Edge Routing with Ordered Bundles

• S. Bereg, A. E. Holroyd, L. Nachmanson, S. Pupyrev, Proc. 19th Internat. Sympos. on Graph Drawing 2011

• We propose a new approach to edge bundling. At the first stage we route the edge paths so as to minimize a weighted sum of the total length of the paths together with their ink. As this problem is NP-hard, we provide an efficient heuristic that finds an approximate solution. The second stage then separates edges belonging to the same bundle. To achieve this, we provide a new and efficient algorithm that solves a variant of the metro-line crossing minimization problem. The method creates aesthetically pleasing edge routes that give an overview of the global graph structure, while still drawing each edge separately, without intersecting graph nodes, and with few crossings.

Generalizing Geometric Graphs

• E. Brunel, A. Gemsa, M. Krug, I. Rutter, D. Wagner, J. Graph Algorithms Appl. 2014

• Network visualization is essential for understanding the data obtained from huge real-world networks such as flight-networks, the AS-network or social networks. Although we can compute layouts for these networks reasonably fast, even the most recent display media are not capable of displaying these layouts in an adequate way. Moreover, the human viewer may be overwhelmed by the displayed level of detail. The increasing amount of data therefore requires techniques aiming at a sensible reduction of the visual complexity of huge layouts.We consider the problem of computing a generalization of a given layout reducing the complexity of the drawing to an amount that can be displayed without clutter and handled by a human viewer. We take a first step at formulating graph generalization within a mathematical model and we consider the resulting problems from an algorithmic point of view. Although these problems are NP-hard in general, we provide efficient approximation algorithms as well as efficient and effective heuristics. At the end of the paper we showcase some sample generalizations.

Schematization of networks

• S. Cabello, M. de Berg, M. van Kreveld, Computational Geometry: Theory and Applications 2005

• We study the problem of computing schematized versions of network maps, like railroad or highway maps. Every path of the schematized map has two or three links with restricted orientations, and the schematized map must be topologically equivalent to the input map. Our approach can handle several types of schematizations, and certain additional constraints can be added, such as a minimum vertical distance between two paths. Our algorithm runs in O(n log n) time, and experimental results showing the quality of the output are given.

Path Schematization forRoute Sketches

• D. Delling, A. Gemsa, M. Noellenburg, T. Pajor, Proc. Scandinavian Workshop on Algorithm Theory

• Motivated from drawing route sketches, we consider the following path schematization problem. We are given a simple embedded polygonal path P = (v1, ..., vn ) and a set C of admissible edge orientations including the coordinate axes. The problem is to redraw P schematically such that all edges are drawn as line segments that are parallel to one of the specified orientations. We also require that the path preserves the orthogonal order and that it remains intersection-free. Finally, we want the drawing to maximize the number of edges having their preferred edge direction and to minimize the path length.In this paper we first present an efficient two-step approach for schematizing monotone paths. It consists of an O(n2)-time algorithm to assign edge directions optimally and a subsequent linear program to minimize the path length. In order to schematize non-monotone paths we propose a heuristic that first splits the input into k monotone subpaths and then combines the optimal embeddings of the monotone subpaths into a single, intersection-free embedding of the initial path in O(k2 + n) time.

On D-regular Schematization of Embedded Paths

• D. Delling, A. Gemsa, M. Moellenburg, T. Pajor, I. Rutter, Comput. Geom. Theory Appl.

• In the d-regular path schematization problem we are given an embedded path P (e.g., a route in a road network) and an integer d. The goal is to find a d-schematized embedding of P in which the orthogonal order of all vertices in the input is preserved and in which every edge has a slope that is an integer multiple of 90°/d. We show that deciding whether a path can be d-schematized is NP-hard for any integer d. We further model the problem as a mixed-integer linear program. An experimental evaluation indicates that this approach generates reasonable route sketches for real-world data.

Force-directed edge bundling for graph visualization

• D. H. R.  Holten, J. J. van Wijk, Computer Graphics Forum 2009• Graphs depicted as node-link diagrams are widely used to show

relationships between entities. However, node-link diagrams comprised of a large number of nodes and edges often suffer from visual clutter. The use of edge bundling remedies this and reveals high-level edge patterns. Previous methods require the graph to contain a hierarchy for this, or they construct a control mesh to guide the edge bundling process, which often results in bundles that show considerable variation in curvature along the overall bundle direction. We present a new edge bundling method that uses a self-organizing approach to bundling in which edges are modeled as flexible springs that can attract each other. In contrast to previous methods, no hierarchy is used and no control mesh. The resulting bundled graphs show significant clutter reduction and clearly visible high-level edge patterns. Curvature variation is furthermore minimized, resulting in smooth bundles that are easy to follow. Finally, we present a rendering technique that can be used to emphasize the bundling.

A new algorithm for continuous area cartogram construction with

triangulation of regions and restriction on bearing changes of edges

• R. Inoue, E, Shimiz, Cartography and Geographic Information Science 2006

• A continuous area cartogram is a transformed map in which regions are resized relative to their data. It is considered an effective visualization tool for statistical data, and many solutions have been proposed. However, most of these solutions are not mathematically clear or user friendly; further, they do not provide visually elegant area cartograms. An essential condition for the construction of a visually elegant area cartogram is that the resultant region shape should resemble the corresponding regions on geographical maps. Since it is impossible to determine the shape of a region based only on the information of size, area cartogram construction is an ill-posed problem that requires regularization. In this study, we propose a construction algorithm that involves triangulation of regions and regularization through restrictions on the bearing changes of the edges in order to obtain visually clear results. First, we formulate a construction using nonlinear least squares. Then, by linearizing, we derive a simple formula to create area cartograms. The application of our algorithm to the USA population datasets reveals that our algorithm has mathematical clarity and is user friendly.

Automatic generation ofdestination maps

• J. Kopf, M. Agrawala, D.Bargeron, D. Salesin, M. Cohen, ACM Trans. on Graphics 2010

• Destination maps are navigational aids designed to show anyone within a region how to reach a location (the destination). Hand-designed destination maps include only the most important roads in the region and are non-uniformly scaled to ensure that all of the important roads from the highways to the residential streets are visible. We present the first automated system for creating such destination maps based on the design principles used by mapmakers. Our system includes novel algorithms for selecting the important roads based on mental representations of road networks, and for laying out the roads based on a non-linear optimization procedure. The final layouts are labeled and rendered in a variety of styles ranging from informal to more formal map styles. The system has been used to generate over 57,000 destination maps by thousands of users. We report feedback from both a formal and informal user study, as well as provide quantitative measures of success.

Increasing the readability of graph drawings

with centrality-based scaling• D. Merrick, J. Gudmundsson, Proc. Asia-Pacific Symposium on

Information Visualisation 2006• A common problem in visualising some networks is the presence of

localised high density areas in an otherwise sparse graph. Applying common graph drawing algorithms on such networks can result in drawings that are not highly readable in the dense areas. Additionally, networks whose layouts are defined geographically often have dense areas that are located within small geographical regions relative to the size of the entire network. In cases where relationships within these dense areas are of interest, it is desirable to be able to distort the graph layout such that the denser areas are enlarged from their original sizes.In this paper, we propose a technique for enlarging dense areas of a given graph layout, and shrinking sparse areas. This technique is applied to geographical layouts of railway networks and force-directed layouts of non-geographical networks. The results show an increase in readability of dense parts of the networks. In addition, they provide improved starting layouts for schematisation methods which may be used to further increase readability.

Optimization approaches for generalization and data abstraction

• M. Sester, Int. Journal of GIS 2005• The availability of methods for abstracting and generalizing spatial

data is vital for understanding and communicating spatial information. Spatial analysis using maps at different scales is a good example of this. Such methods are needed not only for analogue spatial data sets but even more so for digital data. In order to automate the process of generating different levels of detail of a spatial data set, generalization operations are used. The paper first gives an overview on current approaches for the automation of generalization and data abstraction, and then presents solutions for three generalization problems based on optimization techniques. Least‐Squares Adjustment is used for displacement and shape simplification (here, building groundplans), and Self‐Organizing Maps, a Neural Network technique, is applied for typification, i.e. a density preserving reduction of objects. The methods are validated with several examples and evaluated according to their advantages and disadvantages. Finally, a scenario describes how these methods can be combined to automatically yield a satisfying result for integrating two data sets of different scales.

A Probabilistic Model for Road Selection in Mobile Maps

• T. C. van Dijk, J.-H. Haunert, Web and Wireless Geographical Information Systems

• Mobile devices provide an interesting context for map drawing. This paper presents a novel road-selection algorithm based on PageRank, the algorithm famously used by Google to rank web pages by importance. Underlying the PageRank calculation is a probabilistic model of user behavior. We provide suitable generalizations of this model to road networks. Our implementation of the proposed algorithm handles a sizable map in approximately a tenth of a second on a desktop PC. Therefore, our methods should be feasible on modern mobile devices.

Focus+context metro maps

• Y.-S. Wang, M.T. Chi, IEEE Trans. on Visualization and Computer Graphics 2011

• We introduce a focus+context method to visualize a complicated metro map of a modern city on a small displaying area. The context of our work is with regard the popularity of mobile devices. The best route to the destination, which can be obtained from the arrival time of trains, is highlighted. The stations on the route enjoy larger spaces, whereas the other stations are rendered smaller and closer to fit the whole map into a screen. To simplify the navigation and route planning for visitors, we formulate various map characteristics such as octilinear transportation lines and regular station distances into energy terms. We then solve for the optimal layout in a least squares sense. In addition, we label the names of stations that are on the route of a passenger according to human preferences, occlusions, and consistencies of label positions using the graph cuts method. Our system achieves real-time performance by being able to report instant information because of the carefully designed energy terms. We apply our method to layout a number of metro maps and show the results and timing statistics to demonstrate the feasibility of our technique.