### Festive Chemical Structure Generation : Necklaces and Trees!

So, a student asked me about a homework question that is a sub-problem of the structure generation problem. Basically, it was to count the number of chemical structures with exactly one cycle given the elemental formula. Of course, the best solution here is probably to use the Polyá Enumeration Theorem since all that was asked for was a count (enumeration) of the structures.

Naturally, I have a different way to do this - especially since I don't really understand the mathematics of PET enough to implement it. So:

The image shows a rough overview of how I might list all of the structures with a single cycle. It takes a number of necklaces (one shown), and a number of trees, and glues the one to the other in all possible ways. The word 'necklace' here is specifically the combinatorial object; so  the cyclic sequence CNCNO is the same as CNOCN since you can rotate one to get the other.

One tricky decision here is whether to add multiple bonds to the necklace before or after adding the trees. It seems like this would make a difference to how fast the algorithm was - if you add the bonds afterwards, you might reject many of the possible attachments. Hard to say.

The other aspect to consider is the connection of the parts. If we consider necklaces (or 'cycles') and trees as types of block, then the problem is connecting together blocks into a tree. This is essentially the reverse of the approach detailed in this post - using a block decomposition tree to guide the assembly of the blocks:

Although, now I come to look at it, it seems like the attachment points on the necklace would drive the underlying block-tree. So perhaps this is only relevant for graphs that contain multiple cycles - which starts to become a much more difficult problem!

Happy Holidays, anyway...

One Joy said…
Very easy to understand with your words and very vivid. When I read, I also can't help to imagine the picture in my head. If all chemsitry teacher can teach like you, there will be no students dislike chemistry anymore. BOC Sciences
Here comes another holiday...

### 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) …