Fast Deterministic Single-Linkage 2D-Spatial Cluster Analysis

Cluster analysis is a common task in data mining, machine learning and related fields. There exist a plethora of clustering algorithms designed for this purpose, but many are prohibitively inefficient (e.g. quality-threshold clustering), non-deterministic (k-means) or utilise inherently lossy partitioning models (k-d tree clustering). Single-linkage hierarchical clustering is a form of cluster analysis which unites clusters based on the minimum distance between them, using a given distance metric. Though more complex clustering methods exist, the intuitive nature and ease of implementation of single-linkage hierarchical clustering makes it a reasonably common choice for cluster analysis. However, the general case of single-linkage clustering is O(n3) (though the SLINK algorithm runs in O(n2) time for some special cases [1]). A specific case – and likely the most intuitive case – of cluster analysis is that which is performed on a two-dimensional Euclidean plane. This has many real-world applications, including image analysis/segmentation and medical imaging. This paper presents a quasi-linear time algorithm for single-linkage hierarchical clustering of points in two-dimensional Euclidean space. This concept is not itself novel (see [2, 3]); however, the use of an agglomerative approach as opposed to a fixedthreshold edge filtering provides a concise and effective way to extract a specific number of clusters. The algorithm also guarantees that the maximum distance between any pair of points in a cluster is minimised. BODY Find the Delaunay triangulation of the points with Sweephull, then Kruskal’s MST until the required cluster count is reached. O(n logn).