On Network Coding for Communicating the Sum of Symbols from Multiple Sources

In this thesis, we consider a communication problem over a directed acyclic network of unit capacity links having m sources and n terminals, where each terminal requires the sum of symbols generated at all the sources. We assume that each source generates one i.i.d. random process with uniform distribution over a finite alphabet having an abelian group structure, and the different source processes are independent. We also assume that each node in the network is capable of implementing network coding. We further assume that all the links in the network are error-free and delay free. We call such a directed acyclic network as a sum-network. First, we consider the problem of rate-1 linear network code solutions over finite fields. We construct two special classes of sum-networks. In the first constructed class, for any finite set of primes, there exists a sum-network which has rate-1 linear network code solutions for every code length only over finite fields of characteristics belonging to the given set. In the second constructed class, for any finite set of primes, there exists a sum-network which has rate-1 linear network code solutions for every code length only over finite fields of characteristics, which do not belong to the given set. Then we show that there exists a sum-network, where a scalar linear network code solution not only depends on the characteristic of the finite field, but also on the size of the field. We also show that there exist sum-networks where source-terminal pairs connectivity is insufficient for a rate-1 network code solution for any code length. Next, we show the solvable equivalence of sum-networks with other type of networks. Specifically, we show that communicating the sum and communicating any linear function are solvably (respectively linear solvably) equivalent under fractional network coding (respectively fractional linear network coding) if the considered alphabet is a module over a ring and the component linear functions are invertible. Then we show that there exists a solvably (and linear solvably) equivalent sum-network for any multiple-unicast network (and more generally, for any acyclic directed network where each terminal re9 quires a subset of the symbols generated at all the sources). We also show that there exists a linear solvably equivalent multiple-unicast network for any sum-network. As a consequence of our solvable equivalence results, some important known results, for multiple-unicast and directed acyclic networks where each terminal requires a subset of the symbols generated at all the sources, also hold for sum-networks. Specifically, we show that for any set of polynomials having integer coefficients, there exists a sumnetwork which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F. Then we show that there exists a solvable sum-network over a finite alphabet whose reverse network is not solvable over any finite alphabet. However, we show that a sum-network and its reverse sum-network are solvably equivalent under fractional linear network coding. Similarly, we show the insufficiency of linear network coding and unachievability of the network coding capacity for sum-networks. Finally, we consider the network coding capacity of sum-networks over a finite field. However, some of our results also hold over finite alphabets having more general algebraic structures, such as a module over a ring. The network coding capacity of a sum-network is upper bounded by the minimum of min-cut capacities of all source-terminal pairs over any alphabet. We call this upper bound the min-cut bound. We show that the min-cut bound is always achievable for sum-networks with min{m,n} = 1 over sufficiently large finite fields. Moreover, scalar linear network coding is sufficient to achieve the min-cut bound. For sum-networks with min{m,n} = 2, the network coding capacity over every finite field is known to be equal to the min-cut bound, when the min-cut bound is 1. For the min-cut bound greater than 1, we give a lower bound on the network coding capacity. For sum-networks with min{m,n} ≥ 3, we show that there exist sum-networks where the min-cut bound is not achievable over any alphabet. For this class, when the min-cut bound is 1, we give a lower bound on the network coding capacity and show that for sum-networks with min{m,n} = 3, the lower bound is tight. We conjecture that the network coding capacity of a sum-network with m = n = 3 is either 0, 2/3 or at least 1.