Clustering

Abstract

This chapter is concerned with how to make a list of all the ways of clustering the elements of a finite set, which means recognizing a family of non-void subsets, called clusters, any pair of which is either disjoint or nested. The list of clusterings is greatly shortened, without loss of generality, if we define a cluster as a subset with more than one element and not equal to the whole set. This reduces the number of clusterings of a set of µ elements, by a factor of 2^{µ + 1}, since we have eliminated µ + 1 subsets, each of which could otherwise have been either a cluster or not. Accordingly, a clustering is defined as a family of such clusters, any two of them being disjoint or nested. This includes the case of a clustering whose family of clusters is void.