Reconstruction of planar graphs from their centered triangulations in some spatio-relational models

Abstract

In this paper we continue the investigation of an interoperability problem and its complexity in connection with spatio-relational database models. The result, however, is also interesting from the point of view of graph theory, since the main result is that there is a liaear-time algorithm for deciding wether a given triangulation can be seen as a centered triangulation of a certain planar graph. r INTRODUCTIONIn this paper our primer goal is to complete the solution of one of the interoperability problems investigated in [ I ] . This is, how we can translate the PLA spatio-relational model into a constraint spatial model and into the Worboys' model and vice versa. The first part of this task is completely solved in [ 11, where we gave an algorithm to translate the PLA model into a constraint model and Worboys' model. Here we deal with the reverse direction: from a given Worboys' database how can one reconstruct the PLA database.In the next section we review the necessary background with the models' description. The problem is specified in Section 111. To be more precise, an abstraction is given.Due to space restrictions it is not possible to publish the practical algorithms, rather we restrict ourselves to theoretical algorithms, that have to be refined with appropriate data structure. The new results are contained in Section IV, and finally in the last section we close with some concluding remarks and future research directions. I1 BACKGROUND AND MOTIVATIONIn this section first we introduce the spatio-relational models, followed by two types of them, the PLA model and a restricted version of Worboys' and a constraint model. Then we formalize the interoperability problem. A Spatio-Relutiona~ Data ModelsIn spatio-relational databases the underlying data model is the relationat one. The content of a relation is a subset of the Cartesian product of some possible domain sets, that is, Rc DlxDlx ... xD,. One element of a relation i s called a tuple (or ntuple, in case we want to emphasize the number of domain sets). In the original relational model these domains can be just simple, not structured data. The model itself consists of the name of the relation, and the list of so-called attributes, that are the descriptive properties of an objecttwa@ztaki. hu 2 Hungary each atribute can have its data only from the corresponding domain set: Relation-name(Attribute-1 ,Attribute-2, Attribute-n) Additionally, the a-priori fixed constraints, like idetitifiers have to be represented in the relational model. If this relational model is adequate to represent physical objects, then usually it contains geometrical extent. In this case we name it as a spatial model. It also may contain time extension, then it is a temporal model. The . combination of these two is the spatiotemporal relational model. Depending on the technique of the geometricdtime extent representation, it may be an extended model, like the constraint model. This one may contain mathematical inequalities, and by the help of these, an infinite number of tuples can b...