Polynomial time algorithms for multicast network code construction

IEEE Transactions on Information Theory, 2005

The famous max-flow min-cut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the min-cut separating and . 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.

2006 IEEE Information Theory Workshop, 2006

We investigate the network coding problem in a certain class of minimal multicast networks. In a multicast coding network, a source S needs to deliver h symbols, or packets, to a set of destinations T over an underlying communication network modeled by a graph G. A coding network is said to be h-minimal if it can deliver h symbols from S to the destination nodes, while any proper subnetwork of G can deliver at most h − 1 symbols to the set of destination nodes. This problem is motivated by the requirement to minimize the amount of network resources allocated for a multicast connections.

Electronic Colloquium on …, 2003

Traditionally, communication networks are composed of routing nodes, which relay and duplicate data. Work in recent years has shown that for the case of multicast, an improvement in both rate and code-construction complexity can be gained by replacing these routing nodes by linear coding nodes. These nodes transmit linear combinations of the inputs transmitted to them.

New Directions in Wireless Communications Research, 2009

Network coding, introduced by Ahlswede et al. in their pioneering work [1], has generated considerable research interest in recent years, and numerous subsequent papers, e.g., , have built upon this concept. One of the main advantages of network coding over traditional routed networks is in the area of multicast, where common information is transmitted from a source node to a set of terminal nodes. Ahlswede et al. showed in [1] that network coding can achieve the maximum multicast rate, which is not achievable by routing alone. When coding is used to perform multicast, the problem of establishing minimum cost multicast connection is equivalent to two effectively decoupled problems: one of determining the subgraph to code over and the other of determining the code to use over that subgraph. The latter problem has been studied extensively in , and a variety of methods have been proposed, which include employing simple random linear coding at every node. Such random linear coding schemes are completely decentralized, requiring no coordination between nodes, and can operate under dynamic conditions . These papers, however, all assume the availability of dedicated network resources.

2009

In this work, we study the computational perspective of network coding, focusing on two issues. First, we address the computational complexity of finding a network code for acyclic multicast networks. Second, we address the issue of reducing the amount of computation performed by network nodes. In particular, we consider the problem of finding a network code with the minimum possible number of encoding nodes, i.e., nodes that generate new packets by performing algebraic operations on packets received over incoming links.

IEEE Transactions on Information Theory, 2000

Consider a communication network in which certain source nodes multicast information to other nodes on the network in the multihop fashion where every node can pass on any of its received data to others. We are interested in how fast each node can receive the complete information, or equivalently, what the information rate arriving at each node is. Allowing a node to encode its received data before passing it on, the question involves optimization of the multicast mechanisms at the nodes. Among the simplest coding schemes is linear coding, which regards a block of data as a vector over a certain base field and allows a node to apply a linear transformation to a vector before passing it on. We formulate this multicast problem and prove that linear coding suffices to achieve the optimum, which is the max-flow from the source to each receiving node.

Computing Research Repository, 2007

Random linear network coding is a particularly decentralized approach to the multicast problem. Use of random network codes introduces a non-zero probability however that some sinks will not be able to successfully decode the required sources. One of the main theoretical motivations for random network codes stems from the lower bound on the probability of successful decoding reported by Ho et. al. (2003). This result demonstrates that all sinks in a linearly solvable network can successfully decode all sources provided that the random code field size is large enough. This paper develops a new bound on the probability of successful decoding.

IEEE Transactions on Information Theory, 2003

Consider transmitting a set of information sources through a communication network that consists of a number of nodes. Between certain pair of nodes, there exist communication channels on which information can be transmitted. At a node, one or more information sources may be generated, and each of them is multicast to a set of destination nodes on the network. In this paper, we study the problem of under what conditions a set of mutually independent information sources can be faithfully transmitted through a communication network, for which the connectivity among the nodes and the multicast requirements of the source information are arbitrary except that the connectivity does not form directed cycles. We obtain inner and outer bounds on the zero-error admissible coding rate region in term of the regions 0 and 0 , which are fundamental regions in the entropy space defined by Yeung. The results in this paper can be regarded as zero-error network coding theorems for acyclic communication networks.

2004

We consider a multicast configuration with two sources, and translate the network code design problem to vertex coloring of an appropriately defined graph. This observation enables to derive code design algorithms and alphabet size bounds, as well as establish a connection with a number of well-known results from discrete mathematics that increase our insight in the different trade-offs possible for network coding.

2006