A Geometric Approach to Matrix Ordering

TL;DR

This paper introduces a geometric recursive partitioning method for hypergraphs that improves matrix ordering for parallel LU decomposition and sparse matrix-vector multiplication, offering faster performance with competitive quality.

Contribution

A novel geometric recursive hypergraph partitioning approach that enhances matrix ordering efficiency for parallel and cache-oblivious computations.

Findings

Achieves better fill-in results in LU decomposition for some matrices.

Produces smaller cut sizes for sparse matrix-vector multiplication in certain cases.

On average, 21.6 times faster than Mondriaan in partitioning speed.

Abstract

We present a recursive way to partition hypergraphs which creates and exploits hypergraph geometry and is suitable for many-core parallel architectures. Such partitionings are then used to bring sparse matrices in a recursive Bordered Block Diagonal form (for processor-oblivious parallel LU decomposition) or recursive Separated Block Diagonal form (for cache-oblivious sparse matrix-vector multiplication). We show that the quality of the obtained partitionings and orderings is competitive by comparing obtained fill-in for LU decomposition with SuperLU (with better results for 8 of the 28 test matrices) and comparing cut sizes for sparse matrix-vector multiplication with Mondriaan (with better results for 4 of the 12 test matrices). The main advantage of the new method is its speed: it is on average 21.6 times faster than Mondriaan.