One of the core services provided by OWL reasoners is classification: the discovery of all subclass relationships between class names occurring in an ontology. Discovering these relations can be computationally expensive, particularly if individual subsumption tests are costly or if the number of class names is large. We present a classification algorithm which exploits partial information about subclass relationships to reduce both the number of individual tests and the cost of working with large ontologies. We also describe techniques for extracting such partial information from existing reasoners. Empirical results from a prototypical implementation demonstrate substantial performance improvements compared to existing algorithms and implementations.