K-sets+: A linear-time clustering algorithm for data points with a sparse similarity measure

In this paper, we first propose a new iterative algorithm, called the K-sets⁺ algorithm for clustering data points in a semi-metric space, where the distance measure does not necessarily satisfy the triangular inequality. We show that the K-sets⁺ algorithm converges in a finite number of iterations and it retains the same performance guarantee as the K-sets algorithm for clustering data points in a metric space. We then extend the applicability of the K-sets⁺ algorithm from data points in a semi-metric space to data points that only have a symmetric similarity measure. Such an extension leads to great reduction of computational complexity. In particular, for an n × n similarity matrix with m nonzero elements in the matrix, the computational complexity of the K-sets⁺ algorithm is O((Kn + m)I), where I is the number of iterations. The memory complexity to achieve that computational complexity is O(Kn + m). As such, both the computational complexity and the memory complexity are linear in n when the n × n similarity matrix is sparse, i.e., m = O (n). We also conduct various experiments to show the effectiveness of the K-sets⁺ algorithm by using a synthetic dataset from the stochastic block model and a real network from the WonderNetwork website.