A Vector Implementation of Gaussian Elimination over GF(2): Exploring the Design-Space of Strassen's Algorithm as a Case Study

Abstract

Gaussian elimination is a key algorithm in linear algebra. It has many usages, for instance solving systems of linear equations and determining whether a set of vectors is linearly independent. This algorithm transforms an input matrix into a matrix in row (column) echelon form. The matrix entries and the transformations are defined over algebraic fields either infinite (e.g. the real numbers) or finite (e.g. GF (2)). This work discusses a vector implementation of this algorithm over GF (2). The evaluation develops a case study that searches exhaustively for algorithms over GF (2) similar to Strassen's algorithm (a matrix-multiply algorithm with sub cubic complexity) because the search engine requires solving a huge number of Gaussian eliminations over GF (2). Our vector implementation allows the search engine to complete the exploration in less than nine hours on a commodity processor supporting AVX2, outperforming by 1.92X a scalar-SWAR implementation specialized for the case study and by 7.43X a generic scalar-SWAR implementation. Our results show that, over GF (2), there are 20 algorithms similar to Strassen's.