Schreier-Sims variant to produce a compact/minimal representation I am currently looking into the generation of a base alongside a strong generating set for permutation groups. Naturally, I am digging into the details of the Schreier-Sims algorithm.
After having played around with a few examples in GAP, I got the impression that the choice of a BSGS is anything but unique (after all there generally seems to be plenty of room for arbitrariness in choosing the points from which to generate a base and even for the same base, it seems to be possible to arrive at different generating sets).
However, when one wants to do some further processing with the group, it seems to me as if it would be beneficial to choose a BSGS that uses as few base points and as few generators as possible (as then e.g. iterations over all basis points would get fewer as well).
Thus, my question is whether there exist some variations of the Schreier-Sims algorithm that are tuned to exactly this purpose?
 A: A practical/implementable algorithm to find a minimal base in general does not exist (as far as I am aware) and is a very difficult problem (as Derek Holt points out in the comments, deciding whether a base exists of some given bounded size is NP-complete, this is shown in [1]). An overview of this topic is presented here, From which I shall extract the salient point.
To set up notation, for $G$ a permutation group, let $b(G)$ be the size of the smallest base for $G$. I will assume $G$ acts of $\Omega=\{1,\dots,n\}$, and for $\omega\in\Omega$ write $G_\omega$ for the stabiliser of $\omega$ in $G$.
Fortunately the greedy algorithm for finding a base does a reasonably good job of being efficient. This computes a base $B=(\omega_1,\omega_2,\dots,\omega_k)$ by iteratively choosing $\omega_i\in\Omega$ which has the largest orbit under $$\bigcap_{j=1}^{i-1}G_{\omega_j}.$$ In [1], Blaha proves that this algorithm yields a base of size $O(b(G)\log\log n)$.
As for generating sets, any permutation group admits a generating set of size at most $n-1$, but the naïve algorithms to compute a generating set typically produce $n^2$ generators. The way to fix this is by a process called sifting which is discussed in detail in [2]. Note that this paper assumes the base being used is $(1,2,\dots,n)$, but it is straightforward to adapt it to the case of your chosen greedy base $B=(\omega_1,\omega_2,\dots,\omega_k)$.
Although I haven't read it in detail, I think [3] may offer some further improvements to efficiency of these algorithms.
[1] Blaha: Minimum bases for permutation groups: the greedy approximation, J. Algorithms 13 (1992), 297–306.
[2] Jerrum: A compact representation for permutation groups, J. Algorithms 7, (1986), 60-78.
[3] Knuth: Efficient representation of perm groups, Combinatorica 11 (1991), 33-43.
