Re: [Caml-list] Complexity of Set.union

[Radu Grigore <radugrigore@gmail.com>]

On Fri, 25 Feb 2005 19:47:45 +0000, Jon Harrop <jon@jdh30.plus.com> wrote:

> I ask this because the STL set_union is probably O(n+N) (inserting an already
> sorted range into a set is apparently linear) which is worse than the O((n+N)
> log(n+N)) which you've suggested for OCaml.

The complexity of set_union is indeed O(n+N), see [0]. It is basically
a merge of sorted _sequences_ [1]. I assume n is the size of the small
set and N is the size of the small set and the heights are h=O(lg n),
H=O(lg N). With this the complexity of Set.union is more like O(n
lg(n+N)), at least when all elements in one set are smaller than the
elements of the other set.

> I see. This could be improved in the unsymmetric case, by adding elements from
> the smaller set to the larger set. But the size of the set isn't stored so
> you'd have to make do with adding elements from the shallower set to the
> deeper set. I've no idea what the complexity of that would be...

That is how it works now. As Xavier said the trickiest part is split.

> > Did you mean "of two equal height sets such that all elements of the
> > first set are smaller than all elements of the second set"?
> 
> Yes, that's what I meant. :-)

In that case the current Set.union simply adds elements repeatedly
from the set with small height to the set with big height.

> > That
> > could indeed run in constant time (just join the two sets with a
> > "Node" constructor), but I doubt the current implementation achieves
> > this because of the repeated splitting.

What splitting? I see none in this case.

> Having said that, wouldn't it take the Set.union function O(log n + log N)
> time to prove that the inputs are non-overlapping, because it would have to
> traverse to the min/max elements of both sets?

I agree. Also, such a check looks ugly to me (for a standard library).

-- 
regards,
 radu
http://rgrig.blogspot.com/

[0] http://library.n0i.net/programming/c/cp-iso/lib-algorithms.html#lib.set.union
[1] http://rgrig.blogspot.com/2004/11/merging-lists.html