On Network Coding for Sum-Networks

Abstract

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 sumnetwork 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.

On base field of linear network coding

“…However, no handy necessary condition has been derived in [19] for the nonexistence of linear solutions, and this is a key reason that only a few exemplifying networks with the special property q min < q * max were designed therein. For instance, the nonexistence of a linear solution over GF (17) Recall that when q − 1 is prime, Cauchy-Davenport Theorem asserts that for any two nonempty subsets A and B of Z q−1 , |A + B| ≥ min{|A| + |B| − 1, q − 1}. Thus, for arbitrary We next generalize the Cauchy-Davenport Theorem to be over Z q−1 , where q − 1 can be a composite.…”

“…Specifically, a general method was proposed in [16] to associate a polynomial set with a network, such that the polynomials have common roots over a field if and only if the corresponding network has a linear solution over the same field. In [17], analogous equivalence was also established between the existence of common roots over a finite field for a polynomial set and the existence of a linear solution over the same field for an associated sum-network, in which all receivers attempt to recover the sum of all source data symbols. However, when attention is only paid to multicast networks, these results are not applicable anymore.…”

“…However, when attention is only paid to multicast networks, these results are not applicable anymore. First, since a multicast network is linearly solvable over a large enough field, it is linearly solvable over some field of characteristic p for every prime p. Moreover, the general method proposed in [16] and [17] cannot necessarily construct a multicast network from a polynomial set.…”

Multi-Session Function Computation and Multicasting in Undirected Graphs

“…In this paper, we show that for many function classes of practical interest, noninteractive function computation using computation trees can give near-optimal performance. 6) Function Multicasting: One particular problem that has attracted attention is the case where there are several sources, and each destination demands the sum of the sources [34], [35], [37]. In our terminology, we call this the function multicasting problem; in this case, the function happens to be linear as well.In [37], the case of a directed acyclic communication graph is considered and several negative results demonstrating the insufficiency of scalar and vector linear coding is shown, deriving inspiration from similar results for multiple unicast in directed networks.…”

A Relation Between Network Computation and Functional Index Coding Problems

“…The notion of min-cut bound for the network coding problem [2] was extended to the function computation problem in a directed acyclic network with multiple sources and one sink in [8]. The case of directed acyclic network with multiple sources, multiple sinks and each sink demanding the sum of source messages was studied in [10]; such a network is called a sum-network. Relation between linear solvability of multiple-unicast networks and sum-networks was established.…”