We introduce the Stellar decomposition, a model for efficient topological data structures over a broad range of simplicial and cell complexes. A Stellar decomposition of a complex is a collection of regions indexing the complex’s vertices and cells such that each region has sufficient information to locally reconstruct the star of its vertices, i.e., the cells incident in the region’s vertices. Stellar decompositions are general in that they can compactly represent and efficiently traverse arbitrary complexes with a manifold or non-manifold domain. They are scalable to complexes in high dimension and of large size, and they enable users to easily construct tailored application-dependent data structures using a fraction of the memory required by a corresponding global topological data structure on the complex.

As a concrete realization of this model for spatially embedded complexes, we introduce the Stellar tree, which combines a nested spatial tree with a simple tuning parameter to control the number of vertices in a region. Stellar trees exploit the complex’s spatial locality by reordering vertex and cell indices according to the spatial decomposition and by compressing sequential ranges of indices. Stellar trees are competitive with state-of-the-art topological data structures for manifold simplicial complexes and offer significant improvements for cell complexes and non-manifold simplicial complexes. We conclude with a high-level description of several mesh processing and analysis applications that utilize Stellar trees to process large datasets.

我们介绍了Stellar分解，Stellar分解是在广泛的简单和单元复合物中有效的拓扑数据结构的模型。复杂的恒星分解是索引复杂的顶点和单元格的区域的集合，这样每个区域都具有足够的信息来局部重建恒星它的顶点，即在该区域的顶点中入射的单元。恒星分解的一般性之处在于，它们可以紧凑地表示并有效地遍历具有流形或非流形域的任意复合物。它们可伸缩以适应高维度和大尺寸的复合物，并且使用户能够使用复合物上相应的全局拓扑数据结构所需的一部分内存轻松构建量身定制的与应用程序相关的数据结构。

作为此模型的空间嵌入复合体的具体实现，我们引入了Stellar树，该树将嵌套的空间树与简单的调整参数结合在一起，以控​​制区域中的顶点数量。恒星树通过根据空间分解对顶点和单元格索引进行重新排序并压缩索引的顺序范围来利用复合物的空间局部性。恒星树与最先进的拓扑数据结构在流形简单复合体方面具有竞争性，并为细胞复合体和非流形简单复合体提供了显着的改进。我们以对几种利用Stellar树处理大型数据集的网格处理和分析应用程序的高级描述作为结束。