Showing papers by "Jeff Erickson published in 2005"

Greedy optimal homotopy and homology generators

Abstract: We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2-manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2-manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdiere and Lazarus.

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

Abstract: Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R ⋃ B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ℝ2. Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.

Building spacetime meshes over arbitrary spatial domains

Abstract: We present an algorithm to construct meshes suitable for spacetime discontinuous Galerkin finite-element methods. Our method generalizes and improves the ‘Tent Pitcher’ algorithm of Ungor and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T], our algorithm builds a simplicial mesh of the spacetime domain X × [0, T], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of spacetime elements to the local quality and feature size of the underlying space mesh.

Separating point sets in polygonal environments

Abstract: We consider the separability of two point sets inside a polygon by means of chords or geodesic lines. Specifically, given a set of red points and a set of blue points in the interior of a polygon, we provide necessary and sufficient conditions for the existence of a chord and for the existence of a geodesic path that separate the two sets; when they exist we also derive efficient algorithms for their obtention. We also study the separation of the two sets using the minimum number of pairwise non-crossing chords.

Dense Point Sets Have Sparse Delaunay Triangulations or ... But Not Too Nasty

Abstract: The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of $n$ points in~$\Real^3$ with spread $\Delta$ has complexity $O(\Delta^3)$. This bound is tight in the worst case for all $\Delta = O(\sqrt{n})$. In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of $k$-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any $n$ and $\Delta = O(n)$, we construct a regular triangulation of complexity $\Omega(n\Delta)$ whose $n$ vertices have spread $\Delta$.

Spacetime meshing for discontinuous galerkin methods

Abstract: Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the high-fidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs arising from wave propagation phenomena. To support an accurate and efficient solution procedure using SDG methods and to exploit the flexibility of these methods, we give a meshing algorithm to construct an unstructured simplicial spacetime mesh over an arbitrary simplicial space domain. Our algorithm is the first adaptive spacetime meshing algorithm suitable for efficient solution of nonlinear phenomena using spacetime discontinuous Galerkin finite element methods. Given a triangulated d-dimensional Euclidean space domain M (a simplicial complex) corresponding to time t = 0 and initial conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured simplicial mesh of the ( d + 1)-dimensional spacetime domain Ω. Our algorithm uses a near-optimal number of spacetime elements, each with bounded temporal aspect ratio for any finite prefix of Ω. When d ≤ 2, our algorithm varies the size of spacetime elements to an a posteriori numerical estimate. Certain facets of our mesh satisfy gradient constraints that allow interleaving mesh generation with the SDG salver. Our meshing algorithm thus supports an efficient parallelizable solution strategy by SDG methods.

Lower bounds for external algebraic decision trees

Abstract: We propose a natural extension of algebraic decision trees to the external-memory setting, where the cost of disk operations overwhelms CPU time, and prove a tight lower bound of Ω(n logmn) on the complexity of both sorting and element uniqueness in this model of computation. We also prove a Ω(min{n logm n, N}) lower bound for both problems in a less restrictive model, which requires only that the worst-case internal-memory computation time is finite. Standard reductions immediately generalize these lower bounds to a large number of fundamental computational geometry problems.

Distance-2 Edge Coloring is NP-Complete

Abstract: We prove that it is NP-complete to determine whether there exists a distance-2 edge coloring (strong edge coloring) with 5 colors of a bipartite 2-inductive graph with girth 6 and maximum degree 3.

Meshing in 2d×time for front-tracking dg methods

Abstract: Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve hyperbolic PDEs describing wavelike phenomena, such as fluid flow. In front-tracking SDG methods, the inclined facets of spacetime elements are aligned with the trajectories of moving domain boundaries, phase interfaces, and other singular surfaces. These methods are particularly effective because SDG solution fields naturally accommodate the jumps that occur at singular surfaces. Our goal is to construct front-tracking tetrahedral meshes in 2D×time that partition a spacetime analysis domain Ω ⊂ E × R while satisfying a causality condition that facilitates the SDG solution procedure. Trajectories of ∂Ω and of interior singular surfaces are generally solution-dependent and must be computed as the solution evolves.