|Abstract:||Understanding a linguistic theory within OT requires an exact characterization of the ranking conditions necessitated by data. These conditions determine the formal shape of the grammar while providing the crucial link between the data and its interpretation. We introduce an algorithm (â€˜Fusional Reductionâ€™, FRed) which calculates the necessary and sufficient ranking conditions inherent in any collection of candidates and presents them in a maximally concise and informative way. The algorithm, stemming from the original proposal of Brasoveanu 2003, is set within the fusional ERC theory of Prince 2002a (ROA-500). In this context, the Most Informative Basis and the Skeletal Basis emerge as the two important types of reduced representations of ranking structure. We examine their properties and show how FRed produces them from data. Fine-grained FRed is compared with broad-stroke RCD (Tesar & Smolensky 1993, Tesar 1995 et seq.), and RCD is re-interpreted and embraced within FRed as a simplified, information-losing sub-case. Finally, FRed is compared with other related algorithms in structure, worst-case complexity, and relevance to the analytical enterprise. This paper revises Brasoveanu & Prince 2005.