658 research outputs found
Edge Routing with Ordered Bundles
Edge bundling reduces the visual clutter in a drawing of a graph by uniting
the edges into bundles. We propose a method of edge bundling drawing each edge
of a bundle separately as in metro-maps and call our method ordered bundles. To
produce aesthetically looking edge routes it minimizes a cost function on the
edges. The cost function depends on the ink, required to draw the edges, the
edge lengths, widths and separations. The cost also penalizes for too many
edges passing through narrow channels by using the constrained Delaunay
triangulation. The method avoids unnecessary edge-node and edge-edge crossings.
To draw edges with the minimal number of crossings and separately within the
same bundle we develop an efficient algorithm solving a variant of the
metro-line crossing minimization problem. In general, the method creates clear
and smooth edge routes giving an overview of the global graph structure, while
still drawing each edge separately and thus enabling local analysis
Bundled Crossings Revisited
An effective way to reduce clutter in a graph drawing that has (many)
crossings is to group edges that travel in parallel into \emph{bundles}. Each
edge can participate in many such bundles. Any crossing in this bundled graph
occurs between two bundles, i.e., as a \emph{bundled crossing}. We consider the
problem of bundled crossing minimization: A graph is given and the goal is to
find a bundled drawing with at most bundled crossings. We show that the
problem is NP-hard when we require a simple drawing. Our main result is an FPT
algorithm (in ) when we require a simple circular layout. These results make
use of the connection between bundled crossings and graph genus.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
A Coloring Algorithm for Disambiguating Graph and Map Drawings
Drawings of non-planar graphs always result in edge crossings. When there are
many edges crossing at small angles, it is often difficult to follow these
edges, because of the multiple visual paths resulted from the crossings that
slow down eye movements. In this paper we propose an algorithm that
disambiguates the edges with automatic selection of distinctive colors. Our
proposed algorithm computes a near optimal color assignment of a dual collision
graph, using a novel branch-and-bound procedure applied to a space
decomposition of the color gamut. We give examples demonstrating the
effectiveness of this approach in clarifying drawings of real world graphs and
maps
Edge-Path Bundling: A Less Ambiguous Edge Bundling Approach
Edge bundling techniques cluster edges with similar attributes (i.e. similarity in direction and proximity) together to reduce the visual clutter. All edge bundling techniques to date implicitly or explicitly cluster groups of individual edges, or parts of them, together based on these attributes. These clusters can result in ambiguous connections that do not exist in the data. Confluent drawings of networks do not have these ambiguities, but require the layout to be computed as part of the bundling process. We devise a new bundling method, Edge-Path bundling, to simplify edge clutter while greatly reducing ambiguities compared to previous bundling techniques. Edge-Path bundling takes a layout as input and clusters each edge along a weighted, shortest path to limit its deviation from a straight line. Edge-Path bundling does not incur independent edge ambiguities typically seen in all edge bundling methods, and the level of bundling can be tuned through shortest path distances, Euclidean distances, and combinations of the two. Also, directed edge bundling naturally emerges from the model. Through metric evaluations, we demonstrate the advantages of Edge-Path bundling over other techniques
The bundled crossing number
We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all crossings in a drawing of the graph. We show that the bundled crossing number is closely related to the orientable genus of the graph. If multiple crossings and self-intersections of edges are allowed, the two values are identical; otherwise, the bundled crossing number can be higher than the genus. We then investigate the problem of minimizing the number of bundled crossings. For circular graph layouts with a fixed order of vertices, we present a constant-factor approximation algorithm. When the circular order is not prescribed, we get a 6c/c - 2 -approximation for a graph with n vertices having at least cn edges for c > 2. For general graph layouts, we develop an algorithm with an approximation factor of 6c/c - 3 for graphs with at least cn edges for c > 3. © Springer International Publishing AG 2016
Bundled Crossings Revisited
International audienceAn effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into bundles. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a bundled crossing. We consider the problem of bundled crossing minimization: A graph is given and the goal is to find a bundled drawing with at most k bundled crossings. We show that the problem is NP-hard when we require a simple drawing. Our main result is an FPT algorithm (in k) for simple circular layouts where vertices must be placed on a circle and edges must be drawn inside the circle. These results make use of the connection between bundled crossings and graph genus. We also consider bundling crossings in a given drawing, in particular for storyline visualizations
3D Edge Bundling for Geographical Data Visualization
International audienceVisualization of graphs containing many nodes and edges efficiently is quite challenging since representations generally suffer from visual clutter induced by the large amount of edge crossings and node-edge overlaps. That problem becomes even more important when nodes po- sitions are fixed, such as in geography were nodes posi- tions are set according to geographical coordinates. Edge bundling techniques can help to solve this issue by visu- ally merging edges along common routes but it can also help to reveal high-level edge patterns in the network and therefore to understand its overall organization. In this pa- per, we present a generalization of [18] to reduce the clut- ter in a 3D representation by routing edges into bundles as well as a GPU-based rendering method to emphasize bundles densities while preserving edge color. To visualize geographical networks in the context of the globe, we also provide a new technique allowing to bundle edges around and not across it
Edge routing with ordered bundles
Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling that draws each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes, it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle, we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis. © 2015 Elsevier B.V
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