This thesis consists of two topics related to computational geometry and one topic related to topological data analysis (TDA), which combines fields of computational geometry and algebraic topology for analyzing data. The first part studies the classical problem of finding k nearest neighbors to m query points in a larger set of n reference points in any metric space. The second part is about the construction of a Minimum Spanning Tree (MST) on any finite metric space. The third part extends the key concept of persistence within Topological Data Analysis in a new direction.

翻译：该论文由两个与计算几何有关的专题和一个与地形数据分析有关的专题组成,这两个专题将计算几何和代数表学领域结合起来,用于分析数据;第一部分研究在任何计量空间找到较宽的 n 参照点中的 k 近邻至 m 查询点的典型问题;第二部分涉及在任何有限的计量空间上建造一个最小宽度树(MST),第三部分将地形数据分析中持久性的关键概念扩展至新的方向。