On network coding for acyclic networks with delays
Problems related to network coding for acyclic, instantaneous networks (where the edges of the acyclic graph representing the network are assumed to have zero-delay) have been extensively dealt with in the recent past. The most prominent of these problems include (a) the existence of network codes that achieve maximum rate of transmission, (b) efficient network code constructions, and (c) field size issues. In practice, however, networks have transmission delays. In network coding theory, such networks with transmission delays are generally abstracted by assuming that their edges have integer delays. Note that using enough memory at the nodes of an acyclic network with integer delays can effectively simulate instantaneous behavior, which is probably why only acyclic instantaneous networks have been primarily focused on thus far. In this work, we elaborate on issues ((a), (b) and (c) above) related to network coding for acyclic networks with integer delays, which have till now mostly been overlooked. We show that the delays associated with the edges of the network cannot be ignored, and in fact turn out to be advantageous, disadvantageous or immaterial, depending on the topology of the network and the problem considered i.e., (a), (b) or (c). In the process, we also show that for a single source multicast problem in acyclic networks (instantaneous and with delays), the network coding operations at each node can simply be limited to storing old symbols and coding them over a binary field. Therefore, operations over elements of larger fields are unnecessary in the network, the trade-off being that enough memory exists at the nodes and at the sinks, and that the sinks have more processing power.