Copyright © 2010 Qin Guo et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
From the perspectives of graph theory and combinatorics theory we
obtain some new upper bounds on the number of encoding nodes, which can characterize
the coding complexity of the network coding, both in feasible acyclic and cyclic multicast
networks. In contrast to previous work, during our analysis we first investigate the simple
multicast network with source rate , and then . We find that for feasible acyclic
multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast
networks does not exist. Based on the new upper bound, we improve the computational
complexity given by M. Langberg et al. in 2009. Moreover, these results further support
the feasibility of signatures for network coding.