Scientific Annals of Computer Science vol. 25 (1), 2015, pp. 155–170 doi: 10.7561/SACS.2015.1.155 Several Remarks on Dissimilarities and Ultrametrics Dan A. Simovici 1 Abstract We investigate the relationships between tolerance relations, equiv- alence relations, and ultrametrics. The set of spheres associated to an ultrametric space has a tree structure that reflects a hierarchy on the set of equivalences associated to that space. We show that ev- ery ultrametric defined on a finite space is a linear combination of binary ultrametric and we introduce the notion of ultrametricity for dissimilarities, which has applications in many data mining problems. Keywords: tolerance, equivalence, ultrametric, ultrametricity, tree of spheres 1 Introduction Dissimilarity spaces constitute the natural framework for a number of ex- ploratory techniques in machine learning and data mining such as certain classification methods and clustering algorithms. We examine relationships that exist between various types of dissimilarities and focus on ultrametrics. Ultrametrics are dissimilarities that satisfy a stronger version of the triangular inequality (usually associated with metrics) and they occur in many data mining applications such as agglomerative hierarchical clustering algorithms [4, 5, 2, 6], and have applications in the study of phylogenetic trees in biology [10, 7], p-adic numbers in mathematics [12, 1], and certain physical systems [11], etc. 1 University of Massachusetts Boston, Department of Computer Science, Boston, Massachusetts, E-mail: dsim@cs.umb.edu