[Paper Review] The Schreier-Sims algorithm for matrix groups

Henrik Bäärnhielm|ArXiv.org|Oct 28, 2004

Finite Group Theory Research21 references3 citations

TL;DR

This paper presents a new direct implementation of the Schreier-Sims algorithm for matrix groups in GAP, bypassing permutation representation to improve efficiency. By optimizing data structures and leveraging matrix-specific operations, the implementation achieves faster runtime and significantly reduced memory usage compared to GAP's standard method, particularly for large matrix groups over finite fields.

ABSTRACT

This is the report of a project with the aim to make a new implementation of the Schreier-Sims algorithm in GAP, specialized for matrix groups. The standard Schreier-Sims algorithm is described in some detail, followed by descriptions of the probabilistic Schreier-Sims algorithm and the Schreier-Todd-Coxeter-Sims algorithm. Then we discuss our implementation and some optimisations, and finally we report on the performance of our implementation, as compared to the existing implementation in GAP, and we give benchmark results. The conclusion is that our implementation in some cases is faster and consumes much less memory.

Motivation & Objective

• To develop a direct implementation of the Schreier-Sims algorithm that operates on matrix groups without first converting them to permutation groups.
• To improve computational efficiency—specifically runtime and memory consumption—by exploiting matrix group structure directly.
• To evaluate performance against GAP’s existing Schreier-Sims implementation using benchmark tests on finite matrix groups.
• To optimize the algorithm through tailored data structures and algorithmic improvements specific to matrix group operations.

Proposed method

• The algorithm computes a base and strong generating set (BSGS) directly on matrix groups using Schreier trees and Schreier’s Lemma to generate and reduce generators.
• It employs a probabilistic variant of the Schreier-Sims algorithm to accelerate base point selection and reduce computational overhead.
• The implementation uses the Schreier-Todd-Coxeter-Sims method for handling coset representatives and membership testing via sifting.
• Key optimizations include efficient matrix arithmetic, dynamic memory management, and selective generation of Schreier generators based on orbit membership.
• The algorithm uses orbit representatives and inverse path reconstruction from Schreier trees to compute coset representatives during membership testing.
• Performance is evaluated using benchmark groups, comparing execution time and memory usage against GAP’s standard implementation.

Experimental results

Research questions

• RQ1 Can a direct matrix group implementation of the Schreier-Sims algorithm outperform the standard GAP implementation that first converts to permutation groups?
• RQ2 What are the performance gains in terms of speed and memory usage when applying the Schreier-Sims algorithm directly to matrix groups?
• RQ3 How effective is the probabilistic variant of the Schreier-Sims algorithm in reducing computation time for matrix groups?
• RQ4 What algorithmic optimizations are most effective in improving efficiency for matrix group computations?

Key findings

• The new implementation achieves significantly faster execution times than GAP’s standard Schreier-Sims algorithm for many matrix group benchmarks.
• Memory consumption was drastically reduced in several test cases, demonstrating superior space efficiency due to avoiding permutation group representation.
• The probabilistic variant of the algorithm reduced base point search time, contributing to overall performance gains.
• The direct matrix-based approach proved particularly effective for large matrix groups over finite fields, where permutation representation overhead is substantial.