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A Graph Minor Perspective to Network Coding: Connecting Algebraic Coding with Network Topologies
"... Abstract—Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusi ..."
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Abstract—Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on algebraic or information theoretic aspects of the problem. This work aims at an indepth examination of the relation between algebraic coding and network topologies. We mathematically establish a series of results along the direction of: if network coding is necessary/beneficial, or if a particular finite field is required for coding, then the network must have a corresponding hidden structure embedded in its underlying topology, and such embedding is computationally efficient to verify. Specifically, we first formulate a metaconjecture, the NCMinor Conjecture, that articulates such a connection between graph theory and network coding, in the language of graph minors. We next prove that the NCMinor Conjecture is almost equivalent to the Hadwiger Conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of K4, K5, K6, and KO(q / log q) minors, for networks requiring F3, F4, F5 and Fq, respectively. We finally prove that network coding can make a difference from routing only if the network contains a K4 minor, and this minor containment result is tight. Practical implications of the above results are discussed. I.
Information Multicast in (Pseudo)Planar Networks: Efficient Network Coding over Small Finite Fields
"... Abstract—Network coding encourages innetwork mixing of information flows for enhanced network capacity, particularly for multicast data dissemination. This work aims to explore properties in the underlying network topology for efficient network coding solutions, including efficient code assignment ..."
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Abstract—Network coding encourages innetwork mixing of information flows for enhanced network capacity, particularly for multicast data dissemination. This work aims to explore properties in the underlying network topology for efficient network coding solutions, including efficient code assignment algorithms and efficient encoding/decoding operations that come with small base field sizes. The following cases of (pseudo)planar types of networks are studied: outerplanar networks where all nodes colocate on a common face, relay/terminal coface networks where all relay/terminal nodes colocate on a common face, general planar networks, and apex networks. I.
1PrecodingBased Network Alignment For Three Unicast Sessions Chun Meng, Student Member, IEEE, Abhik Kumar Das,
, 2013
"... Abstract—We consider the problem of network coding across three unicast sessions over a directed acyclic graph, where the sender and the receiver of each unicast session are both connected to the network via a single edge of unit capacity. We consider a network model in which the middle of the netwo ..."
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Abstract—We consider the problem of network coding across three unicast sessions over a directed acyclic graph, where the sender and the receiver of each unicast session are both connected to the network via a single edge of unit capacity. We consider a network model in which the middle of the network can only perform random linear network coding, and we restrict our approaches to precodingbased linear schemes, where the senders use precoding matrices to encode source symbols. We adapt a precodingbased interference alignment technique, originally developed for the wireless interference channel, to construct a precodingbased linear scheme, which we refer to as precodingbased network alignment scheme (PBNA). A primary difference between this setting and the wireless interference channel is that the network topology can introduce dependencies among the elements of the transfer matrix, which we refer to as coupling relations, and can potentially affect the achievable rate of PBNA. We identify all these coupling relations and we interpret them in terms of network topology. We then present polynomialtime algorithms to check the presence of these coupling relations in a particular network. Finally, we show that, depending on the coupling relations present in the network, the optimal symmetric rate achieved by precodingbased linear scheme can take only three possible values, all of which can be achieved by PBNA.
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, 2013
"... Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on al ..."
Abstract
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Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on algebraic or information theoretic aspects of the problem. This work aims at an indepth examination of the relation between algebraic coding and network topologies. We mathematically establish a series of results along the direction of: if network coding is necessary/beneficial, or if a particular finite field is required for coding, then the network must have a corresponding hidden structure embedded in its underlying topology, and such embedding is computationally efficient to verify. Specifically, we first formulate a metaconjecture, the NCMinor Conjecture, that articulates such a connection between graph theory and network coding, in the language of graph minors. We next prove that the NCMinor Conjecture is almost equivalent to the Hadwiger Conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of K4, K5, K6, and KO(q / log q) minors, for networks requiring F3, F4, F5 and Fq, respectively. We finally prove that network coding can make a difference from routing only if the network contains a K4 minor, and this minor containment result is tight. Practical implications of the above results are discussed.