Network coding promises to significantly impact the way communications networks are designed, operated, and understood. The first book to present a unified and intuitive overview of the theory, applications, challenges, and future directions of this emerging field, this is a must-have resource for those working in wireline or wireless networking. *Uses an engineering approach - explains the ideas and practical techniques *Covers mathematical underpinnings, practical algorithms, code selection, security, and network management *Discusses key topics of inter-session (non-multicast) network coding, lossy networks, lossless networks, and subgraph-selection algorithms Starting with basic concepts, models, and theory, then covering a core subset of results with full proofs, Ho and Lun provide an authoritative introduction to network coding that supplies both the background to support research and the practical considerations for designing coded networks. This is an essential resource for graduate students and researchers in electronic and computer engineering and for practitioners in the communications industry.

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Network Coding: An Introduction

Erasure coding techniques are used to increase the reliability of distributed storage systems while minimizing storage overhead. Also of interest is minimization of the bandwidth required to repair the system following a node failure. In a recent paper, Wu et al. characterize the tradeoff between the repair bandwidth and the amount of data stored per node. They also prove the existence of regenerating codes that achieve this tradeoff. In this paper, we introduce Exact Regenerating Codes, which are regenerating codes possessing the additional property of being able to duplicate the data stored at a failed node. Such codes require low processing and communication overheads, making the system practical and easy to maintain. Explicit construction of exact regenerating codes is provided for the minimum bandwidth point on the storage-repair bandwidth tradeoff, relevant to distributed-mail-server applications. A subspace based approach is provided and shown to yield necessary and sufficient conditions on a linear code to possess the exact regeneration property as well as prove the uniqueness of our construction. Also included in the paper, is an explicit construction of regenerating codes for the minimum storage point for parameters relevant to storage in peer-to-peer systems. This construction supports a variable number of nodes and can handle multiple, simultaneous node failures. All constructions given in the paper are of low complexity, requiring low field size in particular.

Explicit construction of optimal exact regenerating codes for distributed storage

We consider the problem of data exchange by a group of closely-located wireless nodes. In this problem each node holds a set of packets and needs to obtain all the packets held by other nodes. Each of the nodes can broadcast the packets in its possession (or a combination thereof) via a noiseless broadcast channel of capacity one packet per channel use. The goal is to minimize the total number of transmissions needed to satisfy the demands of all the nodes, assuming that they can cooperate with each other and are fully aware of the packet sets available to other nodes. This problem arises in several practical settings, such as peer-to-peer systems and wireless data broadcast. In this paper, we establish upper and lower bounds on the optimal number of transmissions and present an efficient algorithm with provable performance guarantees. The effectiveness of our algorithms is established through numerical simulations.

On coding for cooperative data exchange

In this paper, we consider the problem of minimizing the mean completion delay in wireless broadcast for instantly decodable network coding. We first formulate the problem as a stochastic shortest path (SSP) problem. Although finding the packet selection policy using SSP is intractable, we use this formulation to draw the theoretical properties of efficient selection algorithms. Based on these properties, we propose a simple online selection algorithm that efficiently minimizes the mean completion delay of a frame of broadcast packets, compared to the random and greedy selection algorithms with a similar computational complexity. Simulation results show that our proposed algorithm indeed outperforms these random and greedy selection algorithms.

On Minimizing Broadcast Completion Delay for Instantly Decodable Network Coding

In this paper, we consider the problem of minimizing the completion delay for instantly decodable network coding (IDNC) in wireless multicast and broadcast scenarios. We are interested in this class of network coding due to its numerous benefits, such as low decoding delay, low coding and decoding complexities, and simple receiver requirements. We first extend the IDNC graph, which represents all feasible IDNC coding opportunities, to efficiently operate in both multicast and broadcast scenarios. We then formulate the minimum completion delay problem for IDNC as a stochastic shortest path (SSP) problem. Although finding the optimal policy using SSP is intractable, we use this formulation to draw the theoretical guidelines for the policies that can minimize the completion delay in IDNC. Based on these guidelines, we design a maximum weight clique selection algorithm, which can efficiently reduce the IDNC completion delay in polynomial time. We also design a quadratic-time heuristic clique selection algorithm, which can operate in real-time applications. Simulation results show that our proposed algorithms significantly reduce the IDNC completion delay compared to the random and maximum-rate algorithms, and almost achieve the global optimal completion delay performance over all network codes in broadcast scenarios.

pdf/completion-delay-minimization-for-instantly-decodable-4c1ruwwzyp.pdf

Completion delay minimization for instantly decodable network codes

We are concerned with designing feedback-based adaptive network coding schemes with the aim of minimizing decoding delay in each transmission in packet-based erasure networks. We study systems where each packet brings new information to the destination regardless of its order and require the packets to be instantaneously decodable. We first formulate the decoding delay minimization problem as an integer linear program and then propose efficient algorithms for finding its optimal solution(s). We show that our problem formulation is applicable to memoryless erasures as well as Gilbert-Elliott erasures with memory. We then propose a number of heuristic algorithms with worst case linear execution complexity that can be used when an optimal solution cannot be found in a reasonable time. We verify the delay and speed performance of our techniques through numerical analysis. This analysis reveals that by taking channel memory into account in network coding decisions, one can considerably reduce decoding delays.

/pdf/an-optimal-adaptive-network-coding-scheme-for-minimizing-r3vg9cy7ra.pdf

An optimal adaptive network coding scheme for minimizing decoding delay in broadcast erasure channels

We consider the throughput-delay tradeoff in network coded transmission over erasure broadcast channels. Interested in minimizing decoding delay, we formulate the problem of instantly decodable network coding as an integer linear program and propose algorithms to solve it heuristically. In particular, we investigate channels with memory and propose algorithms that can exploit channel erasure dependence to increase throughput and decrease delay.

Adaptive network coding for broadcast channels

We consider multicasting using random linear network coding over a multihop wireless network in the bandwidth limited regime. We address the associated medium access problem and propose a scheduling technique that activates hyperarcs rather than links, as in classical scheduling approaches. We encapsulate the constraints on valid network configurations in a conflict graph model and formulate a joint optimization problem taking into account both the network coding subgraph and the schedule. Next, using Lagrangian relaxation, we decompose the overall problem into two subproblems, a multiple-shortest-paths problem and a maximum weighted stable set (MWSS) problem. We show that if we use a greedy heuristic for the MWSS part of the problem, the overall algorithm is completely distributed. We provide extensive simulation results for both the centralized optimal and the decentralized algorithms. The optimal algorithm improves performance by up to a factor of two over widely used techniques such as orthogonal or two-hop-constrained scheduling. The decentralized algorithm is shown to buy its distributed operation with some throughput losses. Experimental results on randomly generated networks suggest that these losses are not large. Finally, we study the power consumption of our scheme and quantify the tradeoff between power and bandwidth efficiency.

Scheduling for network-coded multicast

Achievable rates are derived for reliable communication in networks with multi-access interference. Capacity is established for some such networks. A network flow formulation is developed that includes rate gains and losses caused by correlating the multi-access channel inputs.

Reliable Communication in Networks with Multi-access Interference

In this paper, we survey the application of linear network coding to a multiple unicasts scenario in directed graphs. We discuss related work concerning the complexity of the construction of capacity-achieving linear network codes. We briefly review the algebraic formulation of the problem and in the remainder of the paper, we investigate two approaches to construct network codes. One is based on state-space realizations and the other is a suboptimal, yet simpler approach based on linear optimization

Approaches to Network Coding for Multiple Unicasts

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