Space filling curves are widely used in computer science. In particular, Hilbert curves and their generalizations to higher dimension are used as an indexing method because of their nice locality properties. This article generalizes this concept to the systematic construction of p-adic versions of Hilbert curves based on special affine transformations of the p-adic Gray code and develops a scaled indexing method for data taken from high-dimensional spaces based on these new curves, which with increasing dimension is shown to be less space consuming than the optimal standard static Hilbert curve index. A measure is derived, which allows to assess the local sparsity of a dataset, and is tested on some real-world data.

中文翻译：

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关于高维数据的 p-Adic 标度空间填充曲线指数的行为

空间填充曲线广泛用于计算机科学。特别是，希尔伯特曲线及其对更高维的概括被用作索引方法，因为它们具有良好的局部性。本文将这一概念推广到基于 p-adic 格雷码的特殊仿射变换的 Hilbert 曲线 p-adic 版本的系统构造，并开发了一种基于这些新曲线的高维空间数据的缩放索引方法。与最佳标准静态希尔伯特曲线指数相比，随着维数的增加，空间消耗更少。推导出一个度量，它允许评估数据集的局部稀疏性，并在一些真实世界的数据上进行测试。