Fast computation of the centralizer of a permutation group in the symmetric group

Abstract

Let G be a permutation group acting on a set Ω. Best known algorithms for computing the centralizer of G in the symmetric group on Ω are all based on the same general approach that involves solving the following two fundamental problems: given a G-orbit Δ of size n, compute the centralizer of the restriction of G to Δ in the symmetric group on Δ; and given two G-orbits Δ and Δ′ each of size n, find an equivalence between the action of G restricted to Δ and the action of G restricted to Δ′ when one exists. If G is given by a generating set X, previous solutions to each of these two problems take O(|X|n2) time.In this paper, we first solve each fundamental problem in O(δn+|X|nlog⁡n) time, where δ is the depth of the breadth-first-search Schreier tree for X rooted at some fixed vertex. For the important class of small-base groups G, we improve the theoretical worst-case time bound of our solutions to O(nlogc⁡n+|X|nlog⁡n) for some constant c. Moreover, if ⌈20log2⁡n⌉ uniformly distributed random elements of G are given in advance, our solutions have, with probability at least 1−1/n, a running time of O(nlog2⁡n+|X|nlog⁡n). We then apply our solutions to obtain a more efficient algorithm for computing the centralizer of G in the symmetric group on Ω. In an experimental evaluation we demonstrate that it is substantially faster than previously known algorithms.