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

• Published in: Networks Vol. 40; no. 3; pp. 155 - 164
• Main Authors: Coudert, D., Ferreira, A., Perennes, S.
• Format: Journal Article
• Language: English
• Published: New York Wiley Subscription Services, Inc., A Wiley Company 01.10.2002; John Wiley & Sons; Wiley
• Subjects: Applied sciences; Computer Science; de Bruijn digraph; Discrete Mathematics; distributed systems; Electric, optical and optoelectronic circuits; Electronics; Exact sciences and technology; free space of optical interconnections; Fundamental areas of phenomenology (including applications); graph isomorphisms; Interconnected networks; Networking and Internet Architecture; Networks and services in france and abroad; Neural networks; Optical computers, logic elements, interconnects, switches; neural networks; Optical elements, devices, and systems; optical networks; optical transpose interconnection system (OTIS); Optics; Physics; Telecommunications; Telecommunications and information theory; Teleprocessing networks. Isdn; Optical system; Optical network; Optical telecommunication; Isomorphism; Communication network; Digraph; Directed graph; Optical interconnection
• ISSN: 0028-3045; 1097-0037
• DOI: 10.1002/net.10043

The de Bruijn digraph B(d, D) has degree d, diameter D, dD vertices, and dD+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(dD+1) to $\Theta(\sqrt{d^{D+1}})$ the number of lenses required for the implementation of B(d, D) by the Optical Transpose Interconnection System proposed by Marsden et al. [Opt Lett 18 (1993), 1083–1085]. © 2002 Wiley Periodicals, Inc.