### Automorphism groups and fragment graphs

Structure generation involves not just graph theory, but group theory. Or, I should say, it does in some of the papers I have read. For example, in this paper by J.L.Faulon, there is the sentence:
"The two main steps are to compute the orbits of the automorphism group of G and to saturate all the atoms of a chosen orbit
which may well be incomprehensible to many readers, except if the reader is a mathematician.

I am no mathematician, but thanks to some books on groups, I now understand both what an automorphism group is and what an orbit is. On the other hand, I also believe that this definition of how the algorithm works is overly complex. A more simple term might just be "fragment sets" - as it is fairly clear, if not mathematically exact. So, for the fragment graph [CH3, CH3, CH2, CH2, CH, CH] the fragment set is [CH3, CH2, CH].

Anyway, here is a short analysis of the automorphism group of the fragment graph [CH2, CH2]. This first image shows the tiny group of permutations that swaps the two fragments:

The notation is taken from an excellent book called "Visual Group Theory" that is also associated with some software called group explorer on sourceforge. It might be quite general, I suppose (and I hope I'm using it right), but it shows the permutation that swaps the fragments as a circled s. This is an automorphism with respect to the edges - in other words, after the swap, there are still bonds between [1, 2], [2, 3], [4, 5], and [5-6].

Another part of the automorphism group is a 'flip' like:

which is a little more complex, but shows how 'flipping' each fragment separately combines to form four possible permutations. If this does not seem particularly tricky, consider what happens if you take the direct product of these two groups:

Assuming I have done it right, this should show most (all?) of the automorphisms of the fragment graph. It does look pretty cool, but I don't think that it gets me any closer to implementing the cursed algorithm :)

Anonymous said…
Anonymous said…
Give never the wolf the wether to keep.

### 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) …