Some results on communicating the sum of sources over a network

Abstract

Abstract-We consider the problem of communicating the sum of m sources to n terminals in a directed acyclic network. Recently, it was shown that for a network of unit capacity links with either m = 2 or n = 2, the sum of the sources can be communicated to the terminals if and only if every source-terminal pair is connected in the network. We show in this paper that for any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set. As a corollary, this gives networks where the sum can not be communicated over any finite field even though every source is connected to every terminal.

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

“…Primarily we show that the characterization of [7] no longer holds when the number of sources and terminals is greater than two. We note that a similar result was obtained recently in an independent manner by [6]. We then turn to obtain encoding schemes for the three source three terminal case (3s/3t).…”

On fractional linear network coding solution of multiple-unicast networks

“…Such networks have been given the name sum-networks. In [11] it was shown that for any finite set of primes there exists a sum-network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field belongs to the given set. In [12] it was shown that for any sum-network, there exists a multiple-unicast network which has a rate 1 solution if and only if the sum-network has a rate 1 solution.…”

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

“…ANALYSIS OF CASE 3 We now prove Theorem 1 under the assumption that G has no nodes of type (3, 3), (2,3) and (3,2). Note that the node labeling procedure presented above assigns a label (c s (v), c t (v)) to a node v where c s (v) (c t (v)) is the number of sources (terminals) that v is connected to.…”

“…. 3, such that it does not have any (3,3), (2,3) or (3, 2) nodes. There does not exist an edge between (2, 2) nodes of different color in G.…”

“…In this case a single path between each source terminal pair is both necessary and sufficient. Subsequently, the work of Langberg & Ramamoorthy [2], showed that the characterization of [1] does not hold in the case of networks with three sources and three terminals (see also [3]) and proposed an achievable region. Reference [2], shows that as long as each source terminal pair is connected by two edge disjoint paths, the terminals can recover the sum.…”