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Abstract:We introduce TeraHAC, a $(1+\epsilon)$-approximate hierarchical agglomerative clustering (HAC) algorithm which scales to trillion-edge graphs. Our algorithm is based on a new approach to computing $(1+\epsilon)$-approximate HAC, which is a novel combination of the nearest-neighbor chain algorithm and the notion of $(1+\epsilon)$-approximate HAC. Our approach allows us to partition the graph among multiple machines and make significant progress in computing the clustering within each partition before any communication with other partitions is needed. We evaluate TeraHAC on a number of real-world and synthetic graphs of up to 8 trillion edges. We show that TeraHAC requires over 100x fewer rounds compared to previously known approaches for computing HAC. It is up to 8.3x faster than SCC, the state-of-the-art distributed algorithm for hierarchical clustering, while achieving 1.16x higher quality. In fact, TeraHAC essentially retains the quality of the celebrated HAC algorithm while significantly improving the running time.

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Abstract:We study the problem of graph clustering under a broad class of objectives in which the quality of a cluster is defined based on the ratio between the number of edges in the cluster, and the total weight of vertices in the cluster. We show that our definition is closely related to popular clustering measures, namely normalized associations, which is a dual of the normalized cut objective, and normalized modularity. We give a linear time constant-approximate algorithm for our objective, which implies the first constant-factor approximation algorithms for normalized modularity and normalized associations.

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Abstract:Graph clustering and community detection are central problems in modern data mining. The increasing need for analyzing billion-scale data calls for faster and more scalable algorithms for these problems. There are certain trade-offs between the quality and speed of such clustering algorithms. In this paper, we design scalable algorithms that achieve high quality when evaluated based on ground truth. We develop a generalized sequential and shared-memory parallel framework based on the LambdaCC objective (introduced by Veldt et al.), which encompasses modularity and correlation clustering. Our framework consists of highly-optimized implementations that scale to large data sets of billions of edges and that obtain high-quality clusters compared to ground-truth data, on both unweighted and weighted graphs. Our empirical evaluation shows that this framework improves the state-of-the-art trade-offs between speed and quality of scalable community detection. For example, on a 30-core machine with two-way hyper-threading, our implementations achieve orders of magnitude speedups over other correlation clustering baselines, and up to 28.44x speedups over our own sequential baselines while maintaining or improving quality.

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Abstract:We study the widely used hierarchical agglomerative clustering (HAC) algorithm on edge-weighted graphs. We define an algorithmic framework for hierarchical agglomerative graph clustering that provides the first efficient $\tilde{O}(m)$ time exact algorithms for classic linkage measures, such as complete- and WPGMA-linkage, as well as other measures. Furthermore, for average-linkage, arguably the most popular variant of HAC, we provide an algorithm that runs in $\tilde{O}(n\sqrt{m})$ time. For this variant, this is the first exact algorithm that runs in subquadratic time, as long as $m=n^{2-\epsilon}$ for some constant $\epsilon > 0$. We complement this result with a simple $\epsilon$-close approximation algorithm for average-linkage in our framework that runs in $\tilde{O}(m)$ time. As an application of our algorithms, we consider clustering points in a metric space by first using $k$-NN to generate a graph from the point set, and then running our algorithms on the resulting weighted graph. We validate the performance of our algorithms on publicly available datasets, and show that our approach can speed up clustering of point datasets by a factor of 20.7--76.5x.

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Abstract:DBSCAN is a popular density-based clustering algorithm. It computes the $\epsilon$-neighborhood graph of a dataset and uses the connected components of the high-degree nodes to decide the clusters. However, the full neighborhood graph may be too costly to compute with a worst-case complexity of $O(n^2)$. In this paper, we propose a simple variant called SNG-DBSCAN, which clusters based on a subsampled $\epsilon$-neighborhood graph, only requires access to similarity queries for pairs of points and in particular avoids any complex data structures which need the embeddings of the data points themselves. The runtime of the procedure is $O(sn^2)$, where $s$ is the sampling rate. We show under some natural theoretical assumptions that $s \approx \log n/n$ is sufficient for statistical cluster recovery guarantees leading to an $O(n\log n)$ complexity. We provide an extensive experimental analysis showing that on large datasets, one can subsample as little as $0.1\%$ of the neighborhood graph, leading to as much as over 200x speedup and 250x reduction in RAM consumption compared to scikit-learn's implementation of DBSCAN, while still maintaining competitive clustering performance.

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