### Millions of Graphs : Slow Yet Correct Generation

My newest version of canonical path augmentation code for generating graphs has reached a new high point - generating 11,716,571 graphs on ten vertices. Of course, it also gets the number of nines (261,080) and the number of eights (11,117) correct as well ... which is great, but I'm cautious about declaring it 'correct'. Especially given the last version did not get the sevens and eights right. See, for example these past failures:

So how does it get the right answer? Well, it now properly uses the method mentioned in this post to only pick canonical deletions that are not cut-vertices. That turns out only to be necessary for graphs on 8 vertices, but you still have to check this for all augmentations, which seems expensive. However, there was a more fundamental problem; consider the example below (basically nicked from Derick Stolee's blog post):

Obviously A and B are isomorphic, yet how do we properly distinguish them? Well, the key is the set of vertices added to - on the image, these are the labels on the edges between graphs : {0}, {1, 3}, etc. When a new graph is created, a vertex is chosen - using canonical labelling, in my case - and the vertices attached to it must be the ones we used to make that augmented graph. I was checking the set of augmented vertices in the automorphism group of the parent, not the child.

So, the canonical checking is now better. I seem to have written a thousand of these methods, but this one (I think!) finally does it right. What I was getting wrong was checking the orbit of the canonical deletion vertex, and not the orbit of the set of vertices it was being connected to. Great! Now what? How long does it take? See this, where the purple line is the new code, and the others are older attempts:

Clearly the problem now is that of verifying the results - it's quite slow to generate these large datasets, and storing them (uncompressed) takes a lot of space. The nines took minutes and megabytes of space, while the tens took hours and over a gigabyte. At this rate, the 11s would take days and 10s of gigabytes. In any case - where do you stop?

Tobias Kind said…
Hi,
I tried to download the code but the old URL gives the good ole 404.
https://github.com/gilleain/generate/blob/master/src/test/scheme3/TimingTests.java

Also for those people interested, but not able to install all JAVA dependencies,
a compiled JAR with *.sh *.bat *.exe would be good (that would be me and others :-).

For these benchmarks it would be also useful to have the machine, CPU
and disk/ramdisk specifications.

Good to read some new stuff (heavy github contributions in April/May 2015)
Cheers
Tobias

### 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) …