OTIS Layouts of De Bruijn Digraphs

The Optical Transpose Interconnection System (OTIS) was proposed by Marsden et al. [Opt. Lett 18 (1993) 1083–1085] to implement very dense one-to-one interconnection between processors in a free space of optical interconnections. The system which allows one-to-one optical communications from p groups of q transmitters to q groups of p receivers, using electronic intragroup communications for each group of consecutive d processors, is denoted by OTIS(p, q, d). H(p, q, d) is the digraph which characterizes the underlying topology of the optical interconnection implemented by OTIS(p, q, d). A digraph has an OTIS(p, q, d) layout if it is isomorphic to H(p, q, d). Based on results of Coudert et al. [Networks 40 (2002) 155–164], we show that De Bruijn digraph B(d, n) has an OTIS(dp ′ , dn+1−p , d) layout if and only if gcd(p′, n + 1) = 1. We also prove that H(p, q, d) is a line digraph if and only if d | gcd(p, q).