The only thing we have to convince ourselves is that there are cases where this actually

helps, which is what we did via an example.

Here it is and I'm going to, we're again doing exactly the same switching between the relaxed

problem and the full problem.

The only thing you can see here, the difference is we're getting higher numbers than two,

which is good because remember a heuristic with higher numbers is better than one with

lower numbers.

Because the optimal heuristic, which is somewhere up there, the heuristic should be close together,

but the relative error is actually small.

Now obviously the heuristic is up here, you get numbers up here, it's better than any

heuristic that's down there.

So big numbers are good.

Why are they good?

Because they give you guidance.

And here the crucial difference is that around the time we had kind of drifted off in the

other heuristic into that huge upper subtree here, we're actually getting a tie here.

And we're using alphabetics here to resolve the tie, which means we go down into this

territory here, but since the heuristic is bigger, we're actually getting into big numbers

fast, which means we continue here and from there on it's a straight line to the plan,

which means there's very little search here, at least in this example.

And of course you have to prove that it works well on other examples as well, but it essentially

does, surprisingly for a very simple thing like that.

We looked at a couple of other examples and that was the end.