RECTANGULAR DUALIZATION OF BICONNECTED PLANAR GRAPHS IN LINEAR TIME AND RELATED APPLICATIONS

Abstract

Rectangular dualization is the computation of the rectangular dual of a planar graph. It was originally introduced to find rectangular topologies for floorplanning of integrated circuits: by a floorplan, a rectangular chip area is partitioned into rectilinear polygons corresponding to the relative location of functional entities of the circuit. Subsequently it found application in many other fields, in particular becoming effective in visualization problems. In this paper we briefly describe two possible applications: visualization of network topologies and design of 3D Virtual Worlds. It is known that the engineering and the optimization of large communication networks is a very challenging problem, as often real networks are very huge, including hundreds and even thousands of nodes and links. In order to help human operators in maintaining and updating the description and documentation of the network structure, the network is described in form of a hierarchy of subnetworks. The rectangular dualization is a very useful technique in graph drawing, especially when applied to the hierarchical drawing of a structured graph. The design of 3D Virtual Worlds is a completely different field. Virtual Worlds visualized in 3D are environments where people meet. Such environments provide a consistent and immersive user interface that facilitates awareness of other participants. In other words, Virtual Worlds support to a certain extent the way humans operate and interact in the real world. In order to guarantee a good user experience, it is important to overcome navigation problems, that are strongly related to the human approach with the service interface, because if not well solved they may break the user immersive experience. The generation of the Virtual World is determined by the graph representing the map of the institution. Rectangular dualization of such a map helps to ease and quicken the navigation. The use of rectangular dualization is strongly limited by the fact that not all planar graphs admit a rectangular dual. However it is possible to apply a minimal set of transformations to the original graph to obtain a graph that admits a rectangular dual representation. In the following sections we first describe a linear-time algorithm which computes the rectangular dual of a planar graph, transforming graphs not satisfying the rectangular dual conditions; then we give an overview of the two applications mentioned above.