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

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

Efficient Generation of Geographically Accurate Transit Maps

We present LOOM (Line-Ordering Optimized Maps), a fully automatic generator of geographically accurate transit maps. The input to LOOM is data about the lines of a given transit network, namely for each line, the sequence of stations it serves and the geographical course the vehicles of this line take. We parse this data from GTFS, the prevailing standard for public transit data. LOOM proceeds in three stages: (1) construct a so-called line graph, where edges correspond to segments of the network with the same set of lines following the same course; (2) construct an ILP that yields a line ordering for each edge which minimizes the total number of line crossings and line separations; (3) based on the line graph and the ILP solution, draw the map. As a naive ILP formulation is too demanding, we derive a new custom-tailored formulation which requires significantly fewer constraints. Furthermore, we present engineering techniques which use structural properties of the line graph to further reduce the ILP size. For the subway network of New York, we can reduce the number of constraints from 229,000 in the naive ILP formulation to about 4,500 with our techniques, enabling solution times of less than a second. Since our maps respect the geography of the transit network, they can be used for tiles and overlays in typical map services. Previous research work either did not take the geographical course of the lines into account, or was concerned with schematic maps without optimizing line crossings or line separations.Comment: 7 page

On Embeddability of Buses in Point Sets

Set membership of points in the plane can be visualized by connecting corresponding points via graphical features, like paths, trees, polygons, ellipses. In this paper we study the \emph{bus embeddability problem} (BEP): given a set of colored points we ask whether there exists a planar realization with one horizontal straight-line segment per color, called bus, such that all points with the same color are connected with vertical line segments to their bus. We present an ILP and an FPT algorithm for the general problem. For restricted versions of this problem, such as when the relative order of buses is predefined, or when a bus must be placed above all its points, we provide efficient algorithms. We show that another restricted version of the problem can be solved using 2-stack pushall sorting. On the negative side we prove the NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201

Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases

Crossing minimization is one of the central problems in graph drawing. Recently, there has been an increased interest in the problem of minimizing crossings between paths in drawings of graphs. This is the metro-line crossing minimization problem (MLCM): Given an embedded graph and a set L of simple paths, called lines, order the lines on each edge so that the total number of crossings is minimized. So far, the complexity of MLCM has been an open problem. In contrast, the problem variant in which line ends must be placed in outermost position on their edges (MLCM-P) is known to be NP-hard. Our main results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We give an $O(\sqrt{\log |L|})$-approximation algorithm for MLCM-P

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

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 $k$ bundled crossings. We show that the problem is NP-hard when we require a simple drawing. Our main result is an FPT algorithm (in $k$) 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

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

Visualisation interactive de graphes (élaboration et optimisation d'algortihmes à coûts computationnels élevés)

Un graphe est un objet mathématique modélisant des relations sur un ensemble d'éléments. Il est utilisé dans de nombreux domaines à des fins de modélisation. La taille et la complexité des graphes manipulés de nos jours entraînentdes besoins de visualisation afin de mieux les analyser. Dans cette thèse, nous présentons différents travaux en visualisation interactive de graphes qui s'attachent à exploiter les architectures de calcul parallèle (CPU et GPU) disponibles sur les stations de travail contemporaines. Un premier ensemble de travaux s'intéresse à des problématiques de dessin de graphes. Dessiner un graphe consiste à le plonger visuellement dans un plan ou un espace. La première contribution dans cette thématique est un algorithmede regroupement d'arêtes en faisceaux appelé Winding Roads.Cet algorithme intuitif, facilement implémentable et parallélisable permet de réduireconsidérablement les problèmes d'occlusion dans un dessin de graphedus aux nombreux croisements d'arêtes.La seconde contribution est une méthode permettant dedessiner un réseau métabolique complet. Ce type deréseau modélise l'ensemble des réactions biochimiquesse produisant dans les cellules d'un organise vivant.L'avantage de la méthode est de prendre en compte la décompositiondu réseau en sous-ensembles fonctionnels ainsi que de respecterles conventions de dessin biologique.Un second ensemble de travaux porte sur des techniques d'infographiepour la visualisation interactive de graphes. La première contribution dans cette thématique est une technique de rendude courbes paramétriques exploitant pleinement le processeur graphique. La seconde contribution est une méthodede rendu nommée Edge splatting permettant de visualiserla densité des faisceaux d'arêtes dans un dessin de grapheavec regroupement d'arêtes. La dernière contribution portesur des techniques permettant de mettre en évidence des sous-graphesd'intérêt dans le contexte global d'une visualisation de graphes.A graph is a mathematical object used to model relations over a set of elements.It is used in numerous fields for modeling purposes. The size and complexityof graphs manipulated today call a need for visualization to better analyze them.In that thesis, we introducedifferent works in interactive graph visualisation which aim at exploiting parallel computing architectures (CPU and GPU) available on contemporary workstations.A first set of works focuses on graph drawing problems.Drawing a graph consists of embedding him in a plane or a space.The first contribution in that theme is an edge bundling algorithmnamed Winding Roads. That intuitive, easyly implementable and parallelizable algorithmallows to considerably reduce clutter due to numerous edge crossings in a graph drawing.The second contribution is a method to draw a complete metabolicnetwork. That kind of network models the whole set of biochemical reactionsoccurring within cells of a living organism. The advantage of the methodis to take into account the decomposition of the network into functionnal subsetsbut also to respect biological drawing conventions.A second set of works focuses on computer graphics techniquesfor interactive graph visualisation. The first contributionin that theme is a technique for rendering parametric curvesthat fully exploits the graphical processor unit. The second contributionis a rendering technique named Edge splatting that allowsto visualize the bundles densities in an edge bundled layout. Thelast contribution introduces some techniques for emphasizingsub-graphs of interest in the global context of a graph visualization.BORDEAUX1-Bib.electronique (335229901) / SudocSudocFranceF