I was going to address this at some point, but to be honest, I've lost interest. So I decided to post a closing note instead of letting this silently slip into oblivion.

The point I was going to make is that your heap code is not exactly efficient. The algorithm description is formulated recursively, but you don't need to implement it that way. Since it's just tail recursion, you can trivially write it iteratively, which would gain a lot of ground.

Still, that would almost certainly only accelerate the code by a constant factor, which does not make it worth the effort for the smallish data sets you're working with.

I confess my surprise to find Perl is that slow. I am no stranger to optimizing Perl, but I've never come across such a stellar disparity between a builtin and explicit code before.