Monday, August 18, 2008

Double cosets

I'm now turning to the task of canonical augmentation, or orderly generation. More simply, the question of generating unique isomorphism representatives of a certain class of finite objects, where the notion of isomorphism comes from the action of some finite permutation group.

One key step in the algorithm is to discover whether or not two pairs of objects are isomorphic via the same isomorphism. More specifically, it needs to be computed whether (P, C) ~ (M, g(C)), where g(C) is the canonical relabeling of C (a superobject of P), and M is defined to be the "canonical parent." This means, is there a permutation h such that h(P) = M and h(C) = g(C). First, this means that P and M must be isomorphic, let's say via d: M = d(P). So once we know d, we can say that we are looking for some permutation which is in the coset dAut(P) and also in the coset gAut(C).

There are two places in the code in Sage which does this, in generating graphs and in generating binary codes. Both are currently implemented ad hoc- graphs are written in Python in a pretty naive implementation, and binary codes actually farm the work out to Gap, which takes over half the CPU time.

This is an example of a double-coset problem, which is one of the types of algorithms I was planning on implementing. Since this can also be used to solve the isomorphism problem substantially faster (at least in the negative-answer case; when they are isomorphic the runtime is the same), I have decided to tackle this problem next.

Once more, thanks to Google for funding my work over the summer.

No comments: