Comments on 'Parallel Algorithms for Hierarchical Clustering and Cluster Validity'

the new sample, that is Thus, in general, our learning algorithm may require more than rlog,(IXI-1)1 + 1 samples to solve the problem in the worst case. We have analyzed the problem of constructing a linear classifier for a finite set X of linearly separable vectors by partially supervised leaming. The proposed learning algorithm consists of two major operations: sample selection and classifier construction. In both operations, only the classification information about the sample set is used. The primary goal of the leaming system is to identify the membership of those vectors outside the sample set. The key issue in the design of a partially supervised learning algorithm is sample selection. The following factors should be considered in the evaluation of a sample selection algorithm: 1) the size of the sample set, 2) the classification error, and 3) the computational complexity. We have shown that the sample set selected by our algorithm is minimal for R' and R2 in the worst case, and the classifier derived from such a sample set produces no classification errors. The main disadvantage of our approach is the high computational cost as it requires the solution of O(1XI3) linear programming problems in the worst case. In this study, we have only demonstrated how linearly separable problems can be solved by partially supervised learning. It would be useful to extend our framework to deal with more complicated systems. Query formulation in linear retrieval models, " J. Abstract-The purpose of this correspondence is to indicate that state-of-the-art hierarchical clustering algorithms have 0 n time complexity and should be referred to in preference to the O(n) algorithms, which were described in many texts in the 1970's. We also point out some further references in the parallelizing of hierarchic clustering algorithms. Li [6] describes parallel implementations of hierarchical clustering algorithms that achieve O(n 2) computational time complexity and thereby improve on the baseline of sequential implementations. The latter are stated to be O(n3), with the exception of the single link method. It is inappropriate to use as one's baseline implementations, which could only be described as 1970's vintage. Surely, it should have been noted that O (n 2) time implementations exist for most of the widely known hierarchical clustering methods. Average time implementations that come close to O (n) are also known. Rohlf [13] discusses an O (n log log n) expected time algorithm for the minimal spanning tree, …