We consider the multiple unicast problem with three source-terminal pairs
over directed acyclic networks with unit-capacity edges. The three siββtiβ
pairs wish to communicate at unit-rate via network coding. The connectivity
between the siββtiβ pairs is quantified by means of a connectivity level
vector, [k1βk2βk3β] such that there exist kiβ edge-disjoint paths between
siβ and tiβ. In this work we attempt to classify networks based on the
connectivity level. It can be observed that unit-rate transmission can be
supported by routing if kiββ₯3, for all i=1,β¦,3. In this work,
we consider, connectivity level vectors such that mini=1,β¦,3βkiβ<3. We present either a constructive linear network coding scheme or an
instance of a network that cannot support the desired unit-rate requirement,
for all such connectivity level vectors except the vector [1Β 2Β 4] (and its
permutations). The benefits of our schemes extend to networks with higher and
potentially different edge capacities. Specifically, our experimental results
indicate that for networks where the different source-terminal paths have a
significant overlap, our constructive unit-rate schemes can be packed along
with routing to provide higher throughput as compared to a pure routing
approach.Comment: To appear in the IEEE/ACM Transactions on Networkin