Optimal realization of hypercubes by three-dimensional space-invariant optical interconnections

It is known that an N-vertex hypercube Q_N can be realized by three-dimensional space-invariant optical interconnections using an optical interconnect module (OIM) with fan-out of size 2 log N-1 and two array planes of area O(N log⁴ N). We show that (8 log N-12)/5 and N(log N+1)/2 are lower bounds for the size of fan-out of OIM and the area of the array plane to realize Q_N, respectively. We also show a realization of Q_N using an OIM with fan-out of size 2 log N and two array planes of area N log N+N/2. Our realization is optimal to within a small constant factor