Results 1  10
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305
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 432 (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...
Topological Persistence and Simplification
, 2000
"... We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast ..."
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Cited by 333 (41 self)
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We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 214 (7 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Incremental Topological Flipping Works for Regular Triangulations
 ALGORITHMICA
, 1996
"... A set of n weighted points in general position in Rd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence an ..."
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Cited by 181 (7 self)
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A set of n weighted points in general position in Rd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at most O(n log n+n ⌈d/2 ⌉). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor log n more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coinflips performed by the algorithm.
An optimal algorithm for intersecting line segments in the plane
 J. ACM
, 1992
"... The main contribution of this work is an O(n log r ~ +k)time algorithm focomputinall k intersections among n line segments in the plane, This time complexity IS easily shown to be optimal. Within the same asymptotic cost, our algorithm can also construct the subdiwslon of the plancdefmed by the s ..."
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Cited by 176 (2 self)
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The main contribution of this work is an O(n log r ~ +k)time algorithm focomputinall k intersections among n line segments in the plane, This time complexity IS easily shown to be optimal. Within the same asymptotic cost, our algorithm can also construct the subdiwslon of the plancdefmed by the segments and compute which segment (if any) lies right above (or below) each intersection and each endpoint. The algorithm has been implemented and performs very well. The storage requirement is on the order of n + k in the worst case, but it is considerably lower in practice. To analyze the complexity of the algorithm, an amortization argument based on a new combinatorial theorem on line arrangements is used.
The Union of Balls and its Dual Shape
, 1993
"... Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived f ..."
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Cited by 172 (12 self)
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Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived from this complex. The algorithms are most relevant in R’3 where unions of finitely many balls are commonly used as models of molecules.
Computing Contour Trees in All Dimensions
, 1999
"... We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al. ..."
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Cited by 159 (9 self)
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We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al.
On the Definition and the Construction of Pockets in Macromolecules
, 1995
"... The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm base ..."
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Cited by 103 (26 self)
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The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm based on alpha complexes. The algorithm is implemented and applied to proteins with known threedimensional conformations.