On Approximating the Sum-Rate for Multiple-Unicasts

We study upper bounds on the sum-rate of multiple-unicasts. We approximate the Generalized Network Sharing Bound (GNS cut) of the multiple-unicasts network coding problem with $k$ independent sources. Our approximation algorithm runs in polynomial time and yields an upper bound on the joint source entropy rate, which is within an $O(\log^2 k)$ factor from the GNS cut. It further yields a vector-linear network code that achieves joint source entropy rate within an $O(\log^2 k)$ factor from the GNS cut, but \emph{not} with independent sources: the code induces a correlation pattern among the sources. Our second contribution is establishing a separation result for vector-linear network codes: for any given field $\mathbb{F}$ there exist networks for which the optimum sum-rate supported by vector-linear codes over $\mathbb{F}$ for independent sources can be multiplicatively separated by a factor of $k^{1-\delta}$, for any constant ${\delta>0}$, from the optimum joint entropy rate supported by a code that allows correlation between sources. Finally, we establish a similar separation result for the asymmetric optimum vector-linear sum-rates achieved over two distinct fields $\mathbb{F}_{p}$ and $\mathbb{F}_{q}$ for independent sources, revealing that the choice of field can heavily impact the performance of a linear network code.Comment: 10 pages; Shorter version appeared at ISIT (International Symposium on Information Theory) 2015; some typos correcte

A Geometric Framework for Investigating the Multiple Unicast Network Coding Conjecture

The multiple unicast network coding conjecture states that for multiple unicast sessions in an undirected network, network coding is equivalent to routing. Simple and intuitive as it appears, the conjecture has remained open since its proposal in 2004 [1], [2], and is now a well-known unsolved problem in the field of network coding. Based on a recently proposed tool of space information flow [3]-[5], we present a geometric framework for analyzing the multiple unicast conjecture. The framework consists of four major steps, in which the conjecture is transformed from its throughput version to cost version, from the graph domain to the space domain, and then from high dimension to 1-D, where it is to be eventually proved. We apply the geometric framework to derive unified proofs to known results of the conjecture, as well as new results previously unknown. A possible proof to the conjecture based on this framework is outlined.published_or_final_versio

On the multiple unicast capacity of 3-source, 3-terminal directed acyclic networks

We consider the multiple unicast problem with three source-terminal pairs over directed acyclic networks with unit-capacity edges. The three $s_i-t_i$ pairs wish to communicate at unit-rate via network coding. The connectivity between the $s_i - t_i$ pairs is quantified by means of a connectivity level vector, $[k_1 k_2 k_3]$ such that there exist $k_i$ edge-disjoint paths between $s_i$ and $t_i$. In this work we attempt to classify networks based on the connectivity level. It can be observed that unit-rate transmission can be supported by routing if $k_i \geq 3$, for all $i = 1, \dots, 3$. In this work, we consider, connectivity level vectors such that $\min_{i = 1, \dots, 3} k_i < 3$. We present either a constructive linear network coding scheme or an instance of a network that cannot support the desired unit-rate requirement, for all such connectivity level vectors except the vector $[1~2~4]$ (and its permutations). The benefits of our schemes extend to networks with higher and potentially different edge capacities. Specifically, our experimental results indicate that for networks where the different source-terminal paths have a significant overlap, our constructive unit-rate schemes can be packed along with routing to provide higher throughput as compared to a pure routing approach.Comment: To appear in the IEEE/ACM Transactions on Networkin

An asymptotically optimal push-pull method for multicasting over a random network

We consider allcast and multicast flow problems where either all of the nodes or only a subset of the nodes may be in session. Traffic from each node in the session has to be sent to every other node in the session. If the session does not consist of all the nodes, the remaining nodes act as relays. The nodes are connected by undirected links whose capacities are independent and identically distributed random variables. We study the asymptotics of the capacity region (with network coding) in the limit of a large number of nodes, and show that the normalized sum rate converges to a constant almost surely. We then provide a decentralized push-pull algorithm that asymptotically achieves this normalized sum rate without network coding.Comment: 13 pages, extended version of paper presented at the IEEE International Symposium on Information Theory (ISIT) 2012, minor revision to text to address review comments, to appear in IEEE Transactions in information theor

Network Coding: Exploiting Broadcast and Superposition in Wireless Networks

In this thesis we investigate improvements in efficiency of wireless communication networks, based on methods that are fundamentally different from the principles that form the basis of state-of-the-art technology. The first difference is that broadcast and superposition are exploited instead of reducing the wireless medium to a network of point-to-point links. The second difference is that the problem of transporting information through the network is not treated as a flow problem. Instead we allow for network coding to be used.\ud \ud First, we consider multicast network coding in settings where the multicast configuration changes over time. We show that for certain problem classes a universal network code can be constructed. One application is to efficiently tradeoff throughput against cost.\ud \ud Next, we deal with increasing energy efficiency by means of network coding in the presence of broadcast. It is demonstrated that for multiple unicast traffic in networks with nodes arranged on two and three dimensional rectangular lattices, network coding can reduce energy consumption by factors of four and six, respectively, compared to routing.\ud \ud Finally, we consider the use of superposition by allowing nodes to decode sums of messages. We introduce different deterministic models of wireless networks, representing various ways of handling broadcast and superposition. We provide lower and upper bounds on the transport capacity under these models. For networks with nodes arranged on a hexagonal lattice it is found that the capacity under a model exploiting both broadcast and superposition is at least 2.5 times, and no more than six times, the transport capacity under a model of point-to-point links