A Hybrid MPI/OpenMP Parallelization of $K$ -Means Algorithms Accelerated Using the Triangle Inequality

The standard formulation of the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-means clustering (Lloyd’s method) performs many unnecessary distance calculations. In this paper, we focus on four approaches that use the triangle inequality to avoid unnecessary distance calculations. These approaches are Drake’s, Elkan’s, Annulus, and Yinyang algorithms. We propose a hybrid MPI/OpenMP parallelization of these algorithms in which the dataset and the corresponding data structures storing bounds on distances are evenly divided among MPI processes. Then, in the assignment step of a <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-means iteration, each MPI process computes the assignment of its portion of data using OpenMP threads. In the update step of the iteration, the cluster centroids are computed using a hierarchical all-reduce operation. In the computational experiments, we compared the strong scalability of these four algorithms with the scalability of Lloyd’s algorithm, parallelized using the same approach. The results indicate that all four algorithms maintain an advantage in computing time over Lloyd’s algorithm. A comparison with two software packages, whose sources are publicly available, in the same computing environment, shows that our implementations are more efficient.