Skip to main content

Comparing the EquivalentClassPartitioner and PartitionRefiner : Fullerenes

comment on my previous post reminded me of the "EquivalentClassPartitioner" already in the CDK, written back in 2003 by Junfeng Hao and based on this article by Chang-Yu Hu and Lu Xu. After some testing, it seems they give the same results on various molecules - although, if you only want the equivalence classes the Hu/Xu method is much, much faster. Like 10-100 times faster.

The test molecules I used for comparing speeds are a library of fullerenes that range in size from 20 carbons up to 720. Naturally, I started with the smaller ones, but even there the difference was clear. For instance, here is a table of numbers from the C40 run:



The left-hand column is just the name of the cc1 file, the next two columns are the times for the AtomPartitionRefiner and EquivalentClassPartitioner, and the last two are the order of the automorphism group and the number of equivalence classes. Times are in milliseconds, so clearly the HuXu method is far faster at only one or two ms rather than 30-70 ms. There is presumably some VM speedup going on that accounts for the apparent increase in speed for the first few examples.

It's a similar picture for larger fullerenes : for a typical C92, the PartitionRefiner takes 700 ms and HuXu only 7 ms. There is a speedup for more symmetric molecules - the larger the value of |Aut| (the group order), the faster the PartitionRefiner is. This is to be expected, as it is using the automorphisms to prune the search.

The tests, and some output are in this github repo (fullerene library not included)Finally, here is another image of a fullerene (one of the test cases from the EquivalentClassPartitioner):


because why not. Colourful, is it not?

Comments

Gilleain, do I understand the coloring correctly that this fullerene only has one symmetrical phenyl ring, the outer one?
gilleain said…
That's correct. The gray carbon (39) seems to be 'disordering' the structure from one end. Like a pebble dropped in a still pond. Sort of.

I really should try getting the Schlegel layout stuff working on fullerenes. I got close over the summer, but the optimisation (annealing) part was broken.
J May said…
In my mind the automorphism group is a lot harder to calculate then the equivalent classes so the speed difference is expected. The code looks great, just finished reviewing and about to sign off.
gilleain said…
True - it is harder to calculate. However, I have a sneaking suspicion that you can get the group from the equivalence classes, and that this might be faster. I don't know what the algorithm is, exactly, as I've tried simple ways to do it, that didn't seem to work...

Popular posts from this blog

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…

Common Vertex Matrices of Graphs

There is an interesting set of papers out this year by Milan Randic et al (sorry about the accents - blogger seems to have a problem with accented 'c'...). I've looked at his work before here.

[1] Common vertex matrix: A novel characterization of molecular graphs by counting
[2] On the centrality of vertices of molecular graphs

and one still in publication to do with fullerenes. The central idea here (ho ho) is a graph descriptor a bit like path lengths called 'centrality'. Briefly, it is the count of neighbourhood intersections between pairs of vertices. Roughly this is illustrated here:


For the selected pair of vertices, the common vertices are those at the same distance from each - one at a distance of two and one at a distance of three. The matrix element for this pair will be the sum - 2 - and this is repeated for all pairs in the graph. Naturally, this is symmetric:


At the right of the matrix is the row sum (∑) which can be ordered to provide a graph invarian…

Signatures with user-defined edge colors

A bug in the CDK implementation of my signature library turned out to be due to the fact that the bond colors were hard coded to just recognise the labels {"-", "=", "#" }. The relevant code section even had an XXX above it!

Poor show, but it's finally fixed now. So that means I can handle user-defined edge colors/labels - consider the complete graph (K5) below:

So the red/blue colors here are simply those of a chessboard imposed on top of the adjacency matrix - shown here on the right. You might expect there to be at least two vertex signature classes here : {0, 2, 4} and {1, 3} where the first class has vertices with two blue and two red edges, and the second has three blue and two red.

Indeed, here's what happens for K4 to K7:

Clearly even-numbered complete graphs have just one vertex class, while odd-numbered ones have two (at least?). There is a similar situation for complete bipartite graphs:

Although I haven't explored any more of these…