Book contents
- Frontmatter
- Contents
- 1 Introduction
- 2 Black-Box Groups
- 3 Permutation Groups: A Complexity Overview
- 4 Bases and Strong Generating Sets
- 5 Further Low-Level Algorithms
- 6 A Library of Nearly Linear-Time Algorithms
- 7 Solvable Permutation Groups
- 8 Strong Generating Tests
- 9 Backtrack Methods
- 10 Large-Base Groups
- Bibliography
- Index
3 - Permutation Groups: A Complexity Overview
Published online by Cambridge University Press: 15 August 2009
- Frontmatter
- Contents
- 1 Introduction
- 2 Black-Box Groups
- 3 Permutation Groups: A Complexity Overview
- 4 Bases and Strong Generating Sets
- 5 Further Low-Level Algorithms
- 6 A Library of Nearly Linear-Time Algorithms
- 7 Solvable Permutation Groups
- 8 Strong Generating Tests
- 9 Backtrack Methods
- 10 Large-Base Groups
- Bibliography
- Index
Summary
In this chapter, we start the main topic of this book with an overview of permutation group algorithms.
Polynomial-Time Algorithms
In theoretical computer science, a universally accepted measure of efficiency is polynomial-time computation. In the case of permutation group algorithms, groups are input by a list of generators. Given G = 〈S〉 ≤ Sn, the input is of length |S|n and a polynomial-time algorithm should run in O((|S|n)c) for some fixed constant c. In practice, |S| is usually small: Many interesting groups, including all finite simple groups, can be generated by two elements, and it is rare that in a practical computation a permutation group is given by more than ten generators. On the theoretical side, any G ≤ Sn can be generated by at most n/2 permutations (cf. [McIver and Neumann, 1987]). Moreover, any generating set S can be easily reduced to less than n2 generators in O(|S|n2) time by a deterministic algorithm (cf. Exercise 4.1), and in Theorem 10.1.3 we shall describe how to construct at most n – 1 generators for any G ≤ Sn. Hence, we require that the running time of a polynomial-time algorithm is O(nc + |S|n2) for some constant c.
In this book, we promote a slightly different measure of complexity involving n, |S|, and log |G| (cf. Section 3.2), which better reflects the practical performance of permutation group algorithms.
- Type
- Chapter
- Information
- Permutation Group Algorithms , pp. 48 - 54Publisher: Cambridge University PressPrint publication year: 2003