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Network information flow
 IEEE TRANS. INFORM. THEORY
, 2000
"... We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a pointtopoint communication network on which a number of information sources are to be mulitcast to certain sets of destinations. We assume that the information source ..."
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We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a pointtopoint communication network on which a number of information sources are to be mulitcast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. In this paper, we study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the Maxflow Mincut Theorem for network information flow. Contrary to one’s intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a “fluid” which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems.
Asymmetric Multilevel Diversity Coding and Asymmetric Gaussian Multiple Descriptions
"... We consider the asymmetric multilevel diversity (AMLD) coding problem, where a set of 2 K − 1 information sources, ordered in a decreasing level of importance, is encoded into K messages (or descriptions). There are 2K −1 decoders, each of which has access to a nonempty subset of the encoded messa ..."
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We consider the asymmetric multilevel diversity (AMLD) coding problem, where a set of 2 K − 1 information sources, ordered in a decreasing level of importance, is encoded into K messages (or descriptions). There are 2K −1 decoders, each of which has access to a nonempty subset of the encoded messages. Each decoder is required to reproduce the information sources up to a certain importance level depending on the combination of descriptions available to it. We obtain a single letter characterization of the achievable rate region for the 3description problem. In contrast to symmetric multilevel diversity coding, sourceseparation coding is not sufficient in the asymmetric case, and ideas akin to network coding need to be used strategically. Based on the intuitions gained in treating the AMLD problem, we derive inner and outer bounds for the rate region of the asymmetric Gaussian multiple description (MD) problem with three descriptions. Both the inner and outer bounds have a similar geometric structure to the rate region template of the AMLD coding problem, and moreover, we show that the gap between them is small, which results in an approximate characterization of the asymmetric Gaussian three description rate region.
Rate region for a class of delay mitigating codes and P2P networks
"... Abstract—This paper identifies the relevance of a distributed source coding problem first formulated by Yeung and Zhang in 1999 to two applications in network design: i) the design of delay mitigating codes, and ii) the design of network coded P2P networks. When transmitting timesensitive frames fr ..."
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Abstract—This paper identifies the relevance of a distributed source coding problem first formulated by Yeung and Zhang in 1999 to two applications in network design: i) the design of delay mitigating codes, and ii) the design of network coded P2P networks. When transmitting timesensitive frames from a source to a destination over a multipath network using a collection of coded packets, the decoding requirements determine which subsets of packets will be sufficient for decoding which frames. The rate region of packet sizes consistent with these requirements is shown to be an instance of the aforementioned distributed source coding problem. When encoding file chunks into packets in a peer to peer system, the peers wish to receive their chunks as soon as possible while uploading data at as low a rate as possible. It is shown that the region of encoded packet sizes consistent with the decoding constraints is another instance of the aforementioned distributed source coding problem. These rate regions are placed in the larger context of ratedelay tradeoffs in designing delay mitigating codes and efficient P2P systems. I.
Multilevel Diversity Coding Systems: Rate Regions, Codes, Computation, & Forbidden Minors
, 2014
"... The rate regions of multilevel diversity coding systems (MDCS), a subclass of the broader family of multisource multisink networks with special structure, are investigated. After showing how to enumerate all nonisomorphic MDCS instances of a given size, the Shannon outer bound and several achiev ..."
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The rate regions of multilevel diversity coding systems (MDCS), a subclass of the broader family of multisource multisink networks with special structure, are investigated. After showing how to enumerate all nonisomorphic MDCS instances of a given size, the Shannon outer bound and several achievable inner bounds based on linear codes are given for the rate region of each nonisomorphic instance. For thousands of MDCS instances, the bounds match, and hence exact rate regions are proven. Results gained from these computations are summarized in key statistics involving aspects such as the sufficiency of scalar binary codes, the necessary size of vector binary codes, etc. Also, it is shown how to generate computer aided human readable converse proofs, as well as how to construct the codes for an achievability proof. Based on this large repository of rate regions, a series of results about general MDCS cases that they inspired are introduced and proved. In particular, a series of embedding operations that preserve the property of sufficiency of scalar or vector codes are presented. The utility of these operations is demonstrated by boiling the thousands of MDCS instances for which binary scalar codes are insufficient down to 12 forbidden smallest embedded MDCS instances.
Network Embedding Operations Preserving the Insufficiency of Linear Network Codes
"... Abstract—Operations for extending/embedding a smaller network into a larger network that preserve the insufficiency of classes of linear network codes are presented. Linear network codes over some finite field are said to be sufficient for a network if and only if for every point in the network cod ..."
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Abstract—Operations for extending/embedding a smaller network into a larger network that preserve the insufficiency of classes of linear network codes are presented. Linear network codes over some finite field are said to be sufficient for a network if and only if for every point in the network coding rate region, there exists a code over that finite field to achieve it. Three operations are defined, and it is proven that they have the desired inheritance property, both for scalar linear network codes and for vector linear network codes, separately. Experimental results on the rate regions of multilevel diversity coding systems (MDCS), a subclass of the broader family of multisource multisink networks with special structure, are presented for demonstration. These results demonstrate that these notions of embedding operations enable one to investigate the existences of small numbers of forbidden network minors for sufficiency of linear network codes over a given field. I.