Schreier vector

Abstract: Two new cyclic base change algorithms are presented for a permutation group G acting on n points. One is cleterministic and the other is randomized. When G is a small base permutation group both algorithms have worst case time complexities which are better than existing algorithms in their class. The deterministic algorithm requires O(rz log2 IGI + nlSl log IGI) time. It outputs a Schreier vector data structure which requires O(n log IGI) space and in which every Schreier tree has depth bounded by 2 log IGI. The randomized algorithm returns a Schreier vector data structure for which the sum of the depths of the resulting Schreier trees is O(log IGI). It is shown that the algorithm has probability exceeding 1 – 2/n of using O(n b log2 n) time for b the size of a non-redundant base. As with most ral~domized base change algorithms, it is Las Vegas in the sense that within the same time it can be deterministically verified whet, her t,he answer is correct. In order to achieire this time bound it is necessary that random elements of G be conlpntable in time O(rr log IGI). A final result is a randomized algorithm which given an arbitrary strong generating set S for G constructs a Schreier vector data structure which can be used to compute random elements in O(n log IGI ) time. It is shown that this algorithm has probability exceeding 1 – I/l Gl of using 0(nlog2 IGI + nlSl) time.