Efficient on-line algorithms for Euler diagram region computation

Euler diagrams are an accessible and effective visualisation of data involving simple settheoretic relationships. Sets are represented by closed curves in the plane and often have wellformedness conditions placed on them in order to enhance comprehensibility. The theoretical underpinning for tool support has usually focussed on the problem of generating an Euler diagram from an abstract model. However, the problem of efficient computation of the abstract model from the concrete diagram has not been addressed before, despite this computation being a necessity for computer interpretations of user drawn diagrams. This may be used, together with automated manipulations of the abstract model, for purposes such as semantic information presentation or diagrammatic theorem proving. Furthermore, in interactive settings, the user may update diagrams “on-line” by adding and removing curves, for example, in which case a system requirement is the update of the abstract model (without the necessity of recomputation of the entire abstract model). We define the notion of marked Euler diagrams, together with a method for associating marked points on the diagram with regions in the plane. Utilising these, we provide on-line algorithms which quickly compute the abstract model of a weakly reducible wellformed Euler diagram (constructible as a sequence of additions or removals of curves, keeping a wellformed diagram at each step), and quickly updates both the set of curves in the plane as well as the abstract model according to the on-line operations. Efficiency is demonstrated by comparison with a common, naive algorithm. Furthermore, the methodology enables a straightforward implementation which has subsequently been realised as an application for the user classification domain.