# Block-space GPU Mapping for Embedded Sierpi\'nski Gasket Fractals

## TL;DR

This paper introduces an efficient GPU thread mapping method for embedding Sierpiński gasket fractals in Euclidean space, significantly improving parallel space utilization and performance over traditional bounding-box approaches.

## Contribution

A novel block-space mapping algorithm for GPU fractal embedding that operates in logarithmic time and reduces thread usage compared to bounding-box methods.

## Key findings

- Achieves $ ext{O}( ext{log}_2 ext{log}_2 n)$ mapping time.
- Provides up to 10x speedup for large fractals.
- Reduces thread count to $ ext{O}(n^ ext{H})$ with H≈1.58.

## 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 $\mathcal{O}(\log_2 \log_2(n))$ 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, $\lambda(\omega)$ offers a sub-exponential improvement in parallel space and a monotonically increasing speedup once $n > n_0$. Experimental performance tests show that in practice $\lambda(\omega)$ 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.