Modern hierarchical, agglomerative clustering algorithms
Daniel Müllner
TL;DR
The paper analyzes efficient algorithms for hierarchical, agglomerative clustering under the common stored-dissimilarity setting, where output is a stepwise dendrogram. It introduces a new generic algorithm that supports any distance-update scheme (including inversions) and proves correctness for Rohlf's MST-based single linkage and Murtagh's NN-chain, while offering practical performance comparisons. The work provides concrete recommendations: use MST for single linkage, NN-chain for most other schemes, and the generic algorithm for centroid, median, and flexible updates. It also discusses extensions to vector data and surveys alternative methods, bridging theory with widely used software implementations.
Abstract
This paper presents algorithms for hierarchical, agglomerative clustering which perform most efficiently in the general-purpose setup that is given in modern standard software. Requirements are: (1) the input data is given by pairwise dissimilarities between data points, but extensions to vector data are also discussed (2) the output is a "stepwise dendrogram", a data structure which is shared by all implementations in current standard software. We present algorithms (old and new) which perform clustering in this setting efficiently, both in an asymptotic worst-case analysis and from a practical point of view. The main contributions of this paper are: (1) We present a new algorithm which is suitable for any distance update scheme and performs significantly better than the existing algorithms. (2) We prove the correctness of two algorithms by Rohlf and Murtagh, which is necessary in each case for different reasons. (3) We give well-founded recommendations for the best current algorithms for the various agglomerative clustering schemes.