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49
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...
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 94 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
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...
Approximating center points with iterated Radon points
 Internat. J. Comput. Geom. Appl
, 1996
"... We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d. A point c ∈ IR d is a βcenter point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d + 1)center point; our algori ..."
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Cited by 55 (10 self)
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We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d. A point c ∈ IR d is a βcenter point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d + 1)center point; our algorithm finds an Ω(1/d 2)center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log 2 d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak ɛnets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly. 1
An Optimal Randomized Algorithm for Maximum Tukey Depth
, 2004
"... We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a nondegenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d 3, the expected tim ..."
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Cited by 46 (4 self)
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We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a nondegenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d 3, the expected time bound is O(n ), which is probably optimal as well. The result is obtained using an interesting variant of the author's randomized optimization technique, capable of solving "implicit" linearprogrammingtype problems; some other applications of this technique are briefly mentioned.
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Point Sets With Many KSets
, 1999
"... For any n, k, n 2k > 0, we construct a set of n points in the plane with ne p log k ksets. This improves the bounds of Erd}os, Lovasz, et al. As a consequence, we also improve the lower bound of Edelsbrunner for the number of halving hyperplanes in higher dimensions. 1 Introduction For a ..."
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Cited by 45 (0 self)
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For any n, k, n 2k > 0, we construct a set of n points in the plane with ne p log k ksets. This improves the bounds of Erd}os, Lovasz, et al. As a consequence, we also improve the lower bound of Edelsbrunner for the number of halving hyperplanes in higher dimensions. 1 Introduction For a set P of n points in the ddimensional space, a kset is subset P 0 P such that P 0 = P \H for some halfspace H, and jP 0 j = k. The problem is to determine the maximum number of ksets of an npoint set in the ddimensional space. Even in the most studied two dimensional case, we are very far from the solution, and in higher dimensions even much less is known. The rst results in the two dimensional case are due to Erd}os, Lovasz, Simmons and Straus [L71], [ELSS73]. They established an upper bound O(n p k), and a lower bound (n log k). Despite great interest in this problem [W86], [E87], [S91], [EVW97], [AACS98], partly due to its importance in the analysis of geometric alg...
Constructing Levels in Arrangements and Higher Order Voronoi Diagrams
 SIAM J. COMPUT
, 1994
"... We give simple randomized incremental algorithms for computing the klevel in an arrangement of n hyperplanes in two and threedimensional space. The expected running time of our algorithms is O(nk+nff(n) log n) for the planar case, and O(nk 2 +n log 3 n) for the threedimensional case. Both bo ..."
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Cited by 42 (10 self)
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We give simple randomized incremental algorithms for computing the klevel in an arrangement of n hyperplanes in two and threedimensional space. The expected running time of our algorithms is O(nk+nff(n) log n) for the planar case, and O(nk 2 +n log 3 n) for the threedimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the klevel in an arrangement of discs or xmonotone Jordan curves in the plane. Our approach can also be used to compute the klevel; this yields a randomized algorithm for computing the orderk Voronoi diagram of n points in the plane in expected time O(k(n \Gamma k) log n + n log 3 n).
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
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Cited by 33 (6 self)
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We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...
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.