48,955 research outputs found
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
Optimal Point Placement for Mesh Smoothing
We study the problem of moving a vertex in an unstructured mesh of
triangular, quadrilateral, or tetrahedral elements to optimize the shapes of
adjacent elements. We show that many such problems can be solved in linear time
using generalized linear programming. We also give efficient algorithms for
some mesh smoothing problems that do not fit into the generalized linear
programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was
presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This
is the final version, and will appear in a special issue of J. Algorithms for
papers from SODA '9
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 Stretch Factor of - and -Delaunay Triangulations
In this paper we determine the stretch factor of the -Delaunay and
-Delaunay triangulations, and we show that this stretch is
. Between any two points of such
triangulations, we construct a path whose length is no more than
times the Euclidean distance between and , and this
bound is best possible. This definitively improves the 25-year old bound of
by Chew (SoCG '86). To the best of our knowledge, this is the first
time the stretch factor of the well-studied -Delaunay triangulations, for
any real , is determined exactly
Singular Vertices and the Triangulation Space of the D-sphere
By a sequence of numerical experiments we demonstrate that generic
triangulations of the sphere for contain one {\it singular}
simplex. The mean number of elementary simplices sharing this
simplex increases with the volume of the triangulation according to a simple
power law. The lower dimension subsimplices associated with this
simplex also show a singular behaviour. Possible consequences for the
DT model of four-dimensional quantum gravity are discussed.Comment: 15 pages, 9 figure
Recommended from our members
Graph-theoretical conditions for inscribability and Delaunay realizability
We present new graph-theoretical conditions for inscribable polyhedra and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is inscribable. These results have several consequences:All 4-connected polyhedra are inscribable.All simplical polyhedra in which all vertex degrees are between 4 and 6, inclusive, are inscribable.All triangulations without chords or nonfacial triangles are realizable as Delaunay triangulations.We also strengthen some earlier results about matchings in inscribable polyhedra. Specifically, we show that any nonbipartite inscribable polyhedron has a perfect matching containing any specified edge, and that any bipartite inscribable polyhedron has a perfect matching containing any two specified disjoint edges. We give examples showing that these results are best possible
Random lattice triangulations: Structure and algorithms
The paper concerns lattice triangulations, that is, triangulations of the
integer points in a polygon in whose vertices are also integer
points. Lattice triangulations have been studied extensively both as geometric
objects in their own right and by virtue of applications in algebraic geometry.
Our focus is on random triangulations in which a triangulation has
weight , where is a positive real parameter, and
is the total length of the edges in . Empirically, this
model exhibits a "phase transition" at (corresponding to the
uniform distribution): for distant edges behave essentially
independently, while for very large regions of aligned edges
appear. We substantiate this picture as follows. For sufficiently
small, we show that correlations between edges decay exponentially with
distance (suitably defined), and also that the Glauber dynamics (a local Markov
chain based on flipping edges) is rapidly mixing (in time polynomial in the
number of edges in the triangulation). This dynamics has been proposed by
several authors as an algorithm for generating random triangulations. By
contrast, for we show that the mixing time is exponential. These
are apparently the first rigorous quantitative results on the structure and
dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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