Bounding the Coding Advantage of Combination Network Coding in Undirected Networks

We refer to network coding schemes in which information flows propagate along a combination network topology as combination network coding (CNC). CNC and its variations are the first network coding schemes studied in the literature, and so far still represent arguably the most important class of known structures where network coding is nontrivial. Our main goal in this paper is to seek a thorough understanding on the advantage of CNC in undirected networks, by proving a tight bound on its potential both in improving multicast throughput (the coding advantage) and in reducing multicast cost under a linear link flow cost model (the cost advantage). We prepare three results towards this goal. First, we show that the cost advantage of CNC is upper-bounded by 9/8 under the uniform link cost setting. Second, we show that achieving a larger cost advantage is impossible by considering an arbitrary instead of uniform link cost configuration. Third, we show that in a given network topology, for any form of network coding, the coding advantage under arbitrary link capacity configurations is always upper-bounded by the cost advantage under arbitrary link cost configurations. Combining the three results together, we conclude that the potential for CNC to improve throughput and to reduce routing cost are both upper-bounded by a factor of 9/8. The bound is tight since it is achieved in specific networks. This result can be viewed as a natural step towards improving the bound of 2 proved for the coding advantage of general multicast network coding.