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Choosing Colors for Geometric Graphs via Color Space Embeddings
"... Abstract. Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. However additional work must still be done to draw a graph even after its geometric embedding is specified: assigning display colors to the graph’s vertices. W ..."
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Abstract. Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. However additional work must still be done to draw a graph even after its geometric embedding is specified: assigning display colors to the graph’s vertices. We study the additional aesthetic criterion of assigning distinct colors to vertices of a geometric graph so that the colors assigned to adjacent vertices are as different from one another as possible. We formulate this as a problem involving perceptual metrics in color space and we develop algorithms for solving this problem by embedding the graph in color space. We also present an application of this work to a distributed loadbalancing visualization problem.
A Coloring Solution to the Edge Crossing Problem
"... We introduce the concept of coloring close and crossing edges in graph drawings with perceptually opposing colors making them individually more distinguishable and reducing edgecrossing effects. We define a “closeness ” metric on edges as a combination of distance, angle and crossing. We use the in ..."
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We introduce the concept of coloring close and crossing edges in graph drawings with perceptually opposing colors making them individually more distinguishable and reducing edgecrossing effects. We define a “closeness ” metric on edges as a combination of distance, angle and crossing. We use the inverse of this metric to compute a color embedding in the L*a*b* color space and assign “close ” edges colors that are perceptually far apart. We present the following results: a distance metric on graph edges, a method of coloring graph edges, and anecdotal evidence that this technique can improve the reading of graph edges. Keywords graphs, colors, color embeddings. 1.
Graphical Representations and Infinite Virtual Worlds In Logic and Functional Programming Course
, 2003
"... The assignment scheme of our Logic and Functional Programming course has adopted the generation of graphical representations, quadtrees and octrees to teach the di#erent features of this kind of languages: higher order functions, lazy evaluation, polymorphism, logical variables, constraint sati ..."
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The assignment scheme of our Logic and Functional Programming course has adopted the generation of graphical representations, quadtrees and octrees to teach the di#erent features of this kind of languages: higher order functions, lazy evaluation, polymorphism, logical variables, constraint satisfaction, etc. The use of standard XML vocabularies: SVG for graphics and X3D for virtual worlds enables to have numerous visualization tools. We consider that the incorporation of these exercises facilitates teaching and improves the motivation of students.
Coloring octrees
, 2006
"... An octree is a recursive partition of the unit cube, such that in each step a cube is subdivided into eight smaller cubes. Those cubes that are not further subdivided are the leaves of the octree. We consider the problem of coloring the leaves of an octree using as few colors as possible such that n ..."
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An octree is a recursive partition of the unit cube, such that in each step a cube is subdivided into eight smaller cubes. Those cubes that are not further subdivided are the leaves of the octree. We consider the problem of coloring the leaves of an octree using as few colors as possible such that no two of them get the same color if they share a facet. It turns out that the number of colors needed depends on a parameter that we call unbalancedness. Roughly speaking, this parameter measures how much adjacent cubes differ in size. For most values of this parameter we give tight bounds on the minimum number of colors, and extend the results to higher dimensions.
On Coloring Resilient Graphs
"... Abstract. We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NPhardness and the existence of efficient algorithms for resilient instances. In particular, we study rresiliently kcolorable graphs, which a ..."
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Abstract. We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NPhardness and the existence of efficient algorithms for resilient instances. In particular, we study rresiliently kcolorable graphs, which are those kcolorable graphs that remain kcolorable even after the addition of any r new edges. We prove lower bounds on the NPhardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. This notion of resilience suggests an array of open questions for graph colorability and other combinatorial problems. 1 Introduction and related work An important goal in studying NPcomplete combinatorial problems is to find precise boundaries between tractability and NPhardness. This is often done by adding constraints to the instances being considered until a polynomial time algorithm is found. For instance, while SAT is NPhard, the restricted 2SAT
DiamondKite Adaptive Quadrilateral Meshing∗
"... We describe a family of quadrilateral meshes based on diamonds, rhombi with 60 ◦ and 120 ◦ angles, and kites with 60◦, 90◦, and 120 ◦ angles, that can be adapted to a local size function by local subdivision operations. Our meshes use a number of elements that is within a constant factor of the mini ..."
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We describe a family of quadrilateral meshes based on diamonds, rhombi with 60 ◦ and 120 ◦ angles, and kites with 60◦, 90◦, and 120 ◦ angles, that can be adapted to a local size function by local subdivision operations. Our meshes use a number of elements that is within a constant factor of the minimum possible for any mesh of bounded aspect ratio elements, graded by the same local size function, and is invariant under Laplacian smoothing. The vertices of our meshes form the centers of the circles in a pair of dual circle packings. The same vertex placement algorithm but a different mesh topology gives a pair of dual wellcentered meshes adapted to the given size function. 1