I'm still reading through the Triplet Methods paper, so I'm certain my understanding is not only limited but wrong. Nonetheless, section 7.4.1 ("Reconstruct a network by a sorting network") caught my eye for the simple reason that Algorithm-Networksort exists on CPAN and I am its author.
So if the module can be of use to you, great. If there's a feature that you need from it that's doable, I'd be more than happy to add it to the module. Let me know.
By the way, your link to your jpeg on picpaste doesn't display anything.