We propose a data structure in d-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces. We demonstrate the usefulness of our hyperbolic quadtree data structure by giving an algorithm for constant-approximate closest pair and dynamic constant-approximate nearest neighbours in hyperbolic space of constant dimension d.

翻译：我们提出了一种在d维双曲空间中的数据结构，该结构可视为欧氏空间中四叉树的自然对应物。基于此数据结构，我们提出了双曲点集的一种所谓L序，它是欧氏空间Z序的延伸。通过给出常维数双曲空间中恒定近似最近点对和动态恒定近似最近邻的算法，我们证明了该双曲四叉树数据结构的有效性。