Algorithm Design Using Traversals of the Covering Relation

Abstract

Where posets are used to represent taxonomies, concept lattices, or information ordered databases there is a need to engineer algorithms that search, update, and transform posets. This paper demonstrates an approach to designing such algorithms. It presents a picture of covering relation traversals that characterises these in terms of up-set and down-set expressions involving union, intersection, and difference. It then provides a detailed analysis of three types of covering relation traversal. The approach is demonstrated by describing a suite of derived algorithms. The intention is to express a manner of decomposing mathematical problems into poset traversals, and to provide context to the selection a particular traversal algorithm. This line of work has previously been pursued by [1]. However, the success and influence of Formal Concept Analysis [2] has shifted the emphasis from posets to lattices, and from algorithms that operate on the graph of the partial order to the formal context. This paper contributes a methodology for the renewed investigation of poset algorithms, with the potential to lead to improvements in algorithms such as the online completion to a lattice.