Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

The Sudbury Neutrino Observatory

The Sudbury Neutrino Observatory is a second generation water Cherenkov detector designed to determine whether the currently observed solar neutrino deficit is a result of neutrino oscillations. The detector is unique in its use of D2O as a detection medium, permitting it to make a solar model-independent test of the neutrino oscillation hypothesis by comparison of the charged- and neutral-current interaction rates. In this paper the physical properties, construction, and preliminary operation of the Sudbury Neutrino Observatory are described. Data and predicted operating parameters are provided whenever possible.Comment: 58 pages, 12 figures, submitted to Nucl. Inst. Meth. Uses elsart and epsf style files. For additional information about SNO see http://www.sno.phy.queensu.ca . This version has some new reference

Two-Dimensional Arrangements in CGAL and Adaptive Point Location for Parametric Curves

. Given a collection C of curves in the plane, the arrangement of C is the subdivision of the plane into vertices, edges and faces induced by the curves in C. Constructing arrangements of curves in the plane is a basic problem in computational geometry. Applications relying on arrangements arise in fields such as robotics, computer vision and computer graphics. Many algorithms for constructing and maintaining arrangements under various conditions have been published in papers. However, there are not many implementations of (general) arrangements packages available. We present an implementation of a generic and robust package for arrangements of curves that is part of the CGAL 1 library. We also present an application based on this package for adaptive point location in arrangements of parametric curves.

Flipping to Robustly Delete a Vertex in a Delaunay Tetrahedralization

International Conference on Computational Science and Its Applications (ICCSA 2005), Singapore, May 9-12, 2005We discuss the deletion of a single vertex in a Delaunay tetrahedralization (DT). While some theoretical solutions exist for this problem, the many degeneracies in three dimensions make them impossible to be implemented without the use of extra mechanisms. In this paper, we present an algorithm that uses a sequence of bistellar flips to delete a vertex in a DT, and we present two different mechanisms to ensure its robustness.Department of Computin

Incrementally Constructing and Updating Constrained Delaunay Tetrahedralizations with Finite Precision Coordinates

Summary. Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision coordinates yield vertices on a line that are not perfectly collinear and polygonal facets that are not perfectly flat. We experimentally compare two recent algorithms for inserting a polygonal facet into a CDT: a bistellar flip algorithm of Shewchuk (Proc. 19th Annual Symposium on Computational Geometry, June 2003) and a cavity retriangulation algorithm of Si and Gärtner (Proc. Fourteenth International Meshing Roundtable, September 2005). We modify these algorithms to succeed in practice for polygons whose vertices deviate from exact coplanarity.