On the Parallel Complexity of Minimum Sum of Diameters Clustering

Given a set of n entities to be classified, and a matric of dissimilarities between pairs of them. This paper considers the problem called MINIMUM SUM OF DIAMETERS CLUSTERING PROBLEM, where a partition of the set of entities into k clusters such that the sum of the diameters of these clusters is minimized. Brucker showed that the complexity of the problem is NP-hard, when k >= 3 [1]. For the case of k = 2, Hansen and Jaumard gave an O(n(3) log n) algorithm [2], which Ramnath later improved the running time to O(n(3)) [3]. This paper discusses the parallel complexity of the MINIMUM SUM OF DIAMETERS CLUSTERING PROBLEM. For the case of k = 2, we show that the problem in parallel in fact belongs in class NC.(1) In particular, we show that the parallel complexity of the problem is O(log n) parallel time and n(7) processors on the COMMON CRCW PRAM model. Additionally, we propose the parallel algorithmic technique which can be applied to improve the processor bound by a factor of n. As a result, we show that the problem can be quickly solved in O(log n) parallel time using n(6) processors on the COMMON CRCW PRAM model. In addition, regarding the issue of high processor complexity, we also propose a more practical NC algorithm which can be implemented in O(log(3) n) parallel time using n(3.376) processors on the EREW PRAM model.