A Linear Delay Algorithm for Building Concept Lattices

Concept lattices (also called Galois lattices) have been applied in numerous areas, and several algorithms have been proposed to construct them. Generally, the input for lattice construction algorithms is a binary matrix with size |G||M | representing binary relation I ⊆ G×M . In this paper, we consider polynomial delay algorithms for building concept lattices. Although the concept lattice may be of exponential size, there exist polynomial delay algorithms for building them. The current best delay-time complexity is O(|G||M |). In this paper, we introduce the notion of irregular concepts, the combinatorial structure of which allows us to develop a linear delay lattice construction algorithm, that is, we give an algorithm with delay time of O(|G||M |). Our algorithm avoids the union operation for the attribute set and does not require checking if new concepts are already generated. In addition, we propose a compact representation for concept lattices and a corresponding construction algorithm. Although we are not guaranteed to achieve optimal compression, the compact representation can save significant storage space compared to the full representation normally used for concept lattices.