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

On practical design for joint distributed source and network coding

This paper considers the problem of communicating correlated information from multiple source nodes over a network of noiseless channels to multiple destination nodes, where each destination node wants to recover all sources. The problem involves a joint consideration of distributed compression and network information relaying. Although the optimal rate region has been theoretically characterized, it was not clear how to design practical communication schemes with low complexity. This work provides a partial solution to this problem by proposing a low-complexity scheme for the special case with two sources whose correlation is characterized by a binary symmetric channel. Our scheme is based on a careful combination of linear syndrome-based Slepian-Wolf coding and random linear mixing (network coding). It is in general suboptimal; however, its low complexity and robustness to network dynamics make it suitable for practical implementation

Communicating the sum of sources over a network

We consider the network communication scenario, over directed acyclic networks with unit capacity edges in which a number of sources $s_i$ each holding independent unit-entropy information $X_i$ wish to communicate the sum $\sum{X_i}$ to a set of terminals $t_j$. We show that in the case in which there are only two sources or only two terminals, communication is possible if and only if each source terminal pair $s_i/t_j$ is connected by at least a single path. For the more general communication problem in which there are three sources and three terminals, we prove that a single path connecting the source terminal pairs does not suffice to communicate $\sum{X_i}$. We then present an efficient encoding scheme which enables the communication of $\sum{X_i}$ for the three sources, three terminals case, given that each source terminal pair is connected by {\em two} edge disjoint paths.Comment: 12 pages, IEEE JSAC: Special Issue on In-network Computation:Exploring the Fundamental Limits (to appear

Intra- and Inter-Session Network Coding in Wireless Networks

In this paper, we are interested in improving the performance of constructive network coding schemes in lossy wireless environments.We propose I2NC - a cross-layer approach that combines inter-session and intra-session network coding and has two strengths. First, the error-correcting capabilities of intra-session network coding make our scheme resilient to loss. Second, redundancy allows intermediate nodes to operate without knowledge of the decoding buffers of their neighbors. Based only on the knowledge of the loss rates on the direct and overhearing links, intermediate nodes can make decisions for both intra-session (i.e., how much redundancy to add in each flow) and inter-session (i.e., what percentage of flows to code together) coding. Our approach is grounded on a network utility maximization (NUM) formulation of the problem. We propose two practical schemes, I2NC-state and I2NC-stateless, which mimic the structure of the NUM optimal solution. We also address the interaction of our approach with the transport layer. We demonstrate the benefits of our schemes through simulations

Relay Selection for Wireless Communications Against Eavesdropping: A Security-Reliability Tradeoff Perspective

This article examines the secrecy coding aided wireless communications from a source to a destination in the presence of an eavesdropper from a security-reliability tradeoff (SRT) perspective. Explicitly, the security is quantified in terms of the intercept probability experienced at the eavesdropper, while the outage probability encountered at the destination is used to measure the transmission reliability. We characterize the SRT of conventional direct transmission from the source to the destination and show that if the outage probability is increased, the intercept probability decreases, and vice versa. We first demonstrate that the employment of relay nodes for assisting the source-destination transmissions is capable of defending against eavesdropping, followed by quantifying the benefits of single-relay selection (SRS) as well as of multi-relay selection (MRS) schemes. More specifically, in the SRS scheme, only the single "best" relay is selected for forwarding the source signal to the destination, whereas the MRS scheme allows multiple relays to participate in this process. It is illustrated that both the SRS and MRS schemes achieve a better SRT than the conventional direct transmission, especially upon increasing the number of relays. Numerical results also show that as expected, the MRS outperforms the SRS in terms of its SRT. Additionally, we present some open challenges and future directions for the wireless relay aided physical-layer security.Comment: 16 pages, IEEE Network, 201