Crossings and Planarization

Unified modelling language (UML) diagrams have become increasingly important in engineering and re-engineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display generalizations, associations, aggregations, and compositions in one picture. The combination of directed and undirected relations poses a special challenge to a graph layout tool. Current approaches for the automatic layout of class diagrams are based on the layered graph drawing paradigm. These algorithms produce good results for class diagrams with large and deep structural information, that is, diagrams with a large and deep inheritance hierarchy. However, they do not perform satisfactorily in absence of this information. We suggest to use the topology-shape--metrics paradigm for automatic layout of class diagrams, which has been used very successfully for drawing undirected graphs in orthogonal style. Moreover, we introduce the algorithms UML-Kandinsky and GoVisual fitting into this paradigm. Both algorithms work for class diagrams with rich structural information as well as for class diagrams with few or no structural information. Therefore, they improve the existing algorithms significantly.

Automatic layout of UML class diagrams in orthogonal style

An upward drawing of a DAG G is a drawing of G in which all arcs are drawn as curves increasing monotonically in the vertical direction. In this article, we present a new approach for upward crossing minimization, that is, finding an upward drawing of a DAG G with as few crossings as possible. Our algorithm is based on a two-stage upward planarization approach, which computes a feasible upward planar subgraph in the first step and reinserts the remaining arcs by computing constraint-feasible upward insertion paths. An experimental study shows that the new algorithm leads to much better results than existing algorithms for upward crossing minimization, including the classical Sugiyama approach.

Layer-free upward crossing minimization

An upward drawing of a DAG G is a drawing of G in which all edges are drawn as curves increasing monotonically in the vertical direction. In this paper, we present a new approach for upward crossing minimization, i.e., finding an upward drawing of a DAG G with as few crossings as possible. Our algorithm is based on a two-stage upward planarization approach, which computes a feasible upward planar subgraph in the first step, and re-inserts the remaining edges by computing constraint-feasible upward insertion paths. An experimental study shows that the new algorithm leads to much better results than existing algorithms for upward crossing minimization, including the classical Sugiyama approach.

A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing y-coordinates. The problem to decide whether a graph is upward planar or not is NP-complete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branch-and-bound algorithm over the graph's triconnectivity structure which was able to solve sparse graphs.

In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively.

/pdf/upward-planarity-testing-via-sat-8d3561nau0.pdf

Upward planarity testing via SAT

An Approach for Mixed Upward Planarization

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An Approach for Mixed Upward Planarization