On a fast algorithm for the reconstruction of the hierarchical $\varepsilon$-cluster structure

The concept of a hierarchical $\varepsilon$-cluster structure is defined, and the properties of such structures are studied. The uniqueness of the decomposition of a metric configuration into a hierarchy of $\varepsilon$ clusters is proved for $\varepsilon<1$. The problem of finding a hierarchical $\varepsilon$-cluster structure in a metric configuration is studied. In the general case, the complexity of this problem is $O(N^2)$. An algorithm for solving this problem is proposed that has complexity from $O(N\ln N)$ to $O(N^2)$ on some specific classes of metric configurations.