us to use tools in linear algebra to design new algorithms. Using the algebraic formulation, we obtain faster algorithms for computing single source edge connectivities and all pairs edge connectivities. In some settings, the amortized time to compute the edge connectivity for one pair is sublinear. Through this connection, we have also found an interesting use of expanders and superconcentrators to design fast algorithms for some graph connectivity problems.Key words. graph connectivities, network coding, expander graphs, linear equations, randomized algorithms AMS subject classifications. 05C40, 05C50, 05C85, 68Q25, 68W20 DOI. 10.1137/1108449701. Introduction. Graph connectivity is a basic concept that measures the reliability and efficiency of a graph. The edge connectivity from vertex s to vertex t is defined as the size of a minimum s-t cut or, equivalently, the maximum number of edge disjoint paths from s to t. Computing edge connectivities is a classical and wellstudied problem in combinatorial optimization. Most known algorithms for solving this problem are based on network flow techniques (see, e.g., [34]).Even and Tarjan [12] introduced the fastest algorithm to compute s-t edge connectivity in a simple directed graph and running in O(min{m 1/2 , n 2/3 } · m) time, where m is the number of edges and n is the number of vertices. To compute the edge connectivities for many pairs, however, it is not known how to do it faster than computing edge connectivity for each pair separately, even when the pairs share the source or the sink. For instance, it is not known how to compute all pairs edge connectivities faster than computing s-t edge connectivity for Θ(n 2 ) pairs. This is in contrast to the problem in undirected graphs, where all pairs edge connectivities can be computed inÕ(mn) time by constructing a Gomory-Hu tree [6].Network coding is an innovative method for transmitting information in a computer network. The fundamental result is a max-information-flow min-cut theorem for multicasting [1]: if the edge connectivity from the source vertex s to each sink vertex t i is at least k, then one can transmit k units of information to all sink vertices simultaneously by performing encoding and decoding at the vertices. An elegant algebraic framework has been developed for constructing efficient network coding schemes for multicasting [29,26].In this paper, we use the techniques developed in network coding to obtain a new algebraic formulation for computing edge connectivities. This reduces the problem

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Section: Related Workmentioning

confidence: 99%

“…directed graphs, network coding schemes require convolution codes [11,28], and it is not known how to use these to compute edge connectivities. Our contribution is a simple formulation that can be used to compute edge connectivities.…”

mentioning

confidence: 99%

Graph Connectivities, Network Coding, and Expander Graphs

Cheung

Lau

Leung

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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“…Construction and decoding of convolutional multicast have been studied by Erez and Feder [52,53], Fragouli and Soljanin [54], and Barbero and Ytrehus [55]. The unifying treatment of convolutional codes here is based on Li and Yeung [56] (see also Yeung et al [50]). Li and Ho [57] recently obtained a general abstract formulation of convolutional network codes based on ring theory.…”

Section: Theoremmentioning

confidence: 99%

Network coding theory: An introduction

Yeung

2010

Front. Electr. Electron. Eng. China

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For a long time, store-and-forward had been the transport mode in network communications. In other words, information had been regarded as a commodity that only needs to be routed through the network, possibly with replication at the intermediate nodes.In the late 1990's, a new concept called network coding fundamentally changed the way a network can be operated. Under the paradigm of network coding, information can be processed within the network for the purpose of transmission. It was demonstrated that compared with store-and-forward, the network throughput can generally be increased by employing network coding. Since then, network coding has made significant impact on different branches of information science. The impact of network coding has gone as far as mathematics, physics, and biology. This expository work aims to be an introduction to this fast-growing subject with a detailed discussion of the basic theoretical results.

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“…Erez et al [14][15] [16] proposed a construction algorithm of convolutional network codes for multicast, which can support acyclic and cyclic networks. Li et al [17] refined the notion of convolutional network code, and proposed the concept of convolutional multicast as well as its proof of existence. Fragouli et al [18] [19] adopted line graph to decompose information flow as well as to discuss connection between convolutional code and network coding in cyclic networks.…”

Section: Other Related Work For Network Coding In Cyclic Networkmentioning

confidence: 99%

“…To unify, assume In(S) as the set of imaginary incoming channels to node S with ω-dimension. A delay function t(d, e) [17] maps every adjacent pair (d, e) of channels to a number in {0, 1, 2, · · · , ∞}. For every cycle, there is at least a pair (d, e) with t(d, e) = 0.…”

Section: Notations and Definitionsmentioning

confidence: 99%

Constructing Network Coding in Cyclic Networks with the Best Linear Independence

Huang

Wang

Zhang

et al. 2009

2009 Fourth International Conference on Computer Sciences and Convergence Information Technology

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Capacity of Network coding in cyclic networks can be strictly better than routing. Harvey et al. showed that it exists in the practical scenarios such as the widespread P2P networks. Thus, it is of application significance to investigate how to construct network code in cyclic networks. According to Li et al., there are four levels of network code in cyclic networks, including Convolutional Multicast(CM), Convolutional Broadcast(CB), Convolutional Dispersion(CD), and Basic Convolutional Network Code(BCNC). This paper focuses on the BCNC that is with the best linear independence, which means it can provide the most efficient transmission in that it has the minimum redundance.The contributions of this paper are (1)to improve our previous BCNC algorithm; (2)to show insights of the essential difference between BCNC and CM/CB/CD which is of great theoretical significance to understand how different kinds of cycles affect the network coding construction.

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On Convolutional Network Coding

Cited by 48 publications

References 6 publications

Graph Connectivities, Network Coding, and Expander Graphs

Graph Connectivities, Network Coding, and Expander Graphs

Network coding theory: An introduction

Constructing Network Coding in Cyclic Networks with the Best Linear Independence

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