  The study of data-parallel domain re-organization and thread-mapping
techniques are relevant topics as they can increase the efficiency of GPU
computations when working on spatial discrete domains with non-box-shaped
geometry. In this work we study the potential benefits of applying a succint
data re-organization of a tetrahedral data-parallel domain of size
$\mathcal{O}(n^3)$ combined with an efficient block-space GPU map of the form
$g:\mathbb{N} \rightarrow \mathbb{N}^3$. Results from the analysis suggest that
in theory the combination of these two optimizations produce significant
performance improvement as block-based data re-organization allows a coalesced
one-to-one correspondence at local thread-space while $g(\lambda)$ produces an
efficient block-space spatial correspondence between groups of data and groups
of threads, reducing the number of unnecessary threads from $O(n^3)$ to
$O(n^2\rho^3)$ where $\rho$ is the linear block-size and typically $\rho^3 \ll
n$. From the analysis, we obtained that a block based succint data
re-organization can provide up to $2\times$ improved performance over a linear
data organization while the map can be up to $6\times$ more efficient than a
bounding box approach. The results from this work can serve as a useful guide
for a more efficient GPU computation on tetrahedral domains found in spin
lattice, finite element and special n-body problems, among others.
