The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
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An algorithm for computing a minimal invariant partition of a permutation group due to Atkinson is analysed. The analysis shows the algorithm is O(n2), where n is the degree of the group. The leading coefficient is 3/2. Some suggested speed-ups are also analysed. The speed-ups imply that the cost of testing primitivity is O(n3), with a leading coefficent of 19/16.Careful use of linked lists to represent subsets of the partition leads to an O(nlogn) algorithm for computing a minimal invariant partition, and an O(n2 logn) algorithm for testing primitivity. For both algorithms, the leading coefficient is 1.The use of trees to represent subsets, and the technique of path compression, lead to an O(nG(n)|S|) algorithm, where S is the set of generators for the group, and G(n) is a very slow growing function related to Ackermann's function, for computing the minimal invariant partition. The leading coefficient is such that, for degrees less than 1,000,000, the linked list version gives a smaller bound.