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24
Applications of Random Sampling in Computational Geometry, II
 Discrete Comput. Geom
, 1995
"... We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric ..."
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Cited by 452 (12 self)
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We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divideandconquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...
A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons
, 1991
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Controlled Simplification of Genus for Polygonal Models
, 1997
"... Genusreducing simplifications are important in constructing multiresolution hierarchies for levelofdetailbased rendering, especially for datasets that have several relatively small holes, tunnels, and cavities. We present a genusreducing simplification approach that is complementary to the exis ..."
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Cited by 50 (1 self)
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Genusreducing simplifications are important in constructing multiresolution hierarchies for levelofdetailbased rendering, especially for datasets that have several relatively small holes, tunnels, and cavities. We present a genusreducing simplification approach that is complementary to the existing work on genuspreserving simplifications. We propose a simplification framework in which genusreducing and genuspreserving simplifications alternate to yield much better multiresolution hierarchies than would have been possible by using either one of them. In our approach we first identify the holes and the concavities by extending the concept of # hulls to polygonal meshes under the L1 distance metric and then generate valid triangulations to fill them. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation  Display algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling  Curve, surface, solid, and object represent...
Topology Simplification for Polygonal Virtual Environments
 IEEE Trans. Visualization and Computer Graphics
, 1998
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Randomized Parallel Algorithms For Trapezoidal Diagrams
, 1992
"... We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of inters ..."
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Cited by 23 (0 self)
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We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of intersecting pairs of segments. If the segments form a simple chain, we give an algorithm requiring optimal O(n) expected work and O(logn log log n log n) expected time a , and a simpler algorithm requiring O(n log n) expected work. The serial algorithm corresponding to the latter is among the simplest known algorithms requiring O(n log n) expected operations. For a set of segments forming K chains, we give an algorithm requiring O(A + n log n + K log n) expected work and O(logn log log n log n) expected time. The parallel time bounds require the assumption that enough processors are available, with processor allocations every log n steps. Keywords: randomized, parallel, trapez...
FIST: Fast industrialstrength triangulation of polygons
 Algorithmica
, 1998
"... A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y ..."
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Cited by 19 (4 self)
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A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y
Partitioning Arrangements of Lines I: An Efficient Deterministic Algorithm
, 1990
"... In this paper we consider the following problem: Given a set L of n lines in the plane, partition the plane into O(r²) triangles so that no triangle meets more than O(n/r) lines of L. We present a deterministic algorithm for this problem with O(nr log n log ° r) running time, where co is a constant ..."
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Cited by 18 (3 self)
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In this paper we consider the following problem: Given a set L of n lines in the plane, partition the plane into O(r²) triangles so that no triangle meets more than O(n/r) lines of L. We present a deterministic algorithm for this problem with O(nr log n log ° r) running time, where co is a constant < 3.33.
Efficient Rendering of Trimmed NURBS Surfaces
, 1995
"... : We present an algorithm for interactive display of trimmed NURBS surfaces. The algorithm converts the NURBS surfaces to B'ezier surfaces and NURBS trimming curves into B'ezier curves. It tessellates each trimmed B'ezier surface into triangles and renders them using the triangle ren ..."
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Cited by 16 (6 self)
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: We present an algorithm for interactive display of trimmed NURBS surfaces. The algorithm converts the NURBS surfaces to B'ezier surfaces and NURBS trimming curves into B'ezier curves. It tessellates each trimmed B'ezier surface into triangles and renders them using the triangle rendering capabilities common in current graphics systems. It makes use of tight bounds for uniform tessellation of B'ezier surfaces into cells and traces the trimming curves to compute the trimmed regions of each cell. This is based on tracing trimming curves, intersection computation with the cells, and triangulation of the cells. The resulting technique also makes use of spatial and temporal coherence between successive frames for cell computation and triangulation. Polygonization anomalies like cracks and angularities are avoided as well. The algorithm can display trimmed models described using thousands of B'ezier surfaces at interactive frame rates on the high end graphics systems. Additional Keywords ...
LinearTime Triangulation of a Simple Polygon Made Easier Via Randomization
, 2000
"... We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm ..."
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Cited by 10 (0 self)
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We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of nonoptimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottomup preprocessing phase previous to the topdown construction phase.