Hierarchy cost of hierarchical clusterings

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 root 41 approximate to 2.85 in the one dimensional case. Moreover, there is a hierarchical clustering with hierarchy cost of at most 3 + 2 root 2 approximate to 5.83 in general metric spaces.