A 'Best-of-Breed' approach for designing a fast algorithm for computing fixpoints of Galois Connections

28 The fixpoints of Galois Connections form patterns in binary relational data, such as object29 attribute relations, that are important in a number of data analysis fields, including Formal 30 Concept Analysis (FCA), Boolean factor analysis and frequent itemset mining. However, the 31 large number of such fixpoints present in a typical dataset requires efficient computation 32 to make analysis tractable, particularly since any particular fixpoint may be computed 33 many times. Because they can be computed in a canonical order, testing the canonicity 34 of fixpoints to avoid duplicates has proven to be a key factor in the design of efficient algo35 rithms. The most efficient of these algorithms have been variants of the Close-By-One 36 (CbO) algorithm. In this article, the algorithms CbO, FCbO, In-Close, In-Close2 and a new 37 variant, In-Close3, are presented together for the first time, with in-Close2 and In-Close3 38 being the results of breeding In-Close with FCbO. To allow them to be easily compared, 39 the algorithms are presented in the same style and notation. The important advances in 40 CbO are described and compared graphically using a simple example. For the first time, 41 the algorithms are implemented using the same structures and techniques to provide a 42 level playing field for evaluation. Their performance is tested and compared using a range 43 of data sets and the most important features identified for a CbO ‘Best-of-Breed’. This arti44 cle also presents, for the first time, the ‘partial-closure’ canonicity test. 45 2014 Published by Elsevier Inc. 46