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 : Z2E 7→ Z 2 F is proposed, from Euclidean parallel space E to embedded fractal space F, that maps in O(log 2 log 2 (n)) time and uses no more than O(n) 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 = 2 , reaching approximately 10× of speedup for n = 2 under optimal block configurations. Keywords—GPU computing; thread mapping; block-space fractal domains; Sierpinski gasket;