Optimal transmission schedules for lightwave networks embedded with de Bruijn graphs

Abstract

We consider the problem of embedding a virtual de Bruijn topology, both directed and undirected, in a physical optical passive star time and wavelength division multiplexed (TWDM) network and constructing a schedule to transmit packets along all edges of the virtual topology in the shortest possible time. We develop general graph theoretic results and algorithms and using these build optimal embeddings and optimal transmission schedules, assuming certain conditions on the network parameters. We prove our transmission schedules are optimal over all possible embeddings. As a general framework we use a model of the passive star network with fixed tuned receivers and tunable transmitters. Our transmission schedules are optimal regardless of the tuning time. Our results are also applicable to models with one or more fixed tuned transmitters per node. We give results that minimize the number of tunings needed. For the directed de Bruijn topology a single fixed tuning of the transmitter suffices. For the undirected de Bruijn topology two tunings per cycle (or two fixed tuned transmitters per node) suffice and we prove this is the minimum possible.