De Bruijn isomorphisms and free space optical networks

The de Bruijn digraph B(d, D) is usually defined by words of size D on an alphabet of cardinality d, through a cyclic left shift permutation on the words, after which the rightmost symbol is changed. In this paper we show that any digraph defined on words and alphabets of the same size, through an arbitrary permutation on the alphabet and an arbitrary permutation on the word indices, is isomorphic to the de Bruijn, provided that this latter permutation is cyclic. This work is motivated by the next application. It is known that the optical transpose interconnection system from UCSD can implement the de Bruijn interconnections for n nodes, for a fixed d, with O(n) lenses. We show here how to improve this hardware requirement to Θ(√n)