A Computational Perspective on Network Coding

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. 1. Introduction When network coding was firstly used by Ahlswede et al. [1], the node produced each of its outgoing packets as an arbitrary combination of its incomings, which is referred to as encoding node. Those functions applied by all nodes in the network specify the different network codes, such as linear network codes and random linear network codes via linear function and random linear function, respectively [1–3]. In [3], Li et al. showed that linear network codes are sufficient for achieving the capacity of the network. And in a subsequent work, Koetter and Médard [4] developed an algebraic framework for network coding and studied linear network codes for cyclic networks. Based on this framework, Ho et al. [2] showed that linear network codes can be efficiently constructed through a randomized algorithm. Jaggi et al. [5] presented a deterministic polynomial-time algorithm for finding a feasible network codes in multicast networks. Errors introduced into even a single packet transmitted on the way can propagate and pollute multiple packets making their way to the destination. To prevent the spread of the error packets, signatures for network coding are proposed [6, 7]. Nevertheless, whatever the kind of network codes will be used, the number of encoding nodes plays an important role on the encoding complexity of network coding. In [8, 9], Langberg et al. studied the design of multicast coding networks with a limited number of encoding nodes. And they showed that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by ( is the source rate and is the number of terminal nodes) which is independent of the size of the network. And for the general networks, in which there exist