42 research outputs found
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
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
k-link shortest paths in weighted subdivisions
Abstract. We study the shortest path problem in weighted polygonal subdivisions of the plane, with the additional constraint of an upper bound, k, on the number of links (segments) in the path. We prove structural properties of optimal paths and utilize these results to obtain approximation algorithms that yield a path having O(k) links and weighted length at most (1 + ǫ) times the weighted length of an optimal k-link path, for any fixed ǫ> 0. Some of our results make use of a new solution for the 1-link case, based on computing optimal solutions for a special sum-of-fractionals (SOF) problem. We have implemented a system, based on the CORE library, for computing optimal 1-link paths; we experimentally compare our new solution with a previous method for 1-link optimal paths based on a prune-and-search scheme.