Block-space GPU Mapping for Embedded Sierpiński Gasket Fractals

Abstract: This work studies the problem of GPU thread mapping for a Sierpi\'nski gasket fractal embedded in a discrete Euclidean space of $n \times n$. A block-space map $$\lambda: \mathbb{Z}<em>{\mathbb{E}}^{2} \mapsto \mathbb{Z}</em>{\mathbb{F}}<sup>{2}$$ is proposed, from Euclidean parallel space $\mathbb{E}$ to embedded fractal space $\mathbb{F}$, that maps in $\mathcal{O}(\log_2 \log_2(n))$ time and uses no more than $\mathcal{O}(n<sup>\mathbb{H})$ threads with $\mathbb{H} \approx 1.58...$ being the Hausdorff dimension, making it parallel space efficient. When compared to a bounding-box map, $\lambda(\omega)$ offers a sub-exponential improvement in parallel space and a monotonically increasing speedup once $n &gt; n_0$. Experimental performance tests show that in practice $\lambda(\omega)$ can produce performance improvement at any block-size once $n &gt; n_0 = 2<sup>8$, reaching approximately $10\times$ of speedup for $n=2<sup>{16}$ under optimal block configurations.