A Minimal Set of Shannon-type Inequalities for Functional Dependence Structures

The minimal set of Shannon-type inequalities (referred to as elemental inequalities), plays a central role in determining whether a given inequality is Shannon-type. Often, there arises a situation where one needs to check whether a given inequality is a constrained Shannon-type inequality. Another important application of elemental inequalities is to formulate and compute the Shannon outer bound for multi-source multi-sink network coding capacity. Under this formulation, it is the region of feasible source rates subject to the elemental inequalities and network coding constraints that is of interest. Hence it is of fundamental interest to identify the redundancies induced amongst elemental inequalities when given a set of functional dependence constraints. In this paper, we characterize a minimal set of Shannon-type inequalities when functional dependence constraints are present.Comment: 5 pagers, accepted ISIT201

Scalar Linear Network Coding for Networks with Two Sources

Determining the capacity of networks has been a long-standing issue of interest in the literature. Although for multi-source multi-sink networks it is known that using network coding is advantageous over traditional routing, finding the best coding strategy is not trivial in general. Among different classes of codes that could be potentially used in a network, linear codes due to their simplicity are of particular interest. Although linear codes are proven to be sub-optimal in general, in some cases such as the multicast scenario they achieve the cut-set bound. Since determining the capacity of a network is closely related to the characterization of the entropy region of all its random variables, if one is interested in finding the best linear solution for a network, one should find the region of all linear representable entropy vectors of that network. With this approach, we study the scalar linear solutions over arbitrary network problems with two sources. We explicitly calculate this region for small number of variables and suggest a method for larger networks through finding the best scalar linear solution to a storage problem as an example of practical interest

Scalar Linear Network Coding for Networks with Two Sources

Determining the capacity of networks has been a long-standing issue of interest in the literature. Although for multi-source multi-sink networks it is known that using network coding is advantageous over traditional routing, finding the best coding strategy is not trivial in general. Among different classes of codes that could be potentially used in a network, linear codes due to their simplicity are of particular interest. Although linear codes are proven to be sub-optimal in general, in some cases such as the multicast scenario they achieve the cut-set bound. Since determining the capacity of a network is closely related to the characterization of the entropy region of all its random variables, if one is interested in finding the best linear solution for a network, one should find the region of all linear representable entropy vectors of that network. With this approach, we study the scalar linear solutions over arbitrary network problems with two sources. We explicitly calculate this region for small number of variables and suggest a method for larger networks through finding the best scalar linear solution to a storage problem as an example of practical interest

On the Impact of a Single Edge on the Network Coding Capacity

In this paper, we study the effect of a single link on the capacity of a network of error-free bit pipes. More precisely, we study the change in network capacity that results when we remove a single link of capacity $\delta$. In a recent result, we proved that if all the sources are directly available to a single super-source node, then removing a link of capacity $\delta$ cannot change the capacity region of the network by more than $\delta$ in each dimension. In this paper, we extend this result to the case of multi-source, multi-sink networks for some special network topologies.Comment: Originally presented at ITA 2011 in San Diego, CA. The arXiv version contains an updated proof of Theorem

Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem

We define a notion of network capacity region of networks that generalizes the notion of network capacity defined by Cannons et al. and prove its notable properties such as closedness, boundedness and convexity when the finite field is fixed. We show that the network routing capacity region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. We define the semi-network linear coding capacity region, with respect to a fixed finite field, that inner bounds the corresponding network linear coding capacity region, show that it is a computable rational polytope, and provide exact algorithms and approximation heuristics. We show connections between computing these regions and a polytope reconstruction problem and some combinatorial optimization problems, such as the minimum cost directed Steiner tree problem. We provide an example to illustrate our results. The algorithms are not necessarily polynomial-time.Comment: Appeared in the 2011 IEEE International Symposium on Information Theory, 5 pages, 1 figur