Re: [boost] [graph] planar: face traversal and connected issues

From: Andrew Sutton (asutton_at_[hidden])
Date: 2007-08-30 12:18:46

> I am porting LEDA code to BGL, including code dealing with planar  
> graphs.
>
> I have code like this:
>
> e2 = G.face_cycle_succ(e1);
>
> LEDA documentation says "For an edge e of a map G let  
> face_cycle_succ(e)
> = cyclic_adj_pred(reversal(e))"
>
> I suppose there currently isn't any direct support for such an  
> operation?
>
> As far as I understand, I would have to get the embedding[target(e,g)]
> and then look for e there, and take the predecessor of e, right?
>
> I suppose in an undirected graph edges and their "reversal" (i.e., the
> edge descriptor you get when you get the edge from the other end)  
> are equal?
>
> I think directly supporting this would be a nice addition.

I've never worked with LEDA, so I might be wrong about some of this  
stuff :)

You're looking for a reversal() function? So, given an edge (u,v),  
should return the edge (v,u) assuming that it actually exists. I  
think it would be as easy as:

Edge reversal(Edge e, Graph g)
{
        Vertex u = source(e, g), v = target(e, g);
        return edge(v, u, g).second;
}

And yes... you're right. The reversal of an undirected edge would  
just be itself since the edge (u,v) is the same as the edge (v,u) in  
an undirected graph.

Or something like that... It's an invalid return since there doesn't  
seem to be an implicit null_edge value in Boost.

> Furthermore, I think LEDA can do this in O(1), which we cannot. Well,
> amortized it should be something like O(1), no? The only solution to
> this problem would probably be storing the embedding in the graph  
> object.

There are two ways to get that to run in constant time. First, your  
graph could be an adjacency matrix so the edge(u,v,g) function is  
simply a lookup in a 2d matrix. That's guaranteed O(1). Second, you  
could use an adjacency list and use a hash table to store the out- 
edge list for vertices - of course, that's O(1) on /average/, with O 
(degree(v)) in the worst case. You could use a map to get O(log(degree 
(v))). Otherwise, you're pretty much looking at O(degree(v)) for an  
out-edge list using vectors or lists.

Another way would be to preprocess the graph and store reverse edges  
in an exterior property map. That means you could get away with a  
second variant of the reversal() function:

Edge reversal(Edge e, const Graph& g, EdgeReversalMap erm)
{
        return get(erm, g, e);
}

I think that its definitely a good function to have.

> "edge 	G.new_edge(edge e, node w, int dir=leda::behind)
>    	  	adds a new edge x = (source(e), w) to G. x is inserted in front
> of (dir=leda::before) or behind (dir=leda::behind) edge e into adj$
> \_$edges(source(e)) and appended to in$ \_$edges(w) (if G is directed)
> or adj$ \_$edges(w) (if G is undirected). Here leda::before and
> leda::behind are predefined constants. The operation returns the  
> new edge x.
> Precondition source(e)! = w if G is undirected."
>
> It seems to me that there is only one adjacent edges list for a  
> vertex,
> that represents the planar embedding if it has been correctly computed
> and not invalidated.
>
> Well, what I want to say: Functions that can add edges to an embedding
> would be really cool. Of course, it would be the user's obligation not
> to invalidate the embedding.

I'm not entirely sure what you mean here, but then i'm not too  
familiar with planarity either.

Andrew Sutton
asutton_at_[hidden]