We apply techniques from "Quadrilateral meshing by circle packing" to a magic trick of Houdini: fold a piece of paper so that with one straight cut, you can form your favorite polygon.
We compute the expected numbers of short cycles of each length in certain classes of random graphs used for turbocodes, estimate the probability that there are no such short cycles involving a given vertex, and experimentally verify our estimates. The scarcity of short cycles may help explain the empirically observed accuracy of belief-propagation based error-correction algorithms. Note, the TR, conference, and journal versions of this paper have slightly different titles.
Any four mutually tangent spheres determine three coincident lines through opposite pairs of tangencies. As a consequence, we define two new triangle centers. These centers arose as part of a compass-and-straightedge construction of Soddy circles; also available are some Mathematica calculations of trilinear coordinates for the new triangle centers.
We use random sampling to quickly estimate, for each vertex in a graph, the average distance to all other vertices.
We compute the regression depth of a k-flat in a set of n d-dimensional points, in time O(nd-2), an order of magnitude faster than the best known algorithms for computing the depth of a point or of a hyperplane. The results from this conference paper have been merged into the full version of "Multivariate Regression Depth".
Rule sets for internet routers and firewalls can be represented as sets of prioritized rectangles; the rule to use for a packet is the maximum priority rectangle containing its (source,destination) address pair. We develop efficient data structures for performing these queries, and find an O(n3/2) time algorithm for testing whether a rule set contains any ambiguities.
We show that any graph can be colored in time O(2.415n), by a dynamic programming procedure in which we extend partial colorings on subsets of the vertices by adding one more color for a maximal independent set. The time bound follows from limiting our attention to maximal independent subsets that are small relative to the previously colored subset, and from bounding the number of small maximal independent subsets that can occur in any graph.
We give linear-time quasiconvex programming algorithms for finding a Möbius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, and element size control in conformal structured mesh generation.
We use the ellipsoid method to develop (theoretically) efficient algorithms for optimizing linear functions on intersections of zonotopes, and show how to apply this to train soft-margin support vector classifiers.
We show that any polygon can be cut into kites, connected into a chain by hinges at their vertices, and that this hinged assemblage can be unfolded and refolded to form the mirror image of the polygon.
We unfold any polyhedron with triangular faces into a planar layout in which the triangles are disjoint and are connected in a sequence from vertex to vertex
We consider the problem of subdividing a polyhedral domain in R^3 into cuboids meeting face-to-face. For topological subdivisions (cell complexes in which each cell is combinatorially equivalent to a cube, but may not be embedded as a polyhedron) and simply-connected domains, a simple necessary and sufficient condition for the existence of a hexahedral mesh is known: a domain with quadrilateral faces can be meshed if and only if there is an even number of faces. However, conditions for the existence of polyhedral meshes remain open, as do the topological versions of the problem for more complicated domain topologies.
We examine flips in which a set of mesh cells connected in a similar pattern to a subset of faces of a cube or hypercube is replaced by cells in the pattern of the complementary subset. We show that certain flip types preserve geometric realizability of a mesh, and use this to study the question of whether every topologically meshable domain is geometrically meshable. We also study flip graph connectivity, and prove that the flip graph of quadrilateral meshes has exactly two connected components.
Note that the Meshing Roundtable version was by Bern and Eppstein. Erickson was added as a co-author during the revisions for the journal version.
Which unit-side-length convex polygons can be formed by packing together unit squares and unit equilateral triangles? For instance one can pack six triangles around a common vertex to form a regular hexagon. It turns out that there is a pretty set of 11 solutions. We describe connections from this puzzle to the combinatorics of 3- and 4-dimensional polyhedra, using illustrations from the works of M. C. Escher and others.
We show that geometric thickness and book thickness are not asymptotically equivalent: for every t, there exists a graph with geometric thickness two and book thickness > t.
We define the min-min expectation selection problem (resp. max-min expectation selection problem) to be that of selecting k out of n given discrete probability distributions, to minimize (resp. maximize) the expectation of the minimum value resulting when independent random variables are drawn from the selected distributions. Such problems can be viewed as a simple form of two-stage stochastic programming. We show that if d, the number of values in the support of the distributions, is a constant greater than 2, the min-min expectation problem is NP-complete but admits a fully polynomial time approximation scheme. For d an arbitrary integer, it is NP-hard to approximate the min-min expectation problem with any constant approximation factor. The max-min expectation problem is polynomially solvable for constant d; we leave open its complexity for variable d. We also show similar results for binary selection problems in which we must choose one distribution from each of n pairs of distributions.
Delaunay triangulation has been a staple of triangular mesh generation for a long time. Why? As well as being simple, fast, and visually pleasing, Delaunay triangulations can be shown to be optimal for various measures of mesh quality; for instance, they avoid sharp angles to the maximum extent possible. We review these and other results on construction of meshes that optimize given quality measures, including recent work on postprocessing tetrahedral meshes to eliminate slivers.
Years – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine
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