Each step replaces the two mostsimilar clusters by its union.

As a fundamental unsupervised learning task, hierarchical clustering has been extensively studied in the past decade. In particular, standard metric formulations as hierarchical $k$-center, $k$-means, and $k$-median received a lot of attention and the problems have been studied extensively in different models of computation. Despite all this interest, not many efficient parallel algorithms are known for these problems. In this paper we introduce a new parallel algorithm for the Euclidean hierarchical $k$-median problem that, when using machines with memory $s$ (for $s\in \Omega(\log^2 (n+\Delta+d))$), outputs a hierarchical clustering such that for every fixed value of $k$ the cost of the solution is at most an $O(\min\{d, \log n\} \log \Delta)$ factor larger in expectation than that of an optimal solution. Furthermore, we also get that for all $k$ simultanuously the cost of the solution is at most an $O(\min\{d, \log n\} \log \Delta \log (\Delta d n))$ factor bigger that the corresponding optimal solution.

We acknowledge the reviewer's concern about extending our approach to distributed environments. However, we feel that this should not be a criticism of this paper, whose focus is on multicore systems which pose a unique set of challenges. Furthermore, we have shown that we can handle large graphs without stretching the limits of multicore systems.

Given a similarity graph between items, correlation clustering (CC) groups similar items together and dissimilar ones apart. One of the most popular CC algorithms is KwikCluster: an algorithm that serially clusters neighborhoods of vertices, and obtains a 3-approximation ratio. Unfortunately, in practice KwikCluster requires a large number of clustering rounds, a potential bottleneck for large graphs.

Each step replaces the two most similar clusters by its union.

Random walks are a fundamental primitive used in many machine learning algorithms with several applications in clustering and semi-supervised learning.

In theory, the techniques can speed up matrix operations, however, in practice, they are not fast enough.

Machine learning potentials have achieved great success in accelerating atomistic simulations. Many of them relying on atom-centered local descriptors are natural for parallelization. More recent message passing neural network (MPNN) models have demonstrated their superior accuracy and become increasingly popular. However, efficiently parallelizing MPNN models across multiple nodes remains challenging, limiting their practical applications in large-scale simulations. Here, we propose an efficient parallel algorithm for MPNN models, in which additional data communication is minimized among local atoms only in each MP layer without redundant computation, thus scaling linearly with the layer number. Integrated with our recursively embedded atom neural network model, this algorithm demonstrates excellent strong scaling and weak scaling behaviors in several benchmark systems. This approach enables massive molecular dynamics simulations on MPNN models as fast as on strictly local models for over 100 million atoms, vastly extending the applicability of the MPNN potential to an unprecedented scale. This general parallelization framework can empower various MPNN models to efficiently simulate very large and complex systems.