Comparative study of space filling curves for cache oblivious TU Decomposition

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.