Block-space GPU Mapping for Embedded Sierpiński Gasket Fractals
(1706.04552)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 $\lambda: \mathbb{Z}{\mathbb{E}}{2} \mapsto \mathbb{Z}{\mathbb{F}}{2}$ is proposed, from Euclidean parallel space $\mathbb{E}$ to embedded fractal space $\mathbb{F}$, that maps in $\mathcal{O}(\log2 \log2(n))$ time and uses no more than $\mathcal{O}(n\mathbb{H})$ 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 $n > n0$. Experimental performance tests show that in practice $\lambda(\omega)$ can produce performance improvement at any block-size once $n > n0 = 28$, reaching approximately $10\times$ of speedup for $n=2{16}$ under optimal block configurations.
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