Re^5: Evolving a faster filter? (optimal!)

[LanX]

in reply to Re^4: Evolving a faster filter? (optimal?)
in thread Evolving a faster filter?

Well my argument is sufficient to show that a solution where two adjacent filters don't follow this order can't be optimal, because otherwise swapping those adjacent filters f[i] and f[i+1] would improve the result.

So any optimal solution must follow this strict order criteria. ¹

qed! =)

PS: I'm glad I didn't start implementing the B&B algorithm :-)

UPDATE:

¹) and it's easy to see that all ordered solutions (plural b/c adjacent filters can have the same weight) imply the same total cost.

[tye]

Indeed!

And the number of choices that leaves you with is rather tiny.

Worse (or better), from throwing cases at the code, I bet it is possible to prove that all such orderings are optimal so that the one that sort gives you is optimal.

I think it might not be hard to prove and $a <= $b <= $c implies $a <= $c (assuming non-negative cost and selectivity between 0 and 1). And that should be enough.