Upward planarization and layout

Drawing directed graphs has many applications and occurs whenever a natural flow of information is to be visualized. Given a directed acyclic graph (DAG) G, we are interested in an upward drawing of G, that is, a drawing of G in which all arcs are drawn as curves that are monotonically increasing in the vertical direction. Besides the upward property, it is desirable that the number of arc crossings arising in the drawing should be minimized. In this thesis, we propose a new approach for drawing DAGs based on the idea of upward planarization. We first introduce a novel upward planarization approach for upward crossing minimization of DAGs that utilizes new ideas for subgraph computation and arc reinsertion. In particular, it is the first upward crossing minimization algorithm which does not utilize any layering techniques known from the framework by Sugiyama et al. [STT81] or from the upward planarization algorithm by Eiglsperger et al. [EKE03]. Our approach addresses the main weakness of the classical two step upward crossing minimization approaches, where in the first step a layering of the input DAG is computed and then, in the second step, the number of crossings is minimized by solving the so-called k-level crossing minimization problem (k-LCM). However, choosing an inappropriate layering in the first step can negatively effect the subsequent k-level crossing minimization step, thus causing many unnecessary arc crossings. As shown by experimental evaluations, our new approach—referred to as layer-free upward crossing minimization (LFUP)—outperforms the state-ofthe-art crossing minimization heuristics based on layering even if an exact algorithm for k-level crossing minimization is used. Furthermore, LFUP also outperforms the existing approaches following the idea of upward planarization, that is, the approaches by Di Battista et al. [BPTT89] and Eiglsperger et al. [EKE03]. We also present two extensions for the new approach: an extension for upward planarization of directed hypergraphs and an extension for handling given port constraints, that is, drawing constraints that arise due to the prescribed positions where arcs can be connected to the drawing of the nodes. The upward planarization approach LFUP computes an upward planar representation (UPR) R of the input graph G, where crossings are modeled by dummy nodes. We introduce a new layout approach for realizing UPRs, that is, a drawing algorithm for constructing upward drawings where the arc crossings arising in the drawing are the ones modeled by the dummy nodes in R. Only few algorithms exist for realizing UPRs, most of these algorithms are based on simple ideas and originally developed for drawing planar st-graphs, hence our layout approach constitutes the first approach specialized for realizing UPRs. It offers two main advantages over the popular Sugiyama framework: It benefits from the advantage of the upward planarization approach LFUP, thus producing upward drawings with significantly less arc crossings. i Therefore, the drawing quality increases considerably. Furthermore, while the upward drawings produced by layer-based drawing approaches are often unstructured and appear unnaturally flat, the new layout approach constructs upward drawings that better reflect the structure of the digraphs and give a tidier impression to the viewer.