ParChain: A Framework for Parallel Hierarchical Agglomerative Clustering using Nearest-Neighbor Chain

Abstract

This paper studies the hierarchical clustering problem, where the goal is to produce a dendrogram that represents clusters at varying scales of a data set. We propose the ParChain framework for designing parallel hierarchical agglomerative clustering (HAC) algorithms, and using the framework we obtain novel parallel algorithms for the complete linkage, average linkage, and Ward's linkage criteria. Compared to most previous parallel HAC algorithms, which require quadratic memory, our new algorithms require only linear memory, and are scalable to large data sets. ParChain is based on our parallelization of the nearest-neighbor chain algorithm, and enables multiple clusters to be merged on every round. We introduce two key optimizations that are critical for efficiency: a range query optimization that reduces the number of distance computations required when finding nearest neighbors of clusters, and a caching optimization that stores a subset of previously computed distances, which are likely to be reused.Experimentally, we show that our highly-optimized implementations using 48 cores with two-way hyper-threading achieve 5.8-110.1x speedup over state-of-the-art parallel HAC algorithms and achieve 13.75-54.23x self-relative speedup. Compared to state-ofthe-art algorithms, our algorithms require up to 237.3x less space. Our algorithms are able to scale to data set sizes with tens of millions of points, which existing algorithms are not able to handle.

ParChain

“…Suppose the distance of the furthest point pair between 𝐶 𝑖 and 𝐵 is Δ comp (𝐶 𝑖 , 𝐵) = 𝑑 (𝑝, 𝑞). Since the average Euclidean distance between points in two clusters is not smaller than the distance between their centroids (shown in the full paper [69]), applying this property to 𝐶 𝑖 and {𝑞 } for any point 𝑞 ∈ 𝐵, we see there must exists some 𝑝 ∈ 𝐶 𝑖 such that 𝑑 ( x𝐶 𝑖 , 𝑞 ) ≤ 𝑑 (𝑝 , 𝑞 ). Since (𝑝, 𝑞) is the furthest point pair, we have that 𝑑 ( x𝐶 𝑖 , 𝑞 ) ≤ 𝑑 (𝑝 , 𝑞 ) ≤ 𝑑 (𝑝, 𝑞).…”

“…The nearest neighbor 𝐵 of 𝐶 𝑖 must have its centroid inside the ball, i.e., x𝐶 𝑖 − x𝐵 ≤ Δ avg-1 (𝐶 𝑖 , 𝐵). The proof is provided in the full version of our paper [69].…”