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23
Hierarchical edge bundles: Visualization of adjacency relations in hierarchical data
 IEEE Transactions on Visualization and Computer Graphics
, 2006
"... Abstract—A compound graph is a frequently encountered type of data set. Relations are given between items, and a hierarchy is defined on the items as well. We present a new method for visualizing such compound graphs. Our approach is based on visually bundling the adjacency edges, i.e., nonhierarch ..."
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Cited by 144 (9 self)
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Abstract—A compound graph is a frequently encountered type of data set. Relations are given between items, and a hierarchy is defined on the items as well. We present a new method for visualizing such compound graphs. Our approach is based on visually bundling the adjacency edges, i.e., nonhierarchical edges, together. We realize this as follows. We assume that the hierarchy is shown via a standard tree visualization method. Next, we bend each adjacency edge, modeled as a Bspline curve, toward the polyline defined by the path via the inclusion edges from one node to another. This hierarchical bundling reduces visual clutter and also visualizes implicit adjacency edges between parent nodes that are the result of explicit adjacency edges between their respective child nodes. Furthermore, hierarchical edge bundling is a generic method which can be used in conjunction with existing tree visualization techniques. We illustrate our technique by providing example visualizations and discuss the results based on an informal evaluation provided by potential users of such visualizations.
ForceDirected Edge Bundling for Graph Visualization
, 2009
"... Graphs depicted as nodelink diagrams are widely used to show relationships between entities. However, nodelink diagrams comprised of a large number of nodes and edges often suffer from visual clutter. The use of edge bundling remedies this and reveals highlevel edge patterns. Previous methods requ ..."
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Cited by 37 (0 self)
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Graphs depicted as nodelink diagrams are widely used to show relationships between entities. However, nodelink diagrams comprised of a large number of nodes and edges often suffer from visual clutter. The use of edge bundling remedies this and reveals highlevel 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 selforganizing 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 highlevel 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.
Quasiconvex Analysis of Backtracking Algorithms
, 2003
"... We consider a class of multivariate recurrences frequently arising in the worst case analysis of DavisPutnamstyle exponential time backtracking algorithms for NPhard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to ..."
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Cited by 26 (2 self)
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We consider a class of multivariate recurrences frequently arising in the worst case analysis of DavisPutnamstyle exponential time backtracking algorithms for NPhard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to reduce the problem to that of solving univariate linear recurrences; show how to use quasiconvex programming to determine the weight function yielding the smallest upper bound; and prove that the resulting upper bounds are within a polynomial factor of the true asymptotics of the recurrence. We develop and implement a multiplegradient descent algorithm for the resulting quasiconvex programs, using a realnumber arithmetic package for guaranteed accuracy of the computed worst case time bounds.
GeometryBased edge clustering for graph visualization
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 2008
"... Graphs have been widely used to model relationships among data. For large graphs, excessive edge crossings make the display visually cluttered and thus difficult to explore. In this paper, we propose a novel geometrybased edgeclustering framework that can group edges into bundles to reduce the ove ..."
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Cited by 25 (2 self)
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Graphs have been widely used to model relationships among data. For large graphs, excessive edge crossings make the display visually cluttered and thus difficult to explore. In this paper, we propose a novel geometrybased edgeclustering framework that can group edges into bundles to reduce the overall edge crossings. Our method uses a control mesh to guide the edgeclustering process; edge bundles can be formed by forcing all edges to pass through some control points on the mesh. The control mesh can be generated at different levels of detail either manually or automatically based on underlying graph patterns. Users can further interact with the edgeclustering results through several advanced visualization techniques such as color and opacity enhancement. Compared with other edgeclustering methods, our approach is intuitive, flexible, and efficient. The experiments on some large graphs demonstrate the effectiveness of our method.
Improved Circular Layouts
 GRAPH DRAWING
, 2006
"... Circular graph layout is a drawing scheme where all nodes are placed on the perimeter of a circle. An inherent issue with circular layouts is that the rigid restriction on node placement often gives rise to long edges and an overall dense drawing. We suggest here three independent, complementary tec ..."
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Cited by 15 (0 self)
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Circular graph layout is a drawing scheme where all nodes are placed on the perimeter of a circle. An inherent issue with circular layouts is that the rigid restriction on node placement often gives rise to long edges and an overall dense drawing. We suggest here three independent, complementary techniques for lowering the density and improving the readability of circular layouts. First, a new algorithm is given for placing the nodes on the circle such that edge lengths are reduced. Second, we enhance the circular drawing style by allowing some of the edges to be routed around the exterior of the circle. This is accomplished with an algorithm for optimally selecting such a set of externally routed edges. The third technique reduces density by coupling groups of edges as bundled splines that share part of their route. Together, these techniques are able to reduce clutter, density and crossings compared with existing methods.
EnergyBased Hierarchical Edge Clustering of Graphs
 IN PROC. OF THE 2008 IEEE PACIFIC VISUALIZATION SYMPOSIUM (2008
, 2009
"... Effectively visualizing complex nodelink graphs which depict relationships among data nodes is a challenging task due to the clutter and occlusion resulting from an excessive amount of edges. In this paper, we propose a novel energybased hierarchical edge clustering method for nodelink graphs. Ta ..."
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Cited by 8 (2 self)
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Effectively visualizing complex nodelink graphs which depict relationships among data nodes is a challenging task due to the clutter and occlusion resulting from an excessive amount of edges. In this paper, we propose a novel energybased hierarchical edge clustering method for nodelink graphs. Taking into the consideration of the graph topology, our method first samples graph edges into segments using Delaunay triangulation to generate the control points, which are then hierarchically clustered by energybased optimization. The edges are grouped according to their positions and directions to improve comprehensibility through abstraction and thus reduce visual clutter. The experimental results demonstrate the effectiveness of our proposed method in clustering edges and providing good high level abstractions of complex graphs.
C.: Drawing graphs using modular decomposition
 Graph Drawing. Volume LNCS 3843
, 2005
"... In this paper we present an algorithm for drawing an undirected graph G that takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by traversing the modular decomposition tree of the input graph G on n vertices and m edges in a bottomup fashion u ..."
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Cited by 7 (1 self)
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In this paper we present an algorithm for drawing an undirected graph G that takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by traversing the modular decomposition tree of the input graph G on n vertices and m edges in a bottomup fashion until it reaches the root of the tree, while at the same time intermediate drawings are computed. In order to achieve aesthetically pleasing results, we use grid and circular placement techniques, and utilize an appropriate modification of a wellknown spring embedder algorithm. It turns out, that for some classes of graphs, our algorithm runs in O(n + m) time, while in general, the running time is bounded in terms of the processing time of the spring embedder algorithm. The result is a drawing that reveals the structure of the graph G and preserves certain aesthetic criteria.
Controllable and Progressive Edge Clustering for Large Networks
"... Abstract. Nodelink diagrams are widely used in information visualization to show relationships among data. However, when the size of data becomes very large, nodelink diagrams will become cluttered and visually confusing for users. In this paper, we propose a novel controllable edge clustering met ..."
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Cited by 7 (3 self)
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Abstract. Nodelink diagrams are widely used in information visualization to show relationships among data. However, when the size of data becomes very large, nodelink diagrams will become cluttered and visually confusing for users. In this paper, we propose a novel controllable edge clustering method based on Delaunay triangulation to reduce visual clutter for nodelink diagrams. Our method uses curves instead of straight lines to represent links and these curves can be grouped together according to their relative positions and directions. We further introduce progressive edge clustering to achieve continuous levelofdetails for large networks. 1
Confluent layered drawings
 IN GRAPH DRAWING (PROC. GD ’04)
, 2004
"... We combine the idea of confluent drawings with Sugiyama style drawings, in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed style drawings. The basic idea is to cover a layered graph by complete bipartite s ..."
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Cited by 7 (3 self)
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We combine the idea of confluent drawings with Sugiyama style drawings, in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with treelike structures. The biclique cover problem is reduced to a special edge coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multidepth confluent layered drawings.
Drawing Trees with Perfect Angular Resolution and Polynomial Area
"... Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossingfree straightline drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require e ..."
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Cited by 7 (7 self)
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Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossingfree straightline drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossingfree straightline drawing having perfect angular resolution. 3. Any ordered tree has a crossingfree Lombardistyle drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straightline drawings and what more is achievable with Lombardistyle drawings, with respect to drawings of trees with perfect angular resolution. 1