嵌入Sierpienski Gasket分形的块空间GPU映射
Block-space GPU Mapping for Embedded Sierpiński Gasket Fractals
摘要：
研究了嵌入到$n \times n$的离散欧氏空间中的Sierpienski垫片分形的GPU线程映射问题。提出了一种块空间映射$λ: \mathbb{Z}_{\mathbb{E}}^{2} \mapsto \mathbb{Z}_{\mathbb{F}}^{2}$，从欧几里得并行空间$\mathbb{E}$映射到嵌入分形空间$\mathbb{F}$，该映射在$\mathcal{O}(\log_2 \log_2(n))$时间内完成，使用的线程不超过$\mathcal{O}(n^\mathbb{H})$线程，其中$\mathbb{H} \approx 1.58...$是Hausdorff维数，使其具有并行空间效率。与边界框映射相比，$λ(ω)$在并行空间中提供了次指数级的改进，并且在$n > n_0$之后提供了单调增加的加速比。实验性能测试表明，在实践中，$λ(ω)$可以产生任何块大小的性能提高一次$n > n_0 = 2^8$，达到约$10\times$的$n=2^{16}$在最佳块配置的加速比。
英文摘要：
This work studies the problem of GPU thread mapping for a Sierpiński gasket fractal embedded in a discrete Euclidean space of $n \times n$. A block-space map $λ: \mathbb{Z}_{\mathbb{E}}^{2} \mapsto \mathbb{Z}_{\mathbb{F}}^{2}$ is proposed, from Euclidean parallel space $\mathbb{E}$ to embedded fractal space $\mathbb{F}$, that maps in $\mathcal{O}(\log_2 \log_2(n))$ 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, $λ(ω)$ offers a sub-exponential improvement in parallel space and a monotonically increasing speedup once $n > n_0$. Experimental performance tests show that in practice $λ(ω)$ can produce performance improvement at any block-size once $n > n_0 = 2^8$, reaching approximately $10\times$ of speedup for $n=2^{16}$ under optimal block configurations.
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