Communicating the sum of sources in a 3-sources/3-terminals network

We consider the network communication scenario in which a number of sources si each holding independent information Xi wish to communicate the sum ∑Xi to a set of terminals tj. In this work we consider directed acyclic graphs with unit capacity edges and independent sources of unit-entropy. The case in which there are only two sources or only two terminals was considered by the work of Ramamoorthy [ISIT 2008] where it was shown that communication is possible if and only if each source terminal pair si/tj is connected by at least a single path. In this work we study the communication problem in general, and show that even for the case of three sources and three terminals, a single path connecting source/terminal pairs does not suffice to communicate ∑Xi. We then present an efficient encoding scheme which enables the communication of ∑Xi for the three sources, three terminals case, given that each source terminal pair is connected by two edge disjoint paths. Our encoding scheme includes a structural decomposition of the network at hand which may be found useful for other network coding problems as well

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 information-theoretic view of network management

We present an information-theoretic framework for network management for recovery from nonergodic link failures. Building on recent work in the field of network coding, we describe the input-output relations of network nodes in terms of network codes. This very general concept of network behavior as a code provides a way to quantify essential management information as that needed to switch among different codes (behaviors) for different failure scenarios. We compare two types of recovery schemes, receiver-based and network-wide, and consider two formulations for quantifying network management. The first is a centralized formulation where network behavior is described by an overall code determining the behavior of every node, and the management requirement is taken as the logarithm of the number of such codes that the network may switch among. For this formulation, we give bounds, many of which are tight, on management requirements for various network connection problems in terms of basic parameters such as the number of source processes and the number of links in a minimum source-receiver cut. Our results include a lower bound for arbitrary connections and an upper bound for multitransmitter multicast connections, for linear receiver-based and network-wide recovery from all single link failures. The second is a node-based formulation where the management requirement is taken as the sum over all nodes of the logarithm of the number of different behaviors for each node. We show that the minimum node-based requirement for failures of links adjacent to a single receiver is achieved with receiver-based schemes

Connecting Multiple-unicast and Network Error Correction: Reduction and Unachievability

We show that solving a multiple-unicast network coding problem can be reduced to solving a single-unicast network error correction problem, where an adversary may jam at most a single edge in the network. Specifically, we present an efficient reduction that maps a multiple-unicast network coding instance to a network error correction instance while preserving feasibility. The reduction holds for both the zero probability of error model and the vanishing probability of error model. Previous reductions are restricted to the zero-error case. As an application of the reduction, we present a constructive example showing that the single-unicast network error correction capacity may not be achievable, a result of separate interest.Comment: ISIT 2015. arXiv admin note: text overlap with arXiv:1410.190

Quasi-linear Network Coding

We present a heuristic for designing vector non-linear network codes for non-multicast networks, which we call quasi-linear network codes. The method presented has two phases: finding an approximate linear network code over the reals, and then quantizing it to a vector non-linear network code using a fixed-point representation. Apart from describing the method, we draw some links between some network parameters and the rate of the resulting code

Network coding for undirected information exchange

We consider the information exchange problem where each in a set of terminals transmits information to all other terminals in the set, over an undirected network. We show that the design of only a single network code for multicasting is sufficient to achieve an arbitrary point in the achievable rate region. We also provide an alternative proof for the set of achievable rate tuples