Block-Space GPU Mapping for Embedded Sierpiński Gasket Fractals

This work studies the problem of GPU thread mapping for a Sierpiński gasket fractal embedded in a discrete Euclidean space of n × n. A block-space map λ : Z_E² → Z_F² is proposed, from Euclidean parallel space E to embedded fractal space F, that maps in O(log₂ log₂(n)) time and uses no more than O(n^H) threads with H ≈ 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 > n₀. Experimental performance tests show that in practice λ(ω) can produce performance improvement at any block-size once n > n₀ = 2⁸, reaching approximately 10× of speedup for n = 2¹⁶ under typical block configurations.