The hierarchy cost of a hierarchical clustering measures the quality of the induced k-clusterings compared to optimal k-clusterings. It is defined as the maximal ratio of the cost of an induced k-clustering with respect to k-center to the cost of an optimal k-clustering as k ranges over all possible values. In this article it is shown that there is always an hierarchical clustering with hierarchy cost of at most \(1.25+0.25\sqrt{41} \approx 2.85\) in the one dimensional case. Moreover, there is a hierarchical clustering with hierarchy cost of at most \(3+2\sqrt{2} \approx 5.83\) in general metric spaces.

与最优k聚类相比，层次聚类的层次成本衡量诱导的 k 聚类的质量。它被定义为诱导的k -clustering 相对于k -center 的成本与最佳k -clustering 的成本的最大比率，因为k范围在所有可能的值上。本文表明，在一维情况下，总是存在一个层次聚类，其层次成本最多为\(1.25+0.25\sqrt{41} \approx 2.85\) 。此外，在一般度量空间中存在一个层次结构成本最多为\(3+2\sqrt{2} \approx 5.83\)的层次聚类。