Networks, Matroids, and Non-Shannon Information Inequalities

We define a class of networks, called matroidal networks, which includes as special cases all scalar-linearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the literature, such as the insufficiency of linear network coding and the unachievability of network coding capacity. We also construct a new network, from the Vamos matroid, which we call the Vamos network, and use it to prove that Shannon-type information inequalities are in general not sufficient for computing network coding capacities. To accomplish this, we obtain a capacity upper bound for the Vamos network using a non-Shannon-type information inequality discovered in 1998 by Zhang and Yeung, and then show that it is smaller than any such bound derived from Shannon-type information inequalities. This is the first application of a non-Shannon-type inequality to network coding. We also compute the exact routing capacity and linear coding capacity of the Vamos network. Finally, using a variation of the Vamos network, we prove that Shannon-type information inequalities are insufficient even for computing network coding capacities of multiple-unicast networks.     

Network Coding for Efficient Multicast Routing in Wireless Ad-hoc Networks

Network coding is a powerful coding technique that has been proved to be very effective in achieving the maximum multicast capacity. It is especially suited for new emerging networks such as ad-hoc and sensor networks. In this work, we investigate the multicast routing problem based on network coding and put forward a practical algorithm to obtain the maximum flow multicast routes in ad-hoc networks. The "conflict phenomenon" that occurs in undirected graphs will also be discussed. Given the developed routing algorithm, we will present the condition for a node to be an encoding node along with a corresponding capacity allocation scheme. We will also analyze the statistical characteristics of encoding nodes and maximum flow in ad-hoc networks based on random graph theory.     

Intra-layer network coding for lossy communication of progressive codes

In this paper we discuss layered multicast (LM) of progressive source codes using network coding. LM is absolutely optimal if different sinks in the network are satisfied up to their max-flow. Since absolutely optimal intra-layer network strategies might not exist for general networks, we present conditions under which an absolutely optimal, intra-layer multicast strategy exists for a given network and how that strategy may be efficiently constructed. We also discuss the problem of designing optimal intra-layer multicast strategies for general directed networks.     

Dynamic wavelength assignment for multicast in all-optical WDM networks to maximize the network capacity

We study the problem of wavelength assignment for multicast in order to maximize the network capacity in all-optical wavelength-division multiplexing networks. The motivation behind this work is to minimize the call blocking probability by maximizing the remaining network capacity after each wavelength assignment. While all previous studies on the same objective concentrate only on the unicast case, we study the problem for the multicast case. For a general multicast tree, we prove that the multicast wavelength assignment problem of maximizing the network capacity is NP-hard and propose two efficient greedy algorithms. We also study the same problem for a special network topology, a bidirectional ring network, which is practically the most important topology for optical networks. For bidirectional ring networks, a special multicast tree with at most two leaf nodes is constructed. Polynomial time algorithms for multicast wavelength assignment to maximize the network capacity exist under such a special multicast tree with regard to different splitting capabilities. Our work is the first effort to study the multicast wavelength assignment problem under the objective of maximizing network capacity.     

A Random Linear Network Coding Approach to Multicast

We present a distributed random linear network coding approach for transmission and compression of information in general multisource multicast networks. Network nodes independently and randomly select linear mappings from inputs onto output links over some field. We show that this achieves capacity with probability exponentially approaching 1 with the code length. We also demonstrate that random linear coding performs compression when necessary in a network, generalizing error exponents for linear Slepian-Wolf coding in a natural way. Benefits of this approach are decentralized operation and robustness to network changes or link failures. We show that this approach can take advantage of redundant network capacity for improved success probability and robustness. We illustrate some potential advantages of random linear network coding over routing in two examples of practical scenarios: distributed network operation and networks with dynamically varying connections. Our derivation of these results also yields a new bound on required field size for centralized network coding on general multicast networks     

Cross-Layer Rate Control, Routing and Scheduling Design for Multicast with Network Coding in Ad Hoc Networks

Network coding is a powerful coding technique that has been proved to be very effective in achieving the maximum multicast capacity. It is especially suited for new emerging networks such as ad-hoc and sensor networks. In this paper, we develop a distributed rate control algorithm for multicast session in ad hoc networks. With random network coding, the algorithm can be implemented in a distributed manner, and work at transport layer to adjust source rates and at network layer to carry out network coding. The scheduling element of our algorithm is a dynamic scheduling policy. The stability of the resulted system is established, and simulation results are provided to support our conclusions.     

Maximum flow and network capacity of network coding for ad-hoc networks

Network coding is an effective way to achieve the maximum flow of multicast networks. In this letter, we focus on the statistical properties of the maximum flow or the capacity of network coding for ad-hoc networks based on random graph models. Theoretical analysis shows that the maximum flow can be modelled as extreme order statistics of Gaussian distribution for both wired and wireless ad-hoc networks as the node number is relatively large under a certain condition. We also investigate the effects of the nodes' covering capabilities on the capacity of network coding.     

基于共享路径和网络编码的光组播容量优化

为了降低光组播路由 的光域网络编码代价和提高达到理论最大光组播容量的 概率,提出一种基于共享链路和网络编 码的优化光组播容量方法。首先设计一种从多条源－ 宿最短路径中选择能达到最大光组播容量的最短路径簇,然后在 最短路径簇中计算路径的共享度,选择共享度高的组播路径传输网络编码信息,构造网络编 码次数最少的光组播编码子图, 解决传统的网络编码组 播路由和最大共享度链路组播路由中存在的网络编码次数过多和达到最大光组播容量概率过 低的问 题。仿真结果表明:本文提出的方法具有最低的网络编码代价,能以最大的概率达到光组播 理论最大容量。     

On achieving maximum multicast throughput in undirected networks

The transmission of information within a data network is constrained by the network topology and link capacities. In this paper, we study the fundamental upper bound of information dissemination rates with these constraints in undirected networks, given the unique replicable and encodable properties of information flows. Based on recent advances in network coding and classical modeling techniques in flow networks, we provide a natural linear programming formulation of the maximum multicast rate problem. By applying Lagrangian relaxation on the primal and the dual linear programs (LPs), respectively, we derive a) a necessary and sufficient condition characterizing multicast rate feasibility, and b) an efficient and distributed subgradient algorithm for computing the maximum multicast rate. We also extend our discussions to multiple communication sessions, as well as to overlay and ad hoc network models. Both our theoretical and simulation results conclude that, network coding may not be instrumental to achieve better maximum multicast rates in most cases; rather, it facilitates the design of significantly more efficient algorithms to achieve such optimality.     

A Constant Bound on Throughput Improvement of Multicast Network Coding in Undirected Networks

Recent research in network coding shows that, joint consideration of both coding and routing strategies may lead to higher information transmission rates than routing only. A fundamental question in the field of network coding is: how large can the throughput improvement due to network coding be? In this paper, we prove that in undirected networks, the ratio of achievable multicast throughput with network coding to that without network coding is bounded by a constant ratio of $2$, i.e., network coding can at most double the throughput. This result holds for any undirected network topology, any link capacity configuration, any multicast group size, and any source information rate. This constant bound $2$ represents the tightest bound that has been proved so far in general undirected settings, and is to be contrasted with the unbounded potential of network coding in improving multicast throughput in directed networks.      

The encoding complexity of network coding

In the multicast network coding problem, a source s needs to deliver h packets to a set of k terminals over an underlying communication network G. The nodes of the multicast network can be broadly categorized into two groups. The first group includes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by h/sup 3/k/sup 2/. Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by h/sup 3/k/sup 2/. We show that the number of encoding nodes may depend both on h and k by presenting acyclic coding networks that require /spl Omega/(h/sup 2/k) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2B+1)h/sup 3/k/sup 2/, where B is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an /spl Nscr/P-hard problem.     

Network Coding in Multicast Routing for Wireless Link Breakage Problem

Link breakage is one of the critical problems that limit the performance of multicast routing in wireless networks. To ease the problem, we apply network coding to the routing operation. In our proposal, data packets are encoded by a random coding scheme. By performing a re-encoding process, the coding scheme is able to keep conveying the data in the network even though link breakage occurs (without the need of waiting for retransmission). To route encoded packets in the network, a disjoint-path tree is used, which is the routing structure constructed by combining a number of multicast trees without the overlapping links among them. Simulation results show that our proposal can effectively ease the impact of link breakage, achieving better packet delivery ratio and higher multicast capacity under different scenarios.     

Design of a node architecture for logic-calculation Nased all-optical network coding scheme

Network coding brings many benefits for multicast networks. It is necessary to introduce network coding into optical networks. Nevertheless, the traditional network coding scheme is hard to be implemented in optical networks because of the weak operation capability in photonic domain. In the paper, we focused on realizing two-channel network coding in all-optical multicast networks. An optical network coding scheme which can be realized via logic shift and logic XOR operations in photonic domain was proposed. Moreover, to perform the network coding scheme the coding node structure was designed and the operation principle and processes were illustrated in detail. In the end of the paper, the performance and the cost of different all-optical multicast mode were compared and analyzed.     

A separation theorem for single-source network coding

In this paper, we consider a point-to-point communication network of discrete memoryless channels. In the network, there are a source node and possibly more than one sink node. Information is generated at the source node and is multicast to each sink node. We allow a node to encode its received information before loading it onto an outgoing channel, where the channels are independent of each other. We also allow the nodes to pass along messages asynchronously. In this paper, we characterize the admissibility of single-source multi-sink communication networks. Our result can be regarded as a network generalization of Shannon's result that feedback does not increase the capacity of a discrete memoryless channels (DMCs), and it implies a separation theorem for network coding and channel coding in such a communication network.     

Multicast routing and bandwidth dimensioning in overlay networks

Multicast services can be provided either as a basic network service or as an application-layer service. Higher level multicast implementations often provide more sophisticated features and can provide multicast services at places where no network layer support is available. Overlay multicast networks offer an intermediate option, potentially combining the flexibility and advanced features of application layer multicast with the greater efficiency of network layer multicast. In this paper, we introduce the multicast routing problem specific to the overlay network environment and the related capacity assignment problem for overlay network planning. Our main contributions are the design of several routing algorithms that optimize the end-to-end delay and the interface bandwidth usage at the multicast service nodes within the overlay network. The interface bandwidth is typically a key resource for an overlay network provider, and needs to be carefully managed in order to maximize the number of users that can be served. Through simulations, we evaluate the performance of these algorithms under various traffic conditions and on various network topologies. The results show that our approach is cost-effective and robust under traffic variations.     

The multicast capacity of deterministic relay networks with no interference

The multicast capacity is determined for networks that have deterministic channels with broadcasting at the transmitters and no interference at the receivers. The multicast capacity is shown to have a cut-set interpretation. It is further shown that one cannot always layer channel and network coding in such networks. The proof of the latter result partially generalizes to discrete memoryless broadcast channels and is used to bound the common rate for problems where one achieves a cut bound on throughput.     

Insufficiency of linear coding in network information flow

It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite-field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite-field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution.     

Bounds on the throughput gain of network coding in unicast and multicast wireless networks

Gupta and Kumar established that the per node throughput of ad hoc networks with multi-pair unicast traffic scales with an increasing number of nodes n as lambda(n) = ominus(1/radic(n log n)), thus indicating that performance does not scale well. However, Gupta and Kumar did not consider network coding and wireless broadcasting, which recent works suggest have the potential to significantly improve throughput. Here, we establish bounds on the improvement provided by such techniques. For random networks of any dimension under either the protocol or physical model that were introduced by Gupta and Kumar, we show that network coding and broadcasting lead to at most a constant factor improvement in per node throughput. For the protocol model, we provide bounds on this factor. We also establish bounds on the throughput benefit of network coding and broadcasting for multiple source multicast in random networks. Finally, for an arbitrary network deployment, we show that the coding benefit ratio is at most O(log n) for both the protocol and physical communication models. These results give guidance on the application space of network coding, and, more generally, indicate the difficulty in improving the scaling behavior of wireless networks without modification of the physical layer.     

Binary linear multicast network coding on acyclic networks: principles and applications in wireless communication networks

Conventional linear multicast can be constructed on any acyclic network by increasing the order of the finite field to a sufficiently large amount over which the multicast is defined. In this paper, we first discuss the reciprocal theorem of the conventional linear multicast and design a linear multicast on any give acyclic network with constant finite field by extending the multicast dimension and relaxing the constraint on the information storage. In particular, we propose the binary linear multicast network coding and the linear multicast with binary coefficients. With the proposed method, the computation complexity for network coding at the intermediate nodes can be significantly reduced; therefore cheap network nodes can be deployed in a large scale due to their low cost for wireless communications. In addition, some applications of the proposed binary linear multicast network coding in wireless communication networks are illustrated and validated.     

Capacity of wireless erasure networks

In this paper, a special class of wireless networks, called wireless erasure networks, is considered. In these networks, each node is connected to a set of nodes by possibly correlated erasure channels. The network model incorporates the broadcast nature of the wireless environment by requiring each node to send the same signal on all outgoing channels. However, we assume there is no interference in reception. Such models are therefore appropriate for wireless networks where all information transmission is packetized and where some mechanism for interference avoidance is already built in. This paper looks at multicast problems over these networks. The capacity under the assumption that erasure locations on all the links of the network are provided to the destinations is obtained. It turns out that the capacity region has a nice max-flow min-cut interpretation. The definition of cut-capacity in these networks incorporates the broadcast property of the wireless medium. It is further shown that linear coding at nodes in the network suffices to achieve the capacity region. Finally, the performance of different coding schemes in these networks when no side information is available to the destinations is analyzed.