We have a new version of the paper:
Representing Multidimensional Trees
Changes:
-Rewrote justification for Corollary to Lemma 2
-Changed reasoning for why the N3(d) formula is correct.
-Split Embed section into more subsections
-Modified proof of Lemma 3
-Added paragraph before last paragraph of Embed section
-Added justification for claim in last paragraph of Embed section
-Reworked the explanation of project.
-Added a section at the end regarding rec.
The excavate section needs work, but since we haven't actually implemented the function, I'm not sure I understand it enough to elaborate or make it any clearer.
Jim, is there a reason for the section "Embed" to still be called "Embed"? It seems like since we refer to the process as factoring throughout the paper, we should just change the name of the function and the section, and get rid of the term "embed" completely. What do you think?
I'm of two minds (and on rollerskates) on this.
In one sense, it would be nice to maintain (add)a discussion of embedding, which would take and arbitrary branching tree of arbitrary size and embedd it in a two-branching tree. What we do, though, is a sort of factoring that takes an arbitrarily branching local tree and produces a set of two branching local trees. It isn't really _factoring_ since most of the trees we end up with are actually subtrees of the tree we started with. As we discussed last week, this factoring "induces" and embedding of the arbitrary size trees into a two branching tree (which is to say that it determines a unique mapping of points in the first to points in the second) but we don't currently do this step.
We are, maybe, likely to have to when we look at wrapping embed and excavate around CKY, though, so maybe we should keep it as embed for the moment.