A parallel algorithm for Gaussian elimination over finite fields

TL;DR

This paper introduces a parallel Gaussian elimination algorithm for large matrices over finite fields, subdividing matrices into blocks for enhanced concurrency and efficiency, and demonstrates its effectiveness on very large matrices.

Contribution

It presents a novel block-based parallel Gaussian elimination algorithm that improves concurrency and efficiency for large finite field matrices, including transformation matrix computation.

Findings

Effective on matrices up to 1,000,000 x 1,000,000

Concurrency scales with the number of blocks

Reduces unnecessary data storage during row operations

Abstract

In this paper we describe a parallel Gaussian elimination algorithm for matrices with entries in a finite field. Unlike previous approaches, our algorithm subdivides a very large input matrix into smaller submatrices by subdividing both rows and columns into roughly square blocks sized so that computing with individual blocks on individual processors provides adequate concurrency. The algorithm also returns the transformation matrix, which encodes the row operations used. We go to some lengths to avoid storing any unnecessary data as we keep track of the row operations, such as block columns of the transformation matrix known to be zero. The algorithm is accompanied by a concurrency analysis which shows that the improvement in concurrency is of the same order of magnitude as the number of blocks. An implementation of the algorithm has been tested on matrices as large as $1 000 000\times 1 000 000$ over small finite fields.