Isomorphisms of the de Bruijn digraph and free-space optical networks

The de Bruijn digraph B(d, D) has degree d, diameter D, d(D) vertices, and d(D+1) arcs. It 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 of a given size, through an arbitrary permutation on the alphabet and an arbitrary permutation on the word indices, is isomorphic to the de Bruijn digraph, provided that this latter permutation is cyclic. We use this result to improve from O(d(D+1)) to Theta(rootd(D+1)) the number of lenses required for the implementation of B(d, D) by the Optical Transpose Interconnection System proposed by Marsden et al. (C) 2002 Wiley Periodicals, Inc.