Marcus Krug

Preprocessing speed-up techniques is hard

During the last years, preprocessing-based techniques have been developed to compute shortest paths between two given points in a road network. These speed-up techniques make the computation a matter of microseconds even on huge networks. While there is a vast amount of experimental work in the field, there is still large demand on theoretical foundations. The preprocessing phases of most speed-up techniques leave open some degree of freedom which, in practice, is filled in a heuristical fashion. Thus, for a given speed-up technique, the problem arises of how to fill the according degree of freedom optimally. Until now, the complexity status of these problems was unknown. In this work, we answer this question by showing NP-hardness for the recent techniques.

Manhattan-Geodesic embedding of planar graphs

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is

Minimizing the area for planar straight-line grid drawings

\mathcal{NP}

Orthogeodesic point-set embedding of trees

-hard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is

Synthetic road networks

\mathcal{NP}

Visualizing large hierarchically clustered graphs with a landscape metaphor

-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.

Generalizing geometric graphs

Straight-line grid drawings of bounded size is a classical topic in graph drawing. The Graph Drawing Challenge 2006 dealt with minimizing the area of planar straight-line grid drawings. In this paper, we show that it is NP-complete to decide if a planar graph has a planar straight-line drawing on a grid of given size. Furthermore, we present a new iterative approach to compactify planar straight-line grid drawings. In an experimental study, we evaluate the quality of the compactified drawings with respect to the size of the area as well as to other measures.

Orthogonal graph drawing with flexibility constraints

Abstract Let S be a set of N grid points in the plane, no two of which lie on the same horizontal or vertical line, and let G be a graph with n vertices ( n ⩽ N ). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is a chain of horizontal and vertical segments with bends on grid points whose length is equal to the Manhattan distance of its end vertices. We study the following problem. Given a family F of trees, what is the minimum value f ( n ) such that every n -vertex tree in F admits an orthogeodesic point-set embedding on every set of grid points of size f ( n ) such that no two points lie on the same horizontal or vertical line? We provide polynomial upper bounds on f ( n ) for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L -shaped.

Augmenting

The availability of large graphs that represent huge road networks has led to a vast amount of experimental research that has been custom-tailored for road networks. There are two primary reasons to investigate graph-generators that construct synthetic graphs similar to real-world road-networks: The wish to theoretically explain noticeable experimental results on these networks and to overcome the commercial nature of most datasets that limits scientific use. This is the first work that experimentally evaluates the practical applicability of such generators. To this end we propose a new generator and review the only existing one (which until now has not been tested experimentally). Both generators are examined concerning structural properties and algorithmic behavior. Although both generators prove to be reasonably good models, our new generator outperforms the existing one with respect to both structural properties and algorithmic behavior.

k

Large graphs appear in many application domains. Their analysis can be done automatically by machines, for which the graph size is less of a problem, or, especially for exploration tasks, visually by humans. The graph drawing literature contains many efficient methods for visualizing large graphs, see e.g. [4, Chapter 12], but for large graphs it is often useful to first compute a sequence of coarser and more abstract representations by grouping vertices recursively using a hierarchical clustering algorithm. Then the task is to compute an overview picture of the graph based on a given cluster hierarchy, such that details of the graph, e.g., within clusters, remain visible on demand.