Local hidden variable theory

2013-08-03T09:41:40Z

188.67.18.30: /* Bell tests with no "non-detections" */

The '''Schreier–Sims algorithm''' is an [[algorithm]] in [[computational group theory]] named after mathematicians [[Otto Schreier]] and [[Charles Sims (mathematician)|Charles Sims]]. Once performed, it allows a linear time computation of the [[Order (group theory)|order]] of a finite group, group membership test (is a given permutation contained in a group?), and many other tasks. The algorithm was introduced by Sims in 1970, based on [[Schreier's subgroup lemma]]. The timing was subsequently improved by [[Donald Knuth]] in 1991. Later, an even faster [[Randomized algorithm|randomized]] version of the algorithm was developed.

== Background and timing ==

The algorithm is an efficient method of computing a [[base (group theory)|base]] and [[strong generating set]] (BSGS) of a [[permutation group]]. In particular, an SGS determines the order of a group and makes it easy to test membership in the group. Since the SGS is critical for many algorithms in computational group theory, [[computer algebra system]]s typically rely on the Schreier–Sims algorithm for efficient calculations in groups.

The running time of Schreier–Sims varies on the implementation. Let <math> G \leq S_n </math> be given by <math>t</math> [[Generating set of a group|generators]]. For the [[deterministic]] version of the algorithm, possible running times are:

* <math>O(n^2 \log^3 |G| + tn \log |G|) </math> requiring memory <math>O(n^2 \log |G| + tn)</math>
* <math>O(n^3 \log^3 |G| + tn^2 \log |G|) </math> requiring memory <math>O(n \log^2 |G| + tn) </math>

The use of [[Schreier vector]]s can have a significant influence on the performance of implementations of the Schreier–Sims algorithm.

For [[Monte Carlo algorithm|Monte Carlo]] variations of the Schreier–Sims algorithm, we have the following estimated complexity:

: <math>O(n \log n \log^4 |G| + tn \log |G|)</math> requiring memory <math>O(n \log |G| + tn)</math>

In modern computer algebra systems, such as [[GAP computer algebra system|GAP]] and [[Magma computer algebra system|Magma]], an optimized [[Monte Carlo algorithm]] is typically used.

==References==
* Knuth, Donald E. Efficient representation of perm groups. Combinatorica 11 (1991), no. 1, 33–43.
* Seress, A. Permutation Group Algorithms, Cambridge U Press, 2002.
* Sims, Charles C. Computational methods in the study of permutation groups, in Computational Problems in Abstract Algebra, pp. 169–183, Pergamon, Oxford, 1970.

{{DEFAULTSORT:Schreier-Sims algorithm}}
[[Category:Computational group theory]]
[[Category:Permutation groups]]

formulasearchengine - User contributions [en]

Local hidden variable theory