1,817 research outputs found
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
Triangulating the Real Projective Plane
We consider the problem of computing a triangulation of the real projective
plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a
triangulation of P2 always exists if at least six points in S are in general
position, i.e., no three of them are collinear. We also design an algorithm for
triangulating P2 if this necessary condition holds. As far as we know, this is
the first computational result on the real projective plane
Approximate Euclidean shortest paths in polygonal domains
Given a set of pairwise disjoint simple polygonal obstacles
in defined with vertices, we compute a sketch of
whose size is independent of , depending only on and the
input parameter . We utilize to compute a
-approximate geodesic shortest path between the two given points
in time. Here, is a user
parameter, and is a small positive constant (resulting from the time
for triangulating the free space of using the algorithm in
\cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a
-approximation algorithm to answer two-point Euclidean distance
queries for the case of convex polygonal obstacles.Comment: a few updates; accepted to ISAAC 201
Visibility-Related Problems on Parallel Computational Models
Visibility-related problems find applications in seemingly unrelated and diverse fields such as computer graphics, scene analysis, robotics and VLSI design. While there are common threads running through these problems, most existing solutions do not exploit these commonalities. With this in mind, this thesis identifies these common threads and provides a unified approach to solve these problems and develops solutions that can be viewed as template algorithms for an abstract computational model. A template algorithm provides an architecture independent solution for a problem, from which solutions can be generated for diverse computational models. In particular, the template algorithms presented in this work lead to optimal solutions to various visibility-related problems on fine-grain mesh connected computers such as meshes with multiple broadcasting and reconfigurable meshes, and also on coarse-grain multicomputers.
Visibility-related problems studied in this thesis can be broadly classified into Object Visibility and Triangulation problems. To demonstrate the practical relevance of these algorithms, two of the fundamental template algorithms identified as powerful tools in almost every algorithm designed in this work were implemented on an IBM-SP2. The code was developed in the C language, using MPI, and can easily be ported to many commercially available parallel computers
Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation
We revisit two NP-hard geometric partitioning problems - convex decomposition
and surface approximation. Building on recent developments in geometric
separators, we present quasi-polynomial time algorithms for these problems with
improved approximation guarantees.Comment: 21 pages, 6 figure
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
We provide linear-time algorithms for geometric graphs with sublinearly many
crossings. That is, we provide algorithms running in O(n) time on connected
geometric graphs having n vertices and k crossings, where k is smaller than n
by an iterated logarithmic factor. Specific problems we study include Voronoi
diagrams and single-source shortest paths. Our algorithms all run in linear
time in the standard comparison-based computational model; hence, we make no
assumptions about the distribution or bit complexities of edge weights, nor do
we utilize unusual bit-level operations on memory words. Instead, our
algorithms are based on a planarization method that "zeroes in" on edge
crossings, together with methods for extending planar separator decompositions
to geometric graphs with sublinearly many crossings. Incidentally, our
planarization algorithm also solves an open computational geometry problem of
Chazelle for triangulating a self-intersecting polygonal chain having n
segments and k crossings in linear time, for the case when k is sublinear in n
by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium
on Discrete Algorithms (SODA09
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