Improving the GPU space of computation under triangular domain problems

TL;DR

This paper introduces a new GPU thread-block mapping function optimized for triangular domain problems, significantly reducing wasted computation and improving performance over existing methods on NVIDIA GPUs.

Contribution

The paper proposes a novel mapping function g(lambda) for GPU computation in triangular domains, enhancing efficiency and outperforming existing strategies in speed and resource utilization.

Findings

g(lambda) reduces wasted blocks from O(n^2) to O(n)

Experimental results show 12-15% speedup over bounding box strategy

Outperforms UTM and matches RB in speed, maintaining thread organization

Abstract

There is a stage in the GPU computing pipeline where a grid of thread-blocks is mapped to the problem domain. Normally, this grid is a k-dimensional bounding box that covers a k-dimensional problem no matter its shape. Threads that fall inside the problem domain perform computations, otherwise they are discarded at runtime. For problems with non-square geometry, this is not always the best idea because part of the space of computation is executed without any practical use. Two- dimensional triangular domain problems, alias td-problems, are a particular case of interest. Problems such as the Euclidean distance map, LU decomposition, collision detection and simula- tions over triangular tiled domains are all td-problems and they appear frequently in many areas of science. In this work, we propose an improved GPU mapping function g(lambda), that maps any lambda block to a unique location (i, j) in the triangular domain. The mapping is based on the properties of the lower triangular matrix and it works at a block level, thus not compromising thread organization within a block. The theoretical improvement from using g(lambda) is upper bounded as I < 2 and the number of wasted blocks is reduced from O(n^2) to O(n). We compare our strategy with other proposed methods; the upper-triangular mapping (UTM), the rectangular box (RB) and the recursive partition (REC). Our experimental results on Nvidias Kepler GPU architecture show that g(lambda) is between 12% and 15% faster than the bounding box (BB) strategy. When compared to the other strategies, our mapping runs significantly faster than UTM and it is as fast as RB in practical use, with the advantage that thread organization is not compromised, as in RB. This work also contributes at presenting, for the first time, a fair comparison of all existing strategies running the same experiments under the same hardware.