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 characterize all OTIS(p,q,d) layouts of De Bruijn digraph B(d,n) where both p and q are powers of d. Coudert et al. posed the conjecture that if B(d,n) has an OTIS(p,q,d) layout, then both p and q are powers of d. As an effort to prove this conjecture, we prove that H(p,q,d) is a line digraph if and only if both p and q are multiples of d.