Possibilities of Recursive GPU Mapping for Discrete Orthogonal Simplices

The problem of parallel thread mapping is studied for the case of discrete orthogonal $m$-simplices. The possibility of a $O(1)$ time recursive block-space map $\lambda: \mathbb{Z}^m \mapsto \mathbb{Z}^m$ is analyzed from the point of view of parallel space efficiency and potential performance improvement. The $2$-simplex and $3$-simplex are analyzed as special cases, where constant time maps are found, providing a potential improvement of up to $2\times$ and $6\times$ more efficient than a bounding-box approach, respectively. For the general case it is shown that finding an efficient recursive parallel space for an $m$-simplex depends of the choice of two parameters, for which some insights are provided which can lead to a volume that matches the $m$-simplex for $n>n_0$, making parallel space approximately $m!$ times more efficient than a bounding-box.