# Space-filling Curves for High-performance Data Mining

arXiv: https://arxiv.org/abs/2008.01684 · HTML: https://mlsanity.com/paper/2008.01684/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2020.

## Built on by (most cited first, 3 of 3)

- [Analogous to Evolutionary Algorithm: Designing a Unified Sequence Model](https://mlsanity.com/paper/2105.15089/) (2021)
- [DRBD-Mamba for Robust and Efficient Brain Tumor Segmentation with Analytical Insights](https://mlsanity.com/paper/2510.14383/) (2025)
- [Spatial Priors via Space Filling Curves for Small and Limited Data Vision Transformers](https://mlsanity.com/paper/2606.14757/) (2026)

## Abstract (arXiv)

Space-filling curves like the Hilbert-curve, Peano-curve and Z-order map natural or real numbers from a two or higher dimensional space to a one dimensional space preserving locality. They have numerous applications like search structures, computer graphics, numerical simulation, cryptographics and can be used to make various algorithms cache-oblivious. In this paper, we describe some details of the Hilbert-curve. We define the Hilbert-curve in terms of a finite automaton of Mealy-type which determines from the two-dimensional coordinate space the Hilbert order value and vice versa in a logarithmic number of steps. And we define a context-free grammar to generate the whole curve in a time which is linear in the number of generated coordinate/order value pairs, i.e. a constant time per coordinate pair or order value. We also review two different strategies which enable the generation of curves without the usual restriction to square-like grids where the side-length is a power of two. Finally, we elaborate on a few applications, namely matrix multiplication, Cholesky decomposition, the Floyd-Warshall algorithm, k-Means clustering, and the similarity join.