Reducing a Sorting Network

Question

A sorting network is an arrangement of 2 input comparators, which can sort an input sequence of n elements. For example, here's a sorting network for 9 element input:

Each of the vertical lines is a 2 input comparator, the input sequence enters on the left, and the sorted sequence appears on the right.

My question is: how to prove that if we remove the top or bottom line of any valid n input sorting network, we will end up with a valid sorting network for (n-1) inputs? What about removing any of the middle lines?

I have a feeling that this could be shown using a graph representation of a sorting network, but I can't find a suitable representation.

Answer 1

The top or bottom line can indeed be removed. One way to prove this is using Knuth's 0-1 principle, which states that a sorting network is correct if and only if every sequence of zeros and ones is sorted correctly. Let S be a sorting network and let S' be S with the top line removed. Let x be a 0-1 input to S' . Pass 0x to S . By induction, we can show that the values after k stages agree (except for the removed top line), since all of the gates involving the top line are no-ops. It follows that S' is a correct sorting network.

In general, we cannot remove a middle line. Consider, for example, the network

1 *   *
  |   |
2 * * *
    |
3   *