I am reading the book Permutation Group Algorithms by Á. Seress [1] and I am struggling to fill in detail regarding the time complexity of the Schreier-Sims algorithm for computing the Strong Generating Set of a permutation group, more precisely, constructing Schreier vectors. On page 60 (Section 4.2) the book says:
In the first version, we also have to compute transversals explicitly. For a fixed base point, this may > require O(n) permutation multiplications at cost O(n) each, so the total time spent in transversal computations is O(n^2 \log |G|). In the second version, the bookkeeping necessary to set up the Schreier trees can be done in O(n log |G|) total time. [emphasis added]
The "second version" refers to the implementation where Schreier trees are coded as Schreier vectors.
It would be greatly appreciated if someone knowledgeable could provide hints on how to obtain the improvement to O(n \log n) when Schreier vectors are used.
[1] Seress, Á. (2003). Permutation Group Algorithms (Cambridge Tracts in Mathematics). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511546549