Carsten Gutwenger

A Linear Time Implementation of SPQR-Trees

The data structure SPQR-tree represents the decomposition of a biconnected graph with respect to its triconnected components. SPQR-trees have been introduced by Di Battista and Tamassia [8] and, since then, became quite important in the field of graph algorithms. Theoretical papers using SPQR-trees claim that they can be implemented in linear time using a modification of the algorithm by Hopcroft and Tarjan [15] for decomposing a graph into its triconnected components. So far no correct linear time implementation of either triconnectivity decomposition or SPQR-trees is known to us. Here, we show the incorrectness of the Hopcroft and Tarjan algorithm [15], and correct the faulty parts. We describe the relationship between SPQR-trees and triconnected components and apply the resulting algorithm to the computation of SPQR-trees. Our implementation is publically available in AGD [1].

Inserting an edge into a planar graph

Abstract Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e where all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a combinatorial embedding of a planar graph G where the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, that is able to find a solution with the minimum number of crossings.

A new approach for visualizing UML class diagrams

UML diagrams have become increasingly important in the engineering and reengineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combination of hierarchical and non-hierarchical relations poses a special challenge to a graph layout tool. Existing layout tools treat hierarchical and non-hierarchical relations either alike or as separate tasks in a two-phase process as in, e.g., [Seemann 1997]. We suggest a new approach for visualizing UML class diagrams leading to a balanced mixture of the following aesthetic criteria: Crossing minimization, bend minimization, uniform direction within each class hierarchy, no nesting of one class hierarchy within another, orthogonal layout, merging of multiple inheritance edges, and good edge labelling. We have realized our approach within the graph drawing library GoVisual. Experiments show the superiority to state-of-the-art and industrial standard layouts.

Branch-and-Cut Algorithms for Combinatorial Optimization and Their Implementation in ABACUS

Branch-and-cut (-and-price) algorithms belong to the most successful techniques for solving mixed integer linear programs and combinatorial optimization problems to optimality (or, at least, with certified quality). In this unit, we concentrate on sequential branch-and-cut for hard combinatorial optimization problems, while branch-and-cut for general mixed integer linear programming is treated in [? Martin] and parallel branch-and-cut is treated in [? Ladanyi/Ralphs/Trotter]. After telling our most recent story ofa successful application of branch-and-cut in Section 1, we give in Section 2 a brief review ofthe history, including the contributions of pioneers with an emphasis on the computational aspects of their work. In Section 3, the components ofa generic branch-and-cut algorithm are described and illustrated on the traveling salesman problem. In Section 4, we first elaborate a bit on the important separation problem where we use the traveling salesman problem and the maximum cut problem as examples, then we show how branch-and-cut can be applied to problems with a very large number of variables (branch-and-cut-and-price). Section 5 is devoted to the design and applications of the ABACUS software framework for the implementation of branch-and-cut algorithms. Finally, in Section 6, we make a few remarks on the solution of the exercise consisting of the design of a simple TSP-solver in ABACUS.

An Experimental Study of Crossing Minimization Heuristics

We present an extensive experimental study of heuristics for crossing minimization. The heuristics are based on the planarization approach, so far the most successful framework for crossing minimization. We study the effects of various methods for computing a maximal planar subgraph and for edge re-insertion including post-processing and randomization.

Layer-free upward crossing minimization

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.

Planar Polyline Drawings with Good Angular Resolution

We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree n on a (2n - 5) × (3/2n - 7/2) grid with at most 5n - 15 bends and minimum angle > 2/d. In the constructed drawings, every edge has at most three bends and length O(n). To our best knowledge, this algorithm achieves the best simultaneous bounds concerning the grid size, angular resolution, and number of bends for planar grid drawings of high-degree planar graphs. Besides the nice theoretical features, the practical drawings are aesthetically very pleasing. An implementation of our algorithm is available with the AGD-Library (Algorithms for Graph Drawing) [2, 1]. Our algorithm is based on ideas by Kant for polyline grid drawings for triconnected plane graphs [23]. In particular, our algorithm significantly improves upon his bounds on the angular resolution and the grid size for non-triconnected plane graphs. In this case, Kant could show an angular resolution of 4/3d+7 and a grid size of (2n - 5) × (3n - 6), only.

Automatic layout of UML class diagrams in orthogonal style

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.

Graph Drawing Algorithm Engineering with AGD

We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basically the same functionality. This enables us to exchange components and experiment with various algorithms and implementations of the same type. We give examples for algorithm engineering with AGD for drawing general non-hierarchical graphs and hierarchical graphs.

An experimental evaluation of multilevel layout methods

Applying the multilevel paradigm to energy-based layout algorithms can improve both the quality of the resulting drawings as well as the running time of the layout computation. In order to do this, approaches for the different multilevel phases refinement, placement, layout, and optionally scaling and postprocessing need to be implemented. A number of multilevel layout algorithms have been proposed already, which differ in the way these phases are realized. We present an experimental study that investigates the influence of varying combinations with respect to running time and quality criteria.