### Tree Canonization Simplified

While debugging the methods to make canonical signatures, I learned something about tree isomorphism from various sources, including Prof. Valiente's excellent looking book on trees.

One way of checking isomorphism is canonisation, since two trees are only isomorphic if they have the same canonical form. For simple labelled trees, it looks like there is an almost trivial way to get a canonical string representation. Say we have two trees:

The are rooted, labelled trees. So the conversion to a canonised string proceeds as follows; for each node, lexicographically sort the string form of the labels of its children, and return the concatenated string. In python this looks like:

which...is unreadable. hmmm. Wish there was a better way to get marked-up code into blog posts. Perhaps there is one, and I don't know of it. Anyway, the point is that it is very short.

edit : the code is...

`def printSorted(node): if len(node.children) > 0:   childStrings = [printSorted(child) for child in node.children]   return node.label + "("+ "".join(sorted(childStrings)) + ")" else:   return node.label`

Rich Apodaca said…
Simple and clear. I wonder how can you use this to develop a graph canonicalization algorithm?

For displaying source, you might also check out Gist:

http://gist.github.com/

Or maybe this:

http://stackoverflow.com/questions/687213/what-websites-do-you-use-to-post-temporary-code-snippets
gilleain said…
Egon : Thanks - I think I saw that before, but never went to the trouble of finding out how to use it in blogspot. This post was helpful:

http://manuelmoeg.blogspot.com/2009/03/def-hello-pass.html

although it's a bit odd that google themselves have not integrated a prettifier.
gilleain said…
Rich : From what I have read (and from what I have understood from that) I don't know if this is possible.

I should point out that the method I describe is not my own, but a simplified version of that described by Tarjan and Hopcroft in a paper on, well, graph isomorphism.

Their method only works for planar graphs, and is fiendishly complex - there are at least two papers that were written to explain the algorithm!

In any case, the lexicographically-sorted canonical string form of the labelled tree does not work for signatures in general. Signatures can represent cycles using tree nodes that are numbered - as with smiles notation where you have C1CCCC1 for a 5-membered ring.

### How many isomers of C4H11N are there?

One of the most popular queries that lands people at this blog is about the isomers of C4H11N - which I suspect may be some kind of organic chemistry question on student homework. In any case, this post will describe how to find all members of a small space like this by hand rather than using software.

Firstly, lets connect all the hydrogens to the heavy atoms (C and N, in this case). For example:

Now eleven hydrogens can be distributed among these five heavy atoms in various ways. In fact this is the problem of partitioning a number into a list of other numbers which I've talked about before. These partitions and (possible) fragment lists are shown here:

One thing to notice is that all partitions have to have 5 parts - even if one of those parts is 0. That's not strictly a partition anymore, but never mind. The other important point is that some of the partitions lead to multiple fragment lists - [3, 3, 2, 2, 1] could have a CH+NH2 or an NH+CH2.

The final step is to connect u…

### Generating Dungeons With BSP Trees or Sliceable Rectangles

So, I admit that the original reason for looking at sliceable rectangles was because of this gaming stackoverflow question about generating dungeon maps. The approach described there uses something called a binary split partition tree (BSP Tree) that's usually used in the context of 3D - notably in the rendering engine of the game Doom. Here is a BSP tree, as an example:

In the image, we have a sliced rectangle on the left, with the final rectangles labelled with letters (A-E) and the slices with numbers (1-4). The corresponding tree is on the right, with the slices as internal nodes labelled with 'h' for horizontal and 'v' for vertical. Naturally, only the leaves correspond to rectangles, and each internal node has two children - it's a binary tree.

So what is the connection between such trees and the sliceable dual graphs? Well, the rectangles are related in exactly the expected way:

Here, the same BSP tree is on the left (without some labels), and the slicea…

### Listing Degree Restricted Trees

Although stack overflow is generally just an endless source of questions on the lines of "HALP plz give CODES!? ... NOT homeWORK!! - don't close :(" occasionally you get more interesting ones. For example this one that asks about degree-restricted trees. Also there's some stuff about vertex labelling, but I think I've slightly missed something there.

In any case, lets look at the simpler problem : listing non-isomorphic trees with max degree 3. It's a nice small example of a general approach that I've been thinking about. The idea is to:
Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }For each d_i in D, connect the degrees in all possible ways that make trees.Filter out duplicates within each set generated by some d_i. Hmm. Sure would be nice to have maths formatting on blogger....

Anyway, look at this example for partitioning 12 into 7 parts:

At the top are the partitions, in the middle the trees (colored by degree) …