A Deterministic Polynomial--Time Algorithm for Constructing a Multicast Coding Scheme for Linear Deterministic Relay Networks

We propose a new way to construct a multicast coding scheme for linear deterministic relay networks. Our construction can be regarded as a generalization of the well-known multicast network coding scheme of Jaggi et al. to linear deterministic relay networks and is based on the notion of flow for a unicast session that was introduced by the authors in earlier work. We present randomized and deterministic polynomial--time versions of our algorithm and show that for a network with $g$ destinations, our deterministic algorithm can achieve the capacity in $\left\lceil \log(g+1)\right\rceil$ uses of the network.Comment: 12 pages, 2 figures, submitted to CISS 201

Network coding via evolutionary algorithms

Network coding (NC) is a relatively recent novel technique that generalises network operation beyond traditional store-and-forward routing, allowing intermediate nodes to combine independent data streams linearly. The rapid integration of bandwidth-hungry applications such as video conferencing and HDTV means that NC is a decisive future network technology. NC is gaining popularity since it offers significant benefits, such as throughput gain, robustness, adaptability and resilience. However, it does this at a potential complexity cost in terms of both operational complexity and set-up complexity. This is particularly true of network code construction. Most NC problems related to these complexities are classified as non deterministic polynomial hard (NP-hard) and an evolutionary approach is essential to solve them in polynomial time. This research concentrates on the multicast scenario, particularly: (a) network code construction with optimum network and coding resources; (b) optimising network coding resources; (c) optimising network security with a cost criterion (to combat the unintentionally introduced Byzantine modification security issue). The proposed solution identifies minimal configurations for the source to deliver its multicast traffic whilst allowing intermediate nodes only to perform forwarding and coding. In the method, a preliminary process first provides unevaluated individuals to a search space that it creates using two generic algorithms (augmenting path and linear disjoint path. An initial population is then formed by randomly picking individuals in the search space. Finally, the Multi-objective Genetic algorithm (MOGA) and Vector evaluated Genetic algorithm (VEGA) approaches search the population to identify minimal configurations. Genetic operators (crossover, mutation) contribute to include optimum features (e.g. lower cost, lower coding resources) into feasible minimal configurations. A fitness assignment and individual evaluation process is performed to identify the feasible minimal configurations. Simulations performed on randomly generated acyclic networks are used to quantify the performance of MOGA and VEGA

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

Design and Analysis of Low Complexity Network Coding Schemes

In classical network information theory, information packets are treated as commodities, and the nodes of the network are only allowed to duplicate and forward the packets. The new paradigm of network coding, which was introduced by Ahlswede et al., states that if the nodes are permitted to combine the information packets and forward a function of them, the throughput of the network can dramatically increase. In this dissertation we focused on the design and analysis of low complexity network coding schemes for different topologies of wired and wireless networks. In the first part we studied the routing capacity of wired networks. We provided a description of the routing capacity region in terms of a finite set of linear inequalities. We next used this result to study the routing capacity region of undirected ring networks for two multimessage scenarios. Finally, we used new network coding bounds to prove the optimality of routing schemes in these two scenarios. In the second part, we studied node-constrained line and star networks. We derived the multiple multicast capacity region of node-constrained line networks based on a low complexity binary linear coding scheme. For star networks, we examined the multiple unicast problem and offered a linear coding scheme. Then we made a connection between the network coding in a node-constrained star network and the problem of index coding with side information. In the third part, we studied the linear deterministic model of relay networks (LDRN). We focused on a unicast session and derived a simple capacity-achieving transmission scheme. We obtained our scheme by a connection to the submodular flow problem through the application of tools from matroid theory and submodular optimization theory. We also offered polynomial-time algorithms for calculating the capacity of the network and the optimal coding scheme. In the final part, we considered the multicasting problem in an LDRN and proposed a new way to construct a coding scheme. Our construction is based on the notion of flow for a unicast session in the third part of this dissertation. We presented randomized and deterministic polynomial-time versions of our algorithm

Universal and Robust Distributed Network Codes

Random linear network codes can be designed and implemented in a distributed manner, with low computational complexity. However, these codes are classically implemented over finite fields whose size depends on some global network parameters (size of the network, the number of sinks) that may not be known prior to code design. Also, if new nodes join the entire network code may have to be redesigned. In this work, we present the first universal and robust distributed linear network coding schemes. Our schemes are universal since they are independent of all network parameters. They are robust since if nodes join or leave, the remaining nodes do not need to change their coding operations and the receivers can still decode. They are distributed since nodes need only have topological information about the part of the network upstream of them, which can be naturally streamed as part of the communication protocol. We present both probabilistic and deterministic schemes that are all asymptotically rate-optimal in the coding block-length, and have guarantees of correctness. Our probabilistic designs are computationally efficient, with order-optimal complexity. Our deterministic designs guarantee zero error decoding, albeit via codes with high computational complexity in general. Our coding schemes are based on network codes over ``scalable fields". Instead of choosing coding coefficients from one field at every node, each node uses linear coding operations over an ``effective field-size" that depends on the node's distance from the source node. The analysis of our schemes requires technical tools that may be of independent interest. In particular, we generalize the Schwartz-Zippel lemma by proving a non-uniform version, wherein variables are chosen from sets of possibly different sizes. We also provide a novel robust distributed algorithm to assign unique IDs to network nodes.Comment: 12 pages, 7 figures, 1 table, under submission to INFOCOM 201

Optimal Deterministic Polynomial-Time Data Exchange for Omniscience

We study the problem of constructing a deterministic polynomial time algorithm that achieves omniscience, in a rate-optimal manner, among a set of users that are interested in a common file but each has only partial knowledge about it as side-information. Assuming that the collective information among all the users is sufficient to allow the reconstruction of the entire file, the goal is to minimize the (possibly weighted) amount of bits that these users need to exchange over a noiseless public channel in order for all of them to learn the entire file. Using established connections to the multi-terminal secrecy problem, our algorithm also implies a polynomial-time method for constructing a maximum size secret shared key in the presence of an eavesdropper. We consider the following types of side-information settings: (i) side information in the form of uncoded fragments/packets of the file, where the users' side-information consists of subsets of the file; (ii) side information in the form of linearly correlated packets, where the users have access to linear combinations of the file packets; and (iii) the general setting where the the users' side-information has an arbitrary (i.i.d.) correlation structure. Building on results from combinatorial optimization, we provide a polynomial-time algorithm (in the number of users) that, first finds the optimal rate allocations among these users, then determines an explicit transmission scheme (i.e., a description of which user should transmit what information) for cases (i) and (ii)

Coding for Errors and Erasures in Random Network Coding

The problem of error-control in random linear network coding is considered. A ``noncoherent'' or ``channel oblivious'' model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modelled as the injection into the network of a basis for a vector space $V$ and the collection by the receiver of a basis for a vector space $U$. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space $V \cap U$ is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification

Algebraic List-decoding of Subspace Codes

Subspace codes were introduced in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). Subspace codes are indeed collections of subspaces of a certain vector space over a finite field. The Koetter-Kschischang construction of subspace codes are similar to Reed-Solomon codes in that codewords are obtained by evaluating certain (linearized) polynomials. In this paper, we consider the problem of list-decoding the Koetter-Kschischang subspace codes. In a sense, we are able to achieve for these codes what Sudan was able to achieve for Reed-Solomon codes. In order to do so, we have to modify and generalize the original Koetter-Kschischang construction in many important respects. The end result is this: for any integer $L$, our list-$L$ decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error is at most $L - \frac{L(L+1)}{2}R$ where $R$ is the normalized packet rate. Just as in the case of Sudan's list-decoding algorithm, this exceeds the previously best known error-correction radius $1-R$, demonstrated by Koetter and Kschischang, for low rates $R$