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17
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 algorithms ..."
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Cited by 396 (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
 Comput. Geom. Theory Appl
, 1991
"... This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of t ..."
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Cited by 99 (2 self)
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This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of the same complexity for triangulating polygons. More generally, if S is presented as a plane graph with k connected components, then the expected running time of the algorithm is O(n log n k log n). As a byproduct our algorithm creates a search structure of expected linear size that allows point location queries in the resulting trapezoidation in logarithmic expected time. The analysis of the expected performance is elementary and straightforward. All expectations are with respect to "coinflips" generated by the algorithm and are not based on assumptions about the geometric distribution of the input. Large Portions of the research reported here were conducted while the author visit...
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 45 (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 Transactions on Visualization and Computer Graphics
, 1998
"... We present a topology simplifying approach that can be used for genus reductions, removal of protuberances, and repair of cracks in polygonal models in a unified framework. Our work is complementary to the existing work on geometry simplification of polygonal datasets and we demonstrate that using ..."
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Cited by 25 (1 self)
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We present a topology simplifying approach that can be used for genus reductions, removal of protuberances, and repair of cracks in polygonal models in a unified framework. Our work is complementary to the existing work on geometry simplification of polygonal datasets and we demonstrate that using topology and geometry simplifications together yields superior multiresolution hierarchies than is possible by using either of them alone. Our approach can also address the important issue of repair of cracks in polygonal models as well as for rapid identification and removal of protuberances based on internal accessibility in polygonal models. Our approach is based on identifying holes and cracks by extending the concept of #shapes to polygonal meshes under the L1 distance metric. We then generate valid triangulations to fill them using the intuitive notion of sweeping a L1 cube over the identified regions. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture...
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...
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 rendering capabili ..."
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Cited by 15 (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 ...
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 11 (2 self)
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A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y
LinearTime Triangulation of a Simple Polygon Made Easier Via Randomization
 In Proc. 16th Annu. ACM Sympos. Comput. Geom
, 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. 1 Introduction Polygon triangulation is a classic problem in comp...