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

We consider the problem of multicasting sums over directed acyclic networks with unit capacity edges. A set of source nodes si observe independent unit-entropy source processes Xi and want to communicate Σ Xi to a set of terminals tj. Previous work on this problem has established necessary and sufficient conditions on the si -tj connectivity in the case when there are two sources or two terminals (Ramamoorthy \u2708), and in the case of three sources and three terminals (Langberg-Ramamoorthy \u2709). In particular the latter result establishes that each terminal can recover the sum if there are two edge disjoint paths between each si-tj pair. In this work, we provide a new and significantly simpler proof of this result, and introduce techniques that may be of independent interest in other network coding problems

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

On network coding for sum-networks

A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably) equivalent sum-network for any multiple-unicast network, and thus for any directed acyclic communication network. It is also shown that there exists a linear solvably equivalent multiple-unicast network for every sum-network. It is shown that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F. For any finite or cofinite set of prime numbers, a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set. The insufficiency of linear network coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks. Under fractional vector linear network coding, a sum-network and its reverse network are shown to be equivalent. However, under non-linear coding, it is shown that there exists a solvable sum-network whose reverse network is not solvable.Comment: Accepted to IEEE Transactions on Information Theor

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

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

We consider the problem of multicasting sums over directed acyclic networks with unit capacity edges. A set of source nodes si observe independent unit-entropy source processes Xi and want to communicate Σ Xi to a set of terminals tj. Previous work on this problem has established necessary and sufficient conditions on the si -tj connectivity in the case when there are two sources or two terminals (Ramamoorthy '08), and in the case of three sources and three terminals (Langberg-Ramamoorthy '09). In particular the latter result establishes that each terminal can recover the sum if there are two edge disjoint paths between each si-tj pair. In this work, we provide a new and significantly simpler proof of this result, and introduce techniques that may be of independent interest in other network coding problems.This is a manuscripts of a proceeding from the IEEE International Symposium on Information Theory (2010): 1853, doi:10.1109/ISIT.2010.5513422. Posted with permission.</p