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64
Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (19 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
How good are convex hull algorithms?
, 1996
"... A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facetinducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are esse ..."
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Cited by 82 (8 self)
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A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facetinducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in jHj + jVj. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability todealwell with "degeneracies," or, inability tocontrol the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fatlattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 78 (22 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Product Range Spaces, Sensitive Sampling, and Derandomization
, 1993
"... We introduce the concept of a sensitive Eapproximation, and use it to derive a more efficient algorithm for computing &nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VCdimensional case. This generalizes and simpli ..."
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Cited by 46 (6 self)
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We introduce the concept of a sensitive Eapproximation, and use it to derive a more efficient algorithm for computing &nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VCdimensional case. This generalizes and simplifies results from previous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(n log3 n) deterministic algorithm for computing the diameter of an npoint set in 3dimensional space.
On Bregman Voronoi Diagrams
 in "Proc. 18th ACMSIAM Sympos. Discrete Algorithms
, 2007
"... The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi ..."
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Cited by 42 (22 self)
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The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a byproduct, Bregman Voronoi diagrams allow one to define informationtheoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other
Efficient Approximation and Optimization Algorithms for Computational Metrology
 PROC. 8TH ACMSIAM SYMPOS. DISCRETE ALGORITHMS
, 1997
"... We give efficient algorithms for solving several geometric problems in computational metrology, focusing on the fundamental issues of "flatness" and "roundness." Specifically, we give approximate and exact algorithms for 2 and 3dimensional roundness primitives, deriving results that improve previo ..."
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Cited by 34 (1 self)
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We give efficient algorithms for solving several geometric problems in computational metrology, focusing on the fundamental issues of "flatness" and "roundness." Specifically, we give approximate and exact algorithms for 2 and 3dimensional roundness primitives, deriving results that improve previous approaches in several respects, including problem definition, running time, underlying computational model, and dimensionality of the input. We also study methods for determining the width of a ddimensional point set, which corresponds to the metrology notion of "flatness," giving an approximation method that can serve as a fast exactcomputation filter for this metrology primitive. Finally, we report on experimental results derived from implementation and testing, particularly in 3space, of our approximation algorithms, including several heuristics designed to significantly speedup the computations in practice.
Random Sampling, Halfspace Range Reporting, and Construction of (≤k)Levels in Three Dimensions
 SIAM J. COMPUT
, 1999
"... Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the co ..."
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Cited by 33 (7 self)
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Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the orderk Voronoi diagram in O(n log n + nk log k) expected time.
On efficient representation and computation of reachable sets for hybrid systems
 In HSCC’2003, LNCS 2289
, 2003
"... Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approxim ..."
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Cited by 28 (6 self)
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Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the overapproximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.
Randomized ExternalMemory Algorithms for Some Geometric Problems
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2001
"... We show that the wellknown random incremental construction of Clarkson and Shor [14] can be adapted via gradations to provide efficient externalmemory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computin ..."
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Cited by 27 (3 self)
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We show that the wellknown random incremental construction of Clarkson and Shor [14] can be adapted via gradations to provide efficient externalmemory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line segments in the plane with K pairwise intersections, that requires \Theta( log M=B N ) expected disk accesses, where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to obtain algorithms also for the problems of 3d halfspace intersections, 2d and 3d convex hulls, 2d abstract Voronoi diagrams and batched planar point location, which require an optimal expected number of disk accesses and are simpler than the ones previously known. The results extend to an externalmemory model with multiple disks. Additionally, under reasonable conditions on the parameters N;M;B, these results can be notably simplified originating practical algorithms which still achieve optimal expected bounds.
New Lower Bounds for Convex Hull Problems in Odd Dimensions
 SIAM J. Comput
, 1996
"... We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follow ..."
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Cited by 26 (7 self)
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We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasisimplicial nvertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that ddimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is ⌈d/2⌉hard, in the in the sense of Gajentaan and Overmars.