Fast redshift clustering with the Baire (ultra) metric

If X is endowed with a metric, then this metric can be mapped onto an ultrametric. In practice, endowing X with a metric can be relaxed to a dissimilarity. An often used mapping from metric to ultrametric is by means of an agglomerative hierarchical clustering algorithm. A succession of n 1 pairwise merge steps take place by making use of the closest pair of singletons and/or clusters at each step. Here n is the number of observations, i.e. the cardinality of set X. Closeness between singletons is furnished by whatever distance or dissimilarity is in use. For closeness between singleton or non-singleton clusters, we need to define an inter-cluster distance or dissimilarity. This can be defined with reference to the cluster compactness or other property that we wish to optimize at each step of the algorithm. Since agglomerative hierarchical clustering requires consideration of pairwise dissimilarities at each stage it can be shown that even in the case of the most efficient algorithms, e.g.