Obtaining scalable algorithms for \emph{hierarchical agglomerative clustering} (HAC) is of significant interest due to the massive size of real-world datasets. At the same time, efficiently parallelizing HAC is difficult due to the seemingly sequential nature of the algorithm. In this paper, we address this issue and present ParHAC, the first efficient parallel HAC algorithm with sublinear depth for the widely-used average-linkage function. In particular, we provide a $(1+\epsilon)$-approximation algorithm for this problem on $m$ edge graphs using $\tilde{O}(m)$ work and poly-logarithmic depth. Moreover, we show that obtaining similar bounds for \emph{exact} average-linkage HAC is not possible under standard complexity-theoretic assumptions.We complement our theoretical results with a comprehensive study of the ParHAC algorithm in terms of its scalability, performance, and quality, and compare with several state-of-the-art sequential and parallel baselines. On a broad set of large publicly-available real-world datasets, we find that ParHAC obtains a 50.1x speedup on average over the best sequential baseline, while achieving quality similar to the exact HAC algorithm. We also show that ParHAC can cluster one of the largest publicly available graph datasets with 124 billion edges in a little over three hours using a commodity multicore machine.

We stress that unweighted and weighted in the linkage measure names refer to the linkage methods. Recall that our approach is based on geometric layering, where we group the edges based on their weights and process all edges within the same layer in parallel. A similar idea is used in the Affinity Clustering algorithm of Bateni et al. [ Our algorithm starts by first randomly coloring the active vertices red and blue with equal probability. Directly applying the random-mate approach (e.g., as applied in Let D be initialized to the identity clustering. O (log n) layers are required to represent every weight in this weight range.Lemma 2.1.

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

Obtaining scalable algorithms for \emph{hierarchical agglomerative clustering} (HAC) is of significant interest due to the massive size of real-world datasets. At the same time, efficiently parallelizing HAC is difficult due to the seemingly sequential nature of the algorithm. In this paper, we address this issue and present ParHAC, the first efficient parallel HAC algorithm with sublinear depth for the widely-used average-linkage function. In particular, we provide a (1 \epsilon) -approximation algorithm for this problem on m edge graphs using \tilde{O}(m) work and poly-logarithmic depth. Moreover, we show that obtaining similar bounds for \emph{exact} average-linkage HAC is not possible under standard complexity-theoretic assumptions.We complement our theoretical results with a comprehensive study of the ParHAC algorithm in terms of its scalability, performance, and quality, and compare with several state-of-the-art sequential and parallel baselines.