Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

A greedily routable region (GRR) is a closed subset of $\mathbb R^2$, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.Comment: full version of a paper appearing in ISAAC 201

Recognizing Weighted Disk Contact Graphs

Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be NP-hard. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights.Comment: 24 pages, 21 figures, extended version of a paper to appear at the International Symposium on Graph Drawing and Network Visualization (GD) 201

GraphMaps: Browsing Large Graphs as Interactive Maps

Algorithms for laying out large graphs have seen significant progress in the past decade. However, browsing large graphs remains a challenge. Rendering thousands of graphical elements at once often results in a cluttered image, and navigating these elements naively can cause disorientation. To address this challenge we propose a method called GraphMaps, mimicking the browsing experience of online geographic maps. GraphMaps creates a sequence of layers, where each layer refines the previous one. During graph browsing, GraphMaps chooses the layer corresponding to the zoom level, and renders only those entities of the layer that intersect the current viewport. The result is that, regardless of the graph size, the number of entities rendered at each view does not exceed a predefined threshold, yet all graph elements can be explored by the standard zoom and pan operations. GraphMaps preprocesses a graph in such a way that during browsing, the geometry of the entities is stable, and the viewer is responsive. Our case studies indicate that GraphMaps is useful in gaining an overview of a large graph, and also in exploring a graph on a finer level of detail.Comment: submitted to GD 201

Fast generation of dynamic complex networks with underlying hyperbolic geometry

Complex networks have become increasingly popular for modeling real-world phenomena, ranging from web hyperlinks to interactions between people. Realistic generative network models are important in this context as they avoid privacy concerns of real data and simplify complex network research regarding data sharing, reproducibility, and scalability studies. We study a geometric model creating unitdisk graphs in hyperbolic space. Previous work provided empirical and theoretical evidence that this model creates networks with a hierarchical structure and other realistic features. However, the investigated networks were small, possibly due to a quadratic running time of a straightforward implementation. We provide a faster generator for a representative subset of these networks. Our experiments indicate a time complexity of O((n+m) log n) for our implementation and thus confirm our theoretical considerations. To our knowledge our implementation is the first one with subquadratic running time. The acceleration stems primarily from the reduction of pairwise distance computations through a polar quadtree newly adapted to hyperbolic space. We also extend the generator to an alternative dynamic model which preserves graph properties in expectation. Finally, we generate and evaluate the largest networks of this model published so far. Our parallel implementation computes networks with billions of edges on a shared-memory server in a matter of few minutes. A comprehensive network analysis shows that important features of complex networks, such as a low diameter, power-law degree distribution and a high clustering coefficient, are retained over different graph sizes and densities