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Embedding Stacked Polytopes on a PolynomialSize Grid
, 2011
"... We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) × O(n4) × O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts t ..."
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We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) × O(n4) × O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts to a small 3D polytope, all in linear time. This result solves a question posed by Günter M. Ziegler, and is the first nontrivial subexponential upper bound on the longstanding open question of the grid size necessary to embed arbitrary convex polyhedra, that is, about efficient versions of Steinitz’s 1916 theorem. An immediate consequence of our result is that O(log n)bit coordinates suffice for a greedy routing strategy in planar 3trees.
Drawing 3polytopes with good vertex resolution
 In GD’09, Proc. 17th International Symposium on Graph Drawing, 2009, Lecture Notes in Computer Science
, 2010
"... Abstract. We study the problem how to obtain a small drawing of a 3polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a onedimensional problem, since it is sufficient to guarantee distinct integer xcoordinates. We develop an algorithm that yields an ..."
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Abstract. We study the problem how to obtain a small drawing of a 3polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a onedimensional problem, since it is sufficient to guarantee distinct integer xcoordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained in a 2(n−2)×2×1 box. The constructed embedding can be scaled to a grid embedding whose xcoordinates are contained in [0, 2(n − 2)]. Furthermore, the point set of the embedding has a small spread, which differs from the best possible spread only by a multiplicative constant. 1
3D straightline drawings of ktrees
"... This paper studies the problem of computing 3D crossingfree straightline grid drawings of graphs such that the overall volume is small. We show that every 2tree (and therefore every seriesparallel graph) can be drawn on an integer 3D grid consisting of 15 parallel lines and having linear volume. ..."
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This paper studies the problem of computing 3D crossingfree straightline grid drawings of graphs such that the overall volume is small. We show that every 2tree (and therefore every seriesparallel graph) can be drawn on an integer 3D grid consisting of 15 parallel lines and having linear volume. We extend the study to the problem of drawing general ktrees on a set of parallel grid lines. Lower bounds and upper bounds on the number of such grid lines are presented. The results in this paper extend and improve similar ones already described in the literature.
Small grid embeddings of 3polytopes
, 2009
"... We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadri ..."
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We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadrilateral we can bound the integer coordinates by O(2 5.54n). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte’s ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.
Resolving Loads with Positive Interior Stresses
, 2009
"... We consider the pair (pi, fi) as a force with twodimensional direction vector fi applied at the point pi in the plane. For a given set of forces we ask for a noncrossing geometric graph on the points pi that has the following property: There exists a weight assignment to the edges of the graph, ..."
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We consider the pair (pi, fi) as a force with twodimensional direction vector fi applied at the point pi in the plane. For a given set of forces we ask for a noncrossing geometric graph on the points pi that has the following property: There exists a weight assignment to the edges of the graph, such that for every pi the sum of the weighted edges (seen as vectors) around pi yields −fi. As additional constraint we restrict ourselves to weights that are nonnegative on every edge that is not on the convex hull of the point set. We show that (under a generic assumption) for any reasonable set of forces there is exactly one pointed pseudotriangulation that fulfils the desired properties. Our results will be obtained by linear programming duality over the PPTpolytope. For the case where the forces appear only at convex hull vertices we show that the pseudotriangulation that resolves the load can be computed as weighted Delaunay triangulation. Our observations lead to a new characterization of pointed pseudotriangulations, structures that have been proven to be extremely useful in the design and analysis of efficient geometric algorithms. As an application, we discuss how to compute the maximal locally convex function for a polygon whose corners lie on its convex hull.