Comparative study of space filling curves for cache oblivious TU Decomposition

TL;DR

This paper compares different space-filling curve-based matrix layouts for cache-oblivious parallel TU decomposition, finding Morton-hybrid order to be most efficient in index conversion and overall performance, especially for large matrices.

Contribution

It provides a detailed analysis of index conversion costs for various space-filling curves and demonstrates the superior performance of Morton-hybrid layout in cache-oblivious matrix decomposition.

Findings

Morton-hybrid order has the lowest index conversion cost.

Morton-hybrid layout achieves significant performance improvements for large matrices.

Preliminary experiments show orders of magnitude faster computation with Morton-hybrid layout.

Abstract

We examine several matrix layouts based on space-filling curves that allow for a cache-oblivious adaptation of parallel TU decomposition for rectangular matrices over finite fields. The TU algorithm of \cite{Dumas} requires index conversion routines for which the cost to encode and decode the chosen curve is significant. Using a detailed analysis of the number of bit operations required for the encoding and decoding procedures, and filtering the cost of lookup tables that represent the recursive decomposition of the Hilbert curve, we show that the Morton-hybrid order incurs the least cost for index conversion routines that are required throughout the matrix decomposition as compared to the Hilbert, Peano, or Morton orders. The motivation lies in that cache efficient parallel adaptations for which the natural sequential evaluation order demonstrates lower cache miss rate result in overall faster performance on parallel machines with private or shared caches, on GPU's, or even cloud computing platforms. We report on preliminary experiments that demonstrate how the TURBO algorithm in Morton-hybrid layout attains orders of magnitude improvement in performance as the input matrices increase in size. For example, when $N = 2^{13}$, the row major TURBO algorithm concludes within about 38.6 hours, whilst the Morton-hybrid algorithm with truncation size equal to $64$ concludes within 10.6 hours.