On the parallel complexity of minimum sum of diameters clustering - Juneam, Nopadon; Kantabutra, Sanpawat (2015 International Computer Science and Engineering Conference (ICSEC), 2015)
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.