Cache-oblivious Matrix Multiplication for Exact TU Factorisation

Abstract

We present a cache-oblivious adaptation of matrix multiplication to be incorporated in the parallel TU decomposition for rectangular matrices over finite fields, based on the Morton-hybrid space-filling curve representation. To realise this, we introduce the concepts of alignment and containment of sub-matrices under the Morton-hybrid layout. We redesign the decompositions within the recursive matrix multiplication to force the base case to avoid all jumps in address space, at the expense of extra recursive matrix multiplication (MM) calls. We show that the resulting cache oblivious adaptation has low span, and our experiments demonstrate that its sequential evaluation order demonstrates significant improvement in run-time, despite the recursion overhead. We also observe orders of magnitude reductions in cache misses, which promises to yield a highly I/O efficient multithreaded deployment of this algorithm on parallel machines with private or shared caches.