[Show abstract][Hide abstract] ABSTRACT: We investigate straight-line drawings of topological graphs that consist of a
planar graph plus one edge, also called almost-planar graphs. We present a
characterization of such graphs that admit a straight-line drawing. The
characterization enables a linear-time testing algorithm to determine whether
an almost-planar graph admits a straight-line drawing, and a linear-time
drawing algorithm that constructs such a drawing, if it exists. We also show
that some almost-planar graphs require exponential area for a straight-line
drawing.
[Show abstract][Hide abstract] ABSTRACT: Objective: Aesthetics are important in algorithm design and graph evaluation. This paper presents two user studies that were conducted to investigate the impact of crossing angles on human graph comprehension.
Method and Results: These two studies together demonstrate our newly proposed two-step approach for testing graph aesthetics. The first study is a controlled experiment with purposely-generated graphs. Twenty-two subjects participated in the study and were asked to determine the length of a path which was crossed by a set of parallel edges at different angles. The result of an analysis of variance showed that larger crossing angles induced better task performance. The second study was a non-controlled experiment with general real world graphs. Thirty-seven subjects participated in the study and were asked to find the shortest path of two pre-selected nodes in a set of graph drawings. The results of simple regression tests confirmed the negative effect of small crossing angles. This study also showed that among our four proposed candidates, the minimum crossing angle on the path was the best measure for the aesthetic when path finding is important.
Conclusion: Larger crossing angles make graphs easier to read.
Implications: In situations where crossings cannot be completely removed (for example, graphs are non-planar, or a drawing convention is applied), or where effort needed to remove all crossings cannot be justified, the crossing angle should be maximized to reduce the negative impact of crossings to the minimum.
Journal of Visual Languages & Computing 08/2014; 25(4). DOI:10.1016/j.jvlc.2014.03.001 · 0.89 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A graph is 1-planar if it can be embedded in the plane with at most one crossing per edge. It is known that the problem of testing 1-planarity of a graph is NP-complete. In this paper, we study outer-1-planar graphs. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. We present a linear time algorithm to test whether a given graph is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding in linear time if it exists.
[Show abstract][Hide abstract] ABSTRACT: We introduce a new string matching problem called order-preserving matching on numeric strings, where a pattern matches a text if the text contains a substring of values whose relative orders coincide with those of the pattern. Order-preserving matching is applicable to many scenarios such as stock price analysis and musical melody matching in which the order relations should be matched instead of the strings themselves. Solving order-preserving matching is closely related to the representation of order relations of a numeric string. We define the prefix representation and the nearest neighbor representation of the pattern, both of which lead to efficient algorithms for order-preserving matching. We present efficient algorithms for single and multiple pattern cases. For the single pattern case, we give an O(nlogm) time algorithm and optimize it further to obtain O(n+mlogm) time. For the multiple pattern case, we give an O(nlogm) time algorithm.
[Show abstract][Hide abstract] ABSTRACT: A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete.
In this paper, we consider maximal 1-planar graphs. A graph G is maximal 1-planar if addition of any edge destroys 1-planarity of G. We first study combinatorial properties of maximal 1-planar embeddings. In particular, we show that in a maximal 1-planar embedding, the graph induced by the non-crossing edges is spanning and biconnected.
Using the properties, we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system Φ (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding ξ of G that is consistent with the given rotation system Φ. Our algorithm also produces such an embedding in linear time, if it exists.
[Show abstract][Hide abstract] ABSTRACT: Many automatic graph drawing algorithms implement only one or two aesthetic criteria since most aesthetics conflict with each other. Empirical research has shown that although those algorithms are based on different aesthetics, drawings produced by them have comparable effectiveness.The comparable effectiveness raises a question about the necessity of choosing one algorithm against another for drawing graphs when human performance is a main concern. In this paper, we argue that effectiveness can be improved when algorithms are designed by making compromises between aesthetics, rather than trying to satisfy one or two of them to the fullest. We therefore introduce a new algorithm: BIGANGLE. This algorithm produces drawings with multiple aesthetics being improved at the same time, compared to a classical spring algorithm. A user study comparing these two algorithms indicates that BIGANGLE induces a significantly better task performance and a lower cognitive load, therefore resulting in better graph drawings in terms of human cognitive efficiency.Our study indicates that aesthetics should not be considered separately. Improving multiple aesthetics at the same time, even to small extents, will have a better chance to make resultant drawings more effective. Although this finding is based on a study of algorithms, it also applies in general graph visualization and evaluation.
Journal of Visual Languages & Computing 08/2013; 24(4):262–272. DOI:10.1016/j.jvlc.2011.12.002 · 0.89 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: There is strong empirical evidence that human perception of a graph drawing is negatively correlated with the number of edge crossings. However, recent experiments show that one can reduce the negative effect by ensuring that the edges that cross do so at large angles. These experiments have motivated a number of mathematical and algorithmic studies of “right angle crossing (RAC)” drawings of graphs, where the edges cross each other perpendicularly. In this paper we give an algorithm for constructing RAC drawings of “outer-1-plane” graphs, that is, topological graphs in which each vertex appears on the outer face, and each edge crosses at most one other edge. The drawing algorithm preserves the embedding of the input graph. This is one of the few algorithms available to construct RAC drawings.
International Journal of Computational Geometry & Applications 04/2013; 22(06). DOI:10.1142/S021819591250015X · 0.08 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We introduce a new string matching problem called order-preserving matching
on numeric strings where a pattern matches a text if the text contains a
substring whose relative orders coincide with those of the pattern.
Order-preserving matching is applicable to many scenarios such as stock price
analysis and musical melody matching in which the order relations should be
matched instead of the strings themselves. Solving order-preserving matching
has to do with representations of order relations of a numeric string. We
define prefix representation and nearest neighbor representation, which lead to
efficient algorithms for order-preserving matching. We present efficient
algorithms for single and multiple pattern cases. For the single pattern case,
we give an O(n log m) time algorithm and optimize it further to obtain O(n + m
log m) time. For the multiple pattern case, we give an O(n log m) time
algorithm.
[Show abstract][Hide abstract] ABSTRACT: This paper is motivated by empirical research that has shown that increasing the angle of edge crossings reduces the negative effect of crossings on human readability. We investigate circular graph drawings (where each vertex lies on a circle) with large crossing angles. In particular, we consider the case of right angle crossing (RAC) drawings, where each crossing angle is π/2.
We characterize circular RAC graphs that admit a circular RAC drawing, and present a linear-time algorithm for constructing such a drawing, if it exists. We also describe a quadratic programming approach to construct circular drawings that maximise crossing angles. This method significantly increases crossing angles compared to the traditional equal-spacing algorithm.
[Show abstract][Hide abstract] ABSTRACT: Readability criteria have been commonly used to measure the quality of graph visualizations. In this paper we argue that readability criteria, while necessary, are not sufficient. We propose a new kind of criterion, generically termed faithfulness, for evaluating graph layout methods. We propose a general model for quantifying faithfulness, and contrast it with the well established readability criteria. We use examples of multidimensional scaling, edge bundling and several other visualization metaphors (including matrix-based and map-based visualizations) to illustrate faithfulness.
[Show abstract][Hide abstract] ABSTRACT: A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete. A 1-planar embedding of a graph G is maximal if no edge can be added without violating the 1-planarity of G. In this paper we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding of G that preserves the given rotation system, and our algorithm produces such an embedding in linear time, if it exists.
[Show abstract][Hide abstract] ABSTRACT: Graph streams have been studied extensively, such as for data mining, while fairly limitedly for visualizations. Recently, edge bundling promises to reduce visual clutter in large graph visualizations, though mainly focusing on static graphs. This paper presents a new framework, namely StreamEB, for edge bundling of graph streams, which integrates temporal, neighbourhood, data-driven and spatial compatibility for edges. Amongst these metrics, temporal and neighbourhood compatibility are introduced for the first time. We then present force-directed and tree-based methods for stream edge bundling. The effectiveness of our framework is then demonstrated using US flights data and Thompson-Reuters stock data.
Proceedings of the 20th international conference on Graph Drawing; 09/2012
[Show abstract][Hide abstract] ABSTRACT: Fáry's theorem states that every plane graph can be drawn as a straight-line drawing. A plane graph is a graph embedded in a plane without edge cross-ings. In this paper, we extend Fáry's theorem to non-planar graphs. More specif-ically, we study the problem of drawing 1-plane graphs with straight-line edges. A 1-plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1-plane graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algo-rithm and a drawing algorithm. We also show that there are 1-plane graphs for which every straight-line drawing has exponential area. To our best knowledge, this is the first result to extend Fáry's theorem to non-planar graphs.
[Show abstract][Hide abstract] ABSTRACT: The crossing resolution of a geometric graph is the minimum crossing angle at which any two edges cross each other. In this paper, we present upper and lower bounds to the crossing resolution of the complete geometric graphs.
[Show abstract][Hide abstract] ABSTRACT: W. T. Tutte published a paper in 1963 entitled “How to Draw a Graph”. Tutte’s motivation was mathematical, and his paper can be seen as a contribution to the long tradition of geometric representations of combinatorial objects. Over the following 40 odd years, the motivation for creating visual representations of graphs has changed from mathematical curiosity to Visual Analytics. Current demand for Graph Drawing methods is now high, because of the potential for more human-comprehensible visual forms in industries as diverse as Biotechnology, Homeland Security, and Sensor Networks. Many new methods have been proposed, tested, implemented, and found their way into commercial tools. This paper describes two strands of this history: the force directed approach, and the planarity approach. Both approaches originate in Tutte’s paper.
Expanding the Frontiers of Visual Analytics and Visualization, 01/2012: pages 111-126; , ISBN: 978-1-4471-2803-8
[Show abstract][Hide abstract] ABSTRACT: Dynamic Social network visualization transforms dynamic information in a social network into geometric representations. Most previous work in this field mainly focuses on the evolution of the overall network.
Visual Languages and Human-Centric Computing (VL/HCC), 2012 IEEE Symposium on; 01/2012
[Show abstract][Hide abstract] ABSTRACT: Kozo Sugiyama was born in Gifu Prefecture Japan on September 17, 1945. He received his B.S., M.S., and Dr. Sci. at Nagoya University in 1969, 1971, 1974 respectively. For 23 years from 1974 he was a researcher at Fujitsu. During this time he spent a year at the International Institute for Applied Systems Analysis in Laxenburg in Austria. In the mid 1990s he served as the Director of the Information Processing Society of Japan. In 1997 he moved from Fujitsu to the newly-created Japan Advanced Institute of Science and Technology. His first position there was Professor of the School of Knowledge Science, but he soon became Director of the Center for Knowledge Science, and then Dean of the School of Knowledge Science. His last few years at JAIST were spent as a Vice President of the University.
Graph Drawing - 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21-23, 2011, Revised Selected Papers; 09/2011
[Show abstract][Hide abstract] ABSTRACT: Edge bundling methods became popular for visualising large dense networks; however, most of previous work mainly relies on geometry to define compatibility between the edges. In this paper, we present a new framework for edge bundling, which tightly integrates topology, geometry and importance. In particular, we introduce new edge compatibility measures, namely importance compatibility and topology compatibility. More specifically, we present four variations of force directed edge bundling method based on the framework: Centrality-based bundling, Radial bundling, Topology-based bundling, and Orthogonal bundling. Our experimental results with social networks, biological networks, geographic networks and clustered graphs indicate that our new framework can be very useful to highlight the most important topological skeletal structures of the input networks.
Graph Drawing - 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21-23, 2011, Revised Selected Papers; 01/2011
[Show abstract][Hide abstract] ABSTRACT: A Right Angle Crossing Graph (also called a RAC graph for short) is a graph that has a straight-line drawing where any two crossing edges are orthogonal to each other. A 11-planar graph is a graph that has a drawing where every edge is crossed at most once. This paper studies the combinatorial relationship between the family of RAC graphs and the family of 11-planar graphs. It is proved that: (1) all RAC graphs having maximal edge density belong to the intersection of the two families; and (2) there is no inclusion relationship between the two families. As a by-product of the proof technique, it is also shown that every RAC graph with maximal edge density is the union of two maximal planar graphs.
Graph Drawing - 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21-23, 2011, Revised Selected Papers; 01/2011
[Show abstract][Hide abstract] ABSTRACT: The classical Fáry's theorem from the 1930s states that every planar graph can be drawn as a straight-line drawing. In this paper, we extend Fáry's the-orem to non-planar graphs. More specifically, we study the problem of drawing 1-planar graphs with straight-line edges. A 1-planar graph is a sparse non-planar graph with at most one crossing per edge. We give a characterisation of those 1-planar graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. We also show that there are 1-planar graphs for which every straight-line drawing has exponen-tial area. To our best knowledge, this is the first result to extend Fáry's theorem to non-planar graphs.