An Achievable Region for the Double Unicast Problem Based on a Minimum Cut Analysis

We consider the multiple unicast problem under network coding over directed acyclic networks when there are two source-terminal pairs, s1 - t1 and s2 - t2. Current characterizations of the multiple unicast capacity region in this setting have a large number of inequalities, which makes them hard to explicitly evaluate. In this work we consider a slightly different problem. We assume that we only know certain minimum cut values for the network, e.g., mincut(Si, Tj), where Si ⊆ {si, s2} and Tj ⊆ {t1, t2} for different subsets Si and Tj. Based on these values, we propose an achievable rate region for this problem based on linear codes. Towards this end, we begin by defining a base region where both sources are multicast to both the terminals. Following this we enlarge the region by appropriately encoding the information at the source nodes, such that terminal ti is only guaranteed to decode information from the intended source si, while decoding a linear function of the other source. The rate region takes different forms depending upon the relationship of the different cut values in the network

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

An Achievable Region for the Double Unicast Problem Based on a Minimum Cut Analysis

We consider the multiple unicast problem under network coding over directed acyclic networks when there are two source-terminal pairs, s1-t1 and s2-t2. The capacity region for this problem is not known; furthermore, the outer bounds on the region have a large number of inequalities which makes them hard to explicitly evaluate. In this work we consider a related problem. We assume that we only know certain minimum cut values for the network, e.g., mincut(Si, Tj), where Si ⊆ {s1, s2} and Tj ⊆ {t1, t2} for different subsets Si and Tj. Based on these values, we propose an achievable rate region for this problem using linear network codes. Towards this end, we begin by defining a multicast region where both sources are multicast to both the terminals. Following this we enlarge the region by appropriately encoding the information at the source nodes, such that terminal ti is only guaranteed to decode information from the intended source si, while decoding a linear function of the other source. The rate region depends upon the relationship of the different cut values in the network

Network Coding for Wireless and Wired Networks: Design, Performance and Achievable Rates

Many point-to-point communication problems are relatively well understood in the literature. For example, in addition to knowing what the capacity of a point-to-point channel is, we also know how to construct codes that will come arbitrarily close to the capacity of these channels. However, we know very little about networks. For example, we do not know the capacity of the two-way relay channel which consists of only three transmitters. The situation is not so different in the wired networks except special cases like multicasting. To understand networks better, in this thesis we study network coding which is considered to be a promising technique since the time it was shown to achieve the single-source multicast capacity. First we design and analyze deterministic and random network coding schemes for a cooperative communication setup with multiple sources and destinations. We show that our schemes outperform conventional cooperation in terms of the diversity-multiplexing tradeoff (DMT). Specifically, it can offer the maximum diversity order at the expense of a slightly reduced multiplexing rate. We derive the necessary and sufficient conditions to achieve the maximum diversity order. We show that when the parity-check matrix for a systematic maximum distance separable (MDS) code is used as the network coding matrix, the maximum diversity is achieved. We present two ways to generate full-diversity network coding matrices: namely using the Cauchy matrices and the Vandermonde matrices. We also analyze a selection relaying scheme and prove that a multiplicative diversity order is possible with enough number of relay selection rounds. In addition to the above scheme for wireless networks, we also study wired networks, and apply network coding together with interference alignment. We consider networks with $K$ source nodes and $J$ destination nodes with arbitrary message demands. We first consider a simple network consisting of three source nodes and four destination nodes and show that each user can achieve a rate of one half. Then we extend the result for the general case which states that when the min-cut between each source-destination pair is one, it is possible to achieve a sum rate that is arbitrarily close to the min-cut between the source nodes whose messages are demanded and the destination node where the sum rate is the summation of all the demanded source message rates plus the biggest interferer\u27s rate

A Note on the Multiple Unicast Capacity of Directed Acyclic Networks

We consider the multiple unicast problem under network coding over directed acyclic networks with unit capacity edges. There is a set of n source-terminal (si-ti) pairs that wish to communicate at unit rate over this network. The connectivity between the si-ti pairs is quantified by means of a connectivity level vector, [k1 k2 ... kn] such that there exist ki edge-disjoint paths between si and ti. Our main aim is to characterize the feasibility of achieving this for different values of n and [k1 ... kn]. For 3 unicast connections (n = 3), we characterize several achievable and unachievable values of the connectivity 3-tuple. In addition, in this work, we have found certain network topologies, and capacity characterizations that are useful in understanding the case of general n.This is a manuscript of a proceeding from the IEEE International Conference on Communications (2011), doi:10.1109/icc.2011.5962872. Posted with permission.</p