Abstract-We consider a directed acyclic network where there are two source-terminal pairs and the terminals need to receive the symbols generated at the respective sources. Each source independently generates an i.i.d. random process over the same alphabet. Each edge in the network is error-free, delay-free, and can carry one symbol from the alphabet per use. We give a simple necessary and sufficient condition for being able to simultaneously satisfy the unicast requirements of the two source-terminal pairs at rate-pair (1, 1) using vector network coding. The condition is also sufficient for doing this using only "XOR" network coding and is much simpler compared to the necessary and sufficient conditions known from previous work. Our condition also yields a simple characterization of the capacity region of a double-unicast network which does not support the rate-pair (1, 1).
We consider directed acyclic networks with multiple sources and multiple terminals where each source generates one i.i.d. random process over an abelian group and all the terminals want to recover the sum of these random processes. The different source processes are assumed to be independent. The solvability of such networks has been considered in some previous works. In this paper we investigate on the capacity of such networks, referred as sum-networks, and present some bounds in terms of min-cut, and the numbers of sources and terminals.
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