More on Benes Networks

There's some more information on Benes networks, and a "recursive" formulation of how to construct them, that I thought was in the paper... but it wasn't. So here we go!

One other good thing about this recursive formulation is that it helps give a bit more of an intuitive feel for just why it's a reconfigurable network. The basic idea is that we can route a packet through either of two sub-networks, so if one is blocked we can use the other.

This will be in terms of constructing Benes networks of power-of-two sizes, from 2x2 switch boxes. The 2x2 is a special case; it is simply a single switch box.

Now, the way we make a double-size Benes network is:

1. Make two copies, one above the other
2. Add two more stages, one before and one after our double-height network.
3. Now connect the new switch boxes to the existing network. On the left, the top output from each box goes to the top sub-network and the bottom goes to the bottom, in order (so it's an inverse perfect shuffle; the connections on the right are a mirror image of those on the left.

We can see how this works when we construct a 4x4 Benes network starting with a pair of 2x2 networks. This figure shows the two 2x2 networks on top of each other and the new input and output stages, but without the new connections.

Now we hook up the input and output stages. Notice that this is a special case of the perfect shuffle and inverse perfect shuffle, in which they are the same.

Let's do it again, going from a 4x4 to an 8x8. The first picture shows the two 4x4 Benes networks and the new input and output stages.

And now we put in the reverse and forward perfect shuffles to finish it up.