Journal of Classification

, Volume 1, Issue 1, pp 7–24

Efficient algorithms for agglomerative hierarchical clustering methods

  • William H. E. Day
  • Herbert Edelsbrunner
Authors Of Articles

DOI: 10.1007/BF01890115

Cite this article as:
Day, W.H.E. & Edelsbrunner, H. Journal of Classification (1984) 1: 7. doi:10.1007/BF01890115


Whenevern objects are characterized by a matrix of pairwise dissimilarities, they may be clustered by any of a number of sequential, agglomerative, hierarchical, nonoverlapping (SAHN) clustering methods. These SAHN clustering methods are defined by a paradigmatic algorithm that usually requires 0(n3) time, in the worst case, to cluster the objects. An improved algorithm (Anderberg 1973), while still requiring 0(n3) worst-case time, can reasonably be expected to exhibit 0(n2) expected behavior. By contrast, we describe a SAHN clustering algorithm that requires 0(n2 logn) time in the worst case. When SAHN clustering methods exhibit reasonable space distortion properties, further improvements are possible. We adapt a SAHN clustering algorithm, based on the efficient construction of nearest neighbor chains, to obtain a reasonably general SAHN clustering algorithm that requires in the worst case 0(n2) time and space.

Whenevern objects are characterized byk-tuples of real numbers, they may be clustered by any of a family of centroid SAHN clustering methods. These methods are based on a geometric model in which clusters are represented by points ink-dimensional real space and points being agglomerated are replaced by a single (centroid) point. For this model, we have solved a class of special packing problems involving point-symmetric convex objects and have exploited it to design an efficient centroid clustering algorithm. Specifically, we describe a centroid SAHN clustering algorithm that requires 0(n2) time, in the worst case, for fixedk and for a family of dissimilarity measures including the Manhattan, Euclidean, Chebychev and all other Minkowski metrics.


Algorithm complexityAlgorithm designCentroid clustering methodGeometric modelSAHN clustering method

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • William H. E. Day
    • 1
  • Herbert Edelsbrunner
    • 2
  1. 1.Department of Computer ScienceMemorial University of NewfoundlandSt. John'sCanada
  2. 2.Institute für InformationsverarbeitungTechnische Universität GrazGrazAustria