线性网络编码研究

最大流最小割定理决定了网络的最大吞吐量.近来研究表明,网络编码可以使这-理论在多播方式的网络环境中得以实现.网络编码的提出彻底地改变了计算机通信网络中的信息处理方式,其研究结合了信息论、计算机通信网络、组播技术、多用户信息论和图论等很多方面的知识.文中简要介绍了网络编码的基本原理,线性网络编码的基本概念及其发展,并且提出一种实现线性网络编码的算法.     

On Broadcast Stability of Queue-Based Dynamic Network Coding Over Erasure Channels

The transmission of packets is considered from one source to multiple receivers over single-hop erasure channels. The objective is to evaluate the stability properties of different transmission schemes with and without network coding. First, the throughput limitation of retransmission schemes is discussed and the stability benefits are shown for randomly coded transmissions, which, however, need not optimize the stable throughput for finite coding field size and finite packet block size. Next, a dynamic scheme is introduced for distributing packets among virtual queues depending on the channel feedback and performing linear network coding based on the instantaneous queue contents. The difference of the maximum stable throughput from the min-cut rate is bounded as function of the order of erasure probabilities depending on the complexity allowed for network coding and queue management. This queue-based network coding scheme can asymptotically optimize the stable throughput to the max-flow min-cut bound, as the erasure probabilities go to zero. This is realized for a finite coding field size without accumulating packet blocks at the source to start network coding. The comparison of random and queue-based dynamic network coding with plain retransmissions opens up new questions regarding the tradeoffs of stable throughput, packet delay, overhead, and complexity.      

On Joint MAC and Network Coding in Wireless Ad Hoc Networks

This paper addresses network coding in wireless networks in conjunction with medium access control (MAC). It is known that coding over wired networks enables connections with rates that cannot be achieved by routing. However, the properties of wireless networks (e.g., omnidirectional transmissions, destructive interference, single transceiver per node, finite energy) modify the formulation of time-varying network coding in a way that reflects strong interactions with underlying MAC protocols and deviates from the classical approach used in wired network coding. To perform network coding over conflict-free transmission schedules, predetermined network realizations are separately activated by a time-division mechanism and the content of network flows is derived through network coding to optimize performance measures such as achievable throughput and energy costs. A systematic method is presented to construct linear wireless network codes and interactions with MAC schedules are discussed under wireless assumptions. Network coding is also extended to operate with arbitrary (random or scheduled access based) MAC protocols. Alternatively, conflict-free transmission schedules are jointly constructed with network codes by decomposing wireless networks into subtrees and employing graph coloring on simplified subtree graphs. Finally, network coding and plain routing are compared in terms of throughput, energy and delay performance under different MAC solutions.     

On capacity of random wireless networks with physical-layer network coding

Throughput capacity of a random wireless network has been studied extensively in the literature. Most existing studies were based on the assumption that each transmission involves only one transmitter in order to avoid interference. However, recent studies on physical-layer network coding (PLNC) have shown that such an assumption can be relaxed to improve throughput performance of a wireless network. In PLNC, signals from different senders can be transmitted to the same receiver in the same channel simultaneously. In this paper, we investigate the impact of PLNC on throughput capacity of a random wireless network. Our study reveals that, although PLNC scheme does not change the scaling law, it can improve throughput capacity by a fixed factor. Specifically, for a one-dimensional network, we observe that PLNC can eliminate the effect of interference in some scenarios. A tighter capacity bound is derived for a two-dimensional network. In addition, we also show achievable lower bounds for random wireless networks with network coding and PLNC.     

On the improvement of scaling laws for large-scale MANETs with network coding

This paper investigates the problem of how much benefit network coding can contribute to the network performance in terms of throughput, delay, and storage requirements for mobile ad hoc networks (MANETs), compared to when only replication, storage and forwarding are allowed in relay nodes. We characterize the throughput-delay-storage tradeoffs under different node mobility patterns, i.e., i.i.d. and random walk mobility, with and without network coding. Our results show that when random linear coding instead of replication is used in MANETs, an order improvement on the scaling laws of MANETs can be achieved. Note that previous work showed that network coding could only provide constant improvement on the throughput of static wireless networks. Our work thus differentiates MANETs from static wireless networks by the role network coding plays.     

Polynomial time algorithms for multicast network code construction

The famous max-flow min-cut theorem states that a source node s can send information through a network (V, E) to a sink node t at a rate determined by the min-cut separating s and t. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.     

On the capacity of network coding for random networks

We study the maximum flow possible between a single-source and multiple terminals in a weighted random graph (modeling a wired network) and a weighted random geometric graph (modeling an ad-hoc wireless network) using network coding. For the weighted random graph model, we show that the network coding capacity concentrates around the expected number of nearest neighbors of the source and the terminals. Specifically, for a network with a single source, l terminals, and n relay nodes such that the link capacities between any two nodes is independent and identically distributed (i.i.d.) /spl sim/X, the maximum flow between the source and the terminals is approximately nE[X] with high probability. For the weighted random geometric graph model where two nodes are connected if they are within a certain distance of each other we show that with high probability the network coding capacity is greater than or equal to the expected number of nearest neighbors of the node with the least coverage area.     

Wireless network coding in slotted aloha with two-hop unbalanced traffic

This paper deals with two representative unbalanced traffic cases for two-hop wireless relay access systems employing network coding and a slotted ALOHA protocol. Network coding is a recent and highly regarded technology for capacity enhancement with multiple unicast and multisource multicast networks. We have analyzed the performance of network coding on a two-hop wireless relay access system employing the slotted ALOHA under a balanced bidirectional traffic. The relay nodes will generally undergo this unbalanced multidirectional traffic but the impact of this unbalanced traffic on network coding has not been analyzed. This paper provides closed-form expressions for the throughput and packet delay for two-hop unbalanced bidirectional traffic cases both with and without network coding even if the buffers on nodes are unsaturated. The analytical results are mainly derived by solving queueing systems for the buffer behavior at the relay node. The results show that the transmission probability of the relay node is a design parameter that is crucial to maximizing the achievable throughput of wireless network coding in slotted ALOHA on two-hop unbalanced traffic cases. Furthermore, we show that the throughput is enhanced even if the traffic at the relay node is unbalanced.     

On average throughput and alphabet size in network coding

We examine the throughput benefits that network coding offers with respect to the average throughput achievable by routing, where the average throughput refers to the average of the rates that the individual receivers experience. We relate these benefits to the integrality gap of a standard linear programming formulation for the directed Steiner tree problem. We describe families of configurations over which network coding at most doubles the average throughput, and analyze a class of directed graph configurations with N receivers where network coding offers benefits proportional to /spl radic/N. We also discuss other throughput measures in networks, and show how in certain classes of networks, average throughput bounds can be translated into minimum throughput bounds, by employing vector routing and channel coding. Finally, we show configurations where use of randomized coding may require an alphabet size exponentially larger than the minimum alphabet size required.     

An outer bound for multisource multisink network coding with minimum cost consideration

The max-flow min-cut bound is a fundamental result in the theory of communication networks, which characterizes the optimal throughput for a point-to-point communication network. The recent work of Ahlswede et al. extended it to single-source multisink multicast networks and Li et al. proved that this bound can be achieved by linear codes. Following this line, Erez and Feder as well as Ngai and Yeung proved that the max-flow min-cut bound remains tight in single-source two-sink nonmulticast networks. But the max-flow min-cut bound is in general quite loose (see Yeung, 2002). On the other hand, the admissible rate region of communication networks has been studied by Yeung and Zhang as well as Song and Yeung, but the bounds obtained by these authors are not explicit. In this work, we prove a new explicit outer bound for arbitrary multisource multisink networks and demonstrate its relation with the minimum cost network coding problem. We also determine the capacity region for a special class of three-layer networks.     

基于硬件逻辑的网络编码编解码算法

提出了一种基于硬件逻辑实现的通用网络编码编解码算法。编码算法运用随机线性网络编码对数据分组进行编码,解码算法则运用克莱默法则进行解码。对编码器和解码器的算法和结构进行了详细的设计,并最终运用硬件描述语言在NetFPGA开发板上实现了该设计。测试结果表明,与传统的路由节点相比,使用线速的网络编码编解码器的网络能够达到最大流最小割定理所确定的流量极限,并且端到端的传输延迟稳定在一个很小的常数上。     

Multicast with network coding in application-layer overlay networks

All of the advantages of application-layer overlay networks arise from two fundamental properties: 1) the network nodes in an overlay network, as opposed to lower-layer network elements such as routers and switches, are end systems and have capabilities far beyond basic operations of storing and forwarding; 2) the overlay topology, residing above a densely connected Internet protocol-layer wide-area network, can be constructed and manipulated to suit one's purposes. We seek to improve end-to-end throughput significantly in application-layer multicast by taking full advantage of these unique characteristics. This objective is achieved with two novel insights. First, we depart from the conventional view that overlay nodes can only replicate and forward data. Rather, as end systems, these overlay nodes also have the full capability of encoding and decoding data at the message level using efficient linear codes. Second, we depart from traditional wisdom that the multicast topology from source to receivers needs to be a tree, and propose a novel and distributed algorithm to construct a two-redundant multicast graph (a directed acyclic graph) as the multicast topology, on which network coding is applied. We design our algorithm such that the costs of link stress and stretch are explicitly considered as constraints and minimized. We extensively evaluate our algorithm by provable analytical and experimental results, which show that the introduction of two-redundant multicast graph and network coding may indeed bring significant benefits, essentially doubling the end-to-end throughput in most cases.     

Throughput and Delay in Random Wireless Networks With Restricted Mobility

Grossglauser and Tse (2001) introduced a mobile random network model where each node moves independently on a unit disk according to a stationary uniform distribution and showed that a throughput of Theta(1) is achievable. El Gamal, Mammen, Prabhakar, and Shah (2004) showed that the delay associated with this throughput scales as Theta(nlogn), when each node moves according to an independent random walk. In a later work, Diggavi, Grossglauser, and Tse (2002) considered a random network on a sphere with a restricted mobility model, where each node moves along a randomly chosen great circle on the unit sphere. They showed that even with this one-dimensional restriction on mobility, constant throughput scaling is achievable. Thus, this particular mobility restriction does not affect the throughput scaling. This raises the question whether this mobility restriction affects the delay scaling. This correspondence studies the delay scaling at Theta(1) throughput for a random network with restricted mobility. First, a variant of the scheme presented by Diggavi, Grossglauser, and Tse (2002) is presented and it is shown to achieve Theta(1) throughput using different (and perhaps simpler) techniques. The exact order of delay scaling for this scheme is determined, somewhat surprisingly, to be of Theta(nlogn), which is the same as that without the mobility restriction. Thus, this particular mobility restriction does not affect either the maximal throughput scaling or the corresponding delay scaling of the network. This happens because under this 1-D restriction, each node is in the proximity of every other node in essentially the same manner as without this restriction     

Transmission scheduling in a multi‐channel wireless network with bidirectional relaying links

Using network coding in a wireless network can potentially improve the network throughput. On the other hand, it increases the complexity of resource allocations as the quality of one transmission is affected by the link conditions of the transmitter to multiple receivers. In this work, we study time slot scheduling and channel allocations jointly for a network with bidirectional relaying links, where the two end nodes of each link can exchange data through a relay node. Two scenarios are considered when the relay node forwards packets to the end nodes. In the first scenario, the relay node always forwards network‐coded packets to both end nodes simultaneously; in the second scenario, the relay node opportunistically uses network coding for two‐way relaying and traditional one‐way relaying. For each scenario, an optimization problem is first formulated for maximizing the total network throughput. The optimum scheduling is not causal because it requires future information of channel conditions. We then propose heuristic scheduling schemes. The slot‐based scheduling maximizes the total transmission rate of all the nodes at each time slot, and the node‐based scheduling schedules transmissions based on achievable transmission rates of individual nodes at different channels. The node‐based one has lower complexity than the slot‐based one. Our results indicate that although the node‐based scheduling achieves slightly lower throughput than the slot‐based one, both the proposed scheduling schemes are very effective in the sense that the difference between their throughput and the optimum scheduling is relatively small in different network settings. Copyright © 2015 John Wiley & Sons, Ltd.     

ON THE OPTIMAL MULTI-RATE THROUGHPUT FOR MULTICAST WITH NETWORK CODING

This paper investigates the maximal achievable multi-rate throughput problem of a multicast session at the presence of network coding. Deviating from previous works which focus on single-rate network coding, our work takes the heterogeneity of sinks into account and provides multiple data layers to address the problem. Firstly formulated is the maximal achievable throughput problem with the assumption that the data layers are independent and layer rates are static. It is proved that the problem in this case is, unfortunately, Non-deterministic Polynomial-time （NP）-hard. In addition, our formulation is extended to the problems with dependent layers and dynamic layers. Furthermore, the approximation algorithm which satisfies certain fair- ness is proposed.     

Queuing network models for delay analysis of multihop wireless ad hoc networks

In this paper we analyze the average end-to-end delay and maximum achievable per-node throughput in random access multihop wireless ad hoc networks with stationary nodes. We present an analytical model that takes into account the number of nodes, the random packet arrival process, the extent of locality of traffic, and the back off and collision avoidance mechanisms of random access MAC. We model random access multihop wireless networks as open G/G/1 queuing networks and use the diffusion approximation in order to evaluate closed form expressions for the average end-to-end delay. The mean service time of nodes is evaluated and used to obtain the maximum achievable per-node throughput. The analytical results obtained here from the queuing network analysis are discussed with regard to similarities and differences from the well established information-theoretic results on throughput and delay scaling laws in ad hoc networks. We also investigate the extent of deviation of delay and throughput in a real world network from the analytical results presented in this paper. We conduct extensive simulations in order to verify the analytical results and also compare them against NS-2 simulations.     

Bounds and parameter optimization of medium access control coding for wireless ad hoc and sensor networks

The use of codes to schedule transmissions is an attractive technique able to guarantee a non-zero throughput medium access performance for the nodes of a wireless ad hoc or sensor network regardless of network topology variations. Some authors refer to this technique as topology-transparent scheduling. In this paper, we use the term MAC coding in order to emphasize the exclusive use of codes to achieve topology-transparency within the MAC sub-layer. We present a new upper bound expression on the guaranteed throughput achievable by any linear code used in a MAC coding context. This bound proves to be tighter than the one obtained when the minimum distance of the code is equal to its length. Additionally, we derive new and simple closed analytical expressions for the parameters of maximum distance separable codes that maximize the minimum, average, or joint minimum-average throughput of MAC coding. The optimization methods presented here are also applicable to other codes with available analytical expressions for their minimum distance and distance distribution. Finally, we present system-level simulation results of MAC coding on static and dynamic topologies with mobility and including wireless channel errors. Throughput simulation results are compared with their corresponding analytical expressions and to a random scheduling approach. The results show agreement with analysis and confirm the robustness of MAC coding in maintaining minimum levels of performance with good average performance and graceful degradation.     

Star-Structure Network Coding for Multiple Unicast Sessions in Wireless Mesh Networks

In this paper, first, we propose Star-NC, a new network coding (NC) scheme for multiple unicast sessions in an n-input n-output star structure. Then, we evaluate the network throughput of this coding scheme in wireless mesh network over the traditional non-NC transmission. Our scheme benefits from the proximity of all the nodes around the relay node and employs a more general form of overhearing different from other schemes such as COPE. We found that the gain of our NC scheme depends on both the star size and the routing pattern of the unicast transmissions. Based on this, we identify both the situations which the maximum gain is achievable and a lower bound for the expected value of the gain in the case of random routing pattern. Next, we propose an analytical framework for studying throughput gain of our Star-NC scheme in general wireless network topologies. Our theoretical formulation via linear programming provides a method for finding source-destination routes and utilizing the best choices of our NC scheme to maximize the throughput. Finally, we evaluate our model for various networks, traffic models and routing strategies over coding-oblivious routing. We also compare the throughput gain of our scheme with COPE-type NC scheme. We show that Star-NC exploits new coding opportunities different from COPE-type NC and thus can be used with or without this scheme. The results show that Star-NC has often better performance than COPE for a directional traffic model which is a typical model in wireless mesh networks. Moreover, we found that, joint Star and COPE-type NC has better throughput performance than each of Star or COPE alone.     

Performance analysis of slotted ALOHA and network coding for single-relay multi-user wireless networks

Deployment of wireless relay nodes can enhance system capacity, extend wireless service coverage, and reduce energy consumption in wireless networks. Network coding enables us to mix two or more packets into a single coded packet at relay nodes and improve performances in wireless relay networks. In this paper, we succeed in developing analytical models of the throughput and delay on slotted ALOHA (S-ALOHA) and S-ALOHA with network coding (S-ALOHA/NC) for single-relay multi-user wireless networks with bidirectional data flows. The analytical models involve effects of queue saturation and unsaturation at the relay node. The throughput and delay for each user node can be extracted from the total throughput and delay by using the analytical models. One can formulate various optimization problems on traffic control in order to maximize the throughput, minimize the delay, or achieve fairness of the throughput or the delay. In particular, we clarify that the total throughput is enhanced in the S-ALOHA/NC protocol on condition that the transmission probability at the relay node is set at the value on the boundary between queue saturation and unsaturation. Our analysis provides achievable regions in throughput on two directional data flows at the relay node for both the S-ALOHA and S-ALOHA/NC protocols. As a result, we show that the achievable region in throughput can be enhanced by using network coding and traffic control.     

Lossy distributed source coding using graphs

In this letter, lossy distributed source coding using graphs is considered. This corresponds to the source coding part of the graph-based framework for transmission of analog correlated sources over the multiple-access channel (MAC). Consequently, it is shown that a pair of analog correlated sources can be reliably represented into a bipartite graph by allowing certain amount of distortion. An achievable rate-distortion region for this problem is also provided. Therefore, it can be concluded that, for transmission of any (both discrete and continuous) set of correlated sources over MACs, graphs can be used as discrete interface between source coding and channel coding modules.