We propose a distributed scheme for practical network coding that obviates the need for centralized knowledge of the graph topology, the encoding functions, and the decoding functions, and furthermore obviates the need for information to be communicated synchronously through the network. The result is a practical system for network coding that is robust to random packet loss and delay as well as robust to any changes in the network topology or capacity due to joins, leaves, node or link failures, congestion, and so on. We simulate such a practical network coding system using the network topologies of several commercial Internet Service Providers, and demonstrate that it can achieve close to the theoretically optimal performance.

Abstract not found

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 is arbitrarily larger than if coding is not allowed. Previous work does not show how to construct such coding schemes efficiently even for directed acyclic graphs with edges of unit capacity. We give deterministic polynomial time algorithms and even faster randomized algorithms for solving this problem using linear codes. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.

We consider a randomized network coding approach for multicasting from several sources over a network, in which nodes independently and randomly select linear mappings from inputs onto output links over some field. This approach was first described in [3], which gave, for acyclic delay-free networks, a bound on error probability, in terms of the number of receivers and random coding output links, that decreases exponentially with code length. The proof was based on a result in [2] relating algebraic network coding to network flows. In this paper, we generalize these results to networks with cycles and delay. We also show, for any given acyclic network, a tighter bound in terms of the probability of connection feasibility in a related network problem with unreliable links. From this we obtain a success probability bound for randomized network coding in link-redundant networks with unreliable links, in terms of link failure probability and amount of redundancy.

Some forms of ad-hoc networks need to operate in extremely performance-challenged environments where end-to-end connectivity is rare. Such environments can be found for example in very sparse mobile networks where nodes ”meet ” only occasionally and are able to exchange information, or in wireless sensor networks where nodes sleep most of the time to conserve energy. Forwarding mechanisms in such networks usually resort to some form of intelligent flooding, as for example in probabilistic routing. We propose a communication algorithm that significantly reduces the overhead of probabilistic routing algorithms, making it a suitable building block for a delay-tolerant network architecture. Our forwarding scheme is based on network coding. Nodes do not simply forward packets they overhear but may send out information that is coded over the contents of several packets they received. We show by simulation that this algorithm achieves the reliability and robustness of flooding at a small fraction of the overhead.

Abstract — The minimum energy required to transmit a bit of information through a network characterizes the most economical way to communicate in a network. In this paper, we show that under a layered model of wireless networks, the minimum energyper-bit for multicasting in a mobile ad hoc network can be found by a linear program; the minimum energy-per-bit can be attained by performing network coding. Compared with conventional routing solutions, network coding not only promises a potentially lower energy-per-bit, but also enables the optimal solution to be found in polynomial time, in sharp contrast with the NPhardness of constructing the minimum-energy multicast tree as the optimal routing solution. We further show that the minimum energy multicast formulation is equivalent to a cost minimization with linear edge-based pricing, where the edge prices are the energy-per-bits of the corresponding physical broadcast links. This paper also investigates minimum energy multicasting with routing. Due to the linearity of the pricing scheme, the minimum energy-per-bit for routing is achievable by using a single distribution tree. A characterization of the admissible rate region for routing with a single tree is presented. The minimum energy-per-bit for multicasting with routing is found by an integer linear program. We show that the relaxation of this integer linear program, studied earlier in the Steiner tree literature, can now be interpreted as the optimization for minimum energy multicasting with network coding. In short, this paper presents a unifying study of minimum energy multicasting with network coding and routing. Index Terms — Network coding, routing, multicast, Steiner tree, wireless ad hoc networks, energy efficiency, mobility.

We study the problem of simultaneously disseminating multiple messages in a large network in a decentralized and distributed manner. We consider a network with n nodes and k (k = O(n)) messages spread throughout the network to start with, but not all nodes have all the messages. Our communication model is such that the nodes communicate in discrete-time steps, and in every time-step, each node communicates with a random communication partner chosen uniformly from all the nodes (known as the random phone call model). The system is bandwidth limited and in each time-step, only one message can be transmitted. The goal is to disseminate rapidly all the messages among all the nodes. We study the time required for this dissemination to occur with high probability, and also in expectation. We present a protocol based on random linear coding (RLC) that disseminates all the messages among all the nodes in O(n) time, which is order optimal, if we ignore the small overhead associated with each transmission. The overhead does not depend on the size of the messages and is less than 1 % for k = 100 and messages of size 100 KB. We also consider a store and forward mechanism without coding, which is a natural extension of gossip-based dissemination with one message in the network. We show that, such an uncoded scheme can do no better than a sequential approach (instead of doing it simultaneously) of disseminating the messages which takes Θ(n ln(n)) time, since disseminating a single message in a gossip network takes Θ(ln(n)) time. 1

Recent work in network coding shows that, it is necessary to consider both the routing and coding strategies to achieve optimal throughput of information transmission in data networks. So far, most research on network coding has focused on the model of directed networks, where each communication link has a fixed direction. In this paper, we study the benefits of network coding in undirected networks, where each communication link is bidirectional. Our theoretical results show that, for a single unicast or broadcast session, there are no improvements with respect to throughput due to network coding. In the case of a single multicast session, such an improvement is bounded by a factor of two, as long as half integer routing is permitted. This is dramatically different from previous results obtained in directed networks. We also show that multicast throughput in an undirected network is independent of the selection of the sender within the multicast group. We finally show that, rather than improving the optimal achievable throughput, the benefit of network coding is to significantly facilitate the design of efficient algorithms to compute and achieve such optimal throughput. I.

The problem of reliably reconstructing a function of sources over a multiple-access channel is considered. It is shown that there is no source-channel separation theorem even when the individual sources are independent. Joint sourcechannel strategies are developed that are optimal when the structure of the channel probability transition matrix and the function are appropriately matched. Even when the channel and function are mismatched, these computation codes often outperform separation-based strategies. Achievable distortions are given for the distributed refinement of the sum of Gaussian sources over a Gaussian multiple-access channel with a joint source-channel lattice code. Finally, computation codes are used to determine the multicast capacity of finite field multiple-access networks, thus linking them to network coding.

With the constraints of network topologies and link capacities, achieving the optimal end-to-end throughput in data networks has been known as a fundamental but computationally hard problem. In this paper, we seek efficient solutions to the problem of achieving optimal throughput in data networks, with single or multiple unicast, multicast and broadcast sessions. Although previous approaches lead to solving NP-complete problems, we show the surprising result that, facilitated by the recent advances of network coding, computing the strategies to achieve the optimal end-to-end throughput can be performed in polynomial time. This result holds for one or more communication sessions, as well as in the overlay network model. Supported by empirical studies, we present the surprising observation that in most topologies, applying network coding may not improve the achievable optimal throughput; rather, it facilitates the design of significantly more efficient algorithms to achieve such optimality. Index terms: Graph theory, Information theory, Mathematical programming/optimization, Simulations.