NC algorithms for minimum sum of diameters clustering

Abstract

Given a set of n entities to be classified, and a matric of dissimilarities between pairs of them. This article considers the problem called Minimum Sum of Diameters Clustering Problem, where a partition of the set of entities into κ clusters such that the sum of the diameters of these clusters is minimized. In sequential, Brucker showed that the problem is NP-hard, when κ ≥ 3 [1]. For the case of κ = 2, Hansen and Jaumard gave an O(n3logn) algorithm [2], which Ramnath later improved the running time to O(n3) [3]. In this article, we discuss parallel algorithms for the Minimum Sum of Diameters Clustering Problem, for the case of κ = 2. In particular, we present an NC algorithm that runs in O(logn) parallel time and n7 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, our algorithm can be implemented in O(logn) parallel time using n6 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(log3n) parallel time using n3.376 processors on the EREW PRAM model.