Linkage methods are among the most popular algorithms for hierarchical clustering. Despite their relevance the current knowledge regarding the quality of the clustering produced by these methods is limited. Here, we improve the currently available bounds on the maximum diameter of the clustering obtained by complete-link for metric spaces. One of our new bounds, in contrast to the existing ones, allows us to separate complete-link from single-link in terms of approximation for the diameter, which corroborates the common perception that the former is more suitable than the latter when the goal is producing compact clusters. We also show that our techniques can be employed to derive upper bounds on the cohesion of a class of linkage methods that includes the quite popular average-link.

In several applications of automatic diagnosis and active learning a central problem is the evaluation of a discrete function by adaptively querying the values of its variables until the values read uniquely determine the value of the function. In general, the process of reading the value of a variable might involve some cost, computational or even a fee to be paid for the experiment required for obtaining the value. This cost should be taken into account when deciding the next variable to read. The goal is to design a strategy for evaluating the function incurring little cost (in the worst case or in expectation according to a prior distribution on the possible variables' assignments). Our algorithm builds a strategy (decision tree) which attains a logarithmic approxima- tion simultaneously for the expected and worst cost spent. This is best possible under the assumption that $P \neq NP.$