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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...
Geometric Pattern Matching under Euclidean Motion
, 1993
"... Given two planar sets A and B, we examine the problem of determining the smallest " such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance " of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem i ..."
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Cited by 73 (2 self)
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Given two planar sets A and B, we examine the problem of determining the smallest " such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance " of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem in modelbased computer vision, when the sets A and B contain points, line segments, or (filledin) polygons. We also show how to use our methods to substantially improve on existing algorithms for finding the minimum Hausdorff distance under Euclidean motion. 1 Author's address: Department of Computer Science, Cornell University, Ithaca, NY 14853. This work was supported by the Advanced Research Projects Agency of the Department of Defense under ONR Contract N0001492J1989, and by ONR Contract N0001492J1839, NSF Contract IRI9006137, and AFOSR Contract AFOSR910328. 2 Author's address: Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218. This work was suppo...
Geometric Applications of a Randomized Optimization Technique
 Discrete Comput. Geom
, 1999
"... We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, ..."
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Cited by 53 (6 self)
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We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal kpoint subsets, matching point sets under translation, computing rectilinear pcenters and discrete 1centers, and solving linear programs with k violations. 1 Introduction Consider the classic randomized algorithm for finding the minimum of r numbers minfA[1]; : : : ; A[r]g: Algorithm randmin 1. randomly pick a permutation hi 1 ; : : : ; i r i of h1; : : : ; ri 2. t /1 3. for k = 1; : : : ; r do 4. if A[i k ] ! t then 5. t / A[i k ] 6. return t By a wellknown fact [27, 44], the expected number of times that step 5 is execut...
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.
Efficient PiecewiseLinear Function Approximation Using the Uniform Metric
 Discrete & Computational Geometry
, 1994
"... We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to ..."
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Cited by 38 (0 self)
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We give an O(n log n)time method for finding a best klink piecewiselinear function approximating an npoint planar data set using the wellknown uniform metric to measure the error, ffl 0, of the approximation. Our method is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in "ffl space" followed by several applications of the parametric searching technique. The previous best running time for this problem was O(n 2 ). 1 Introduction Approximating a set S = f(x 1 ; y 1 ); (x 2 ; y 2 ); : : : ; (x n ; y n )g of points in the plane by a function is a classic problem in applied mathematics. The general goals in this area of research are to find a function F belonging to a class of functions F such that each F 2 F is simple to describe, represent, and compute and such that the chosen F approximates S well. For example, one may desire that F be the class of linear or piecewiselinear functions, and, for any parti...
Approximate geometric pattern matching under rigid motions
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 1999
"... We present techniques for matching pointsets in two and three dimensions under rigidbody transformations. We prove bounds on the worstcase performance of these algorithms to be within a small constant factor of optimal and conduct experiments to show that the average performance of these matchin ..."
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Cited by 30 (0 self)
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We present techniques for matching pointsets in two and three dimensions under rigidbody transformations. We prove bounds on the worstcase performance of these algorithms to be within a small constant factor of optimal and conduct experiments to show that the average performance of these matching algorithms is often better than that predicted by the worstcase bounds.
Optimal slope selection via cuttings
 Computational Geometry: Theory and Applications
, 1998
"... Abstract We give an optimal deterministic O(n log n)time algorithm for slope selection. The algorithm borrows from the optimal solution given in [?], but avoids the complicated machinery of the AKS sorting network and parametric searching. This is achieved by redesigning and refining the O(n log2 n ..."
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Cited by 23 (0 self)
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Abstract We give an optimal deterministic O(n log n)time algorithm for slope selection. The algorithm borrows from the optimal solution given in [?], but avoids the complicated machinery of the AKS sorting network and parametric searching. This is achieved by redesigning and refining the O(n log2 n)time algorithm of [?] with the help of additional approximation tools. 1 Optimal Slope Selection The problem is computing the line defined by two of n given points that has the median slope among all \Gamma n2 \Delta such lines. Equivalently, the problem can be stated as that of selecting the medianabscissa vertex of the arrangement A(L) of a set L of n lines [?]. For generality, we set out to compute the vertex with rank I\Lambda from left to right, for any given 1 ^ I \Lambda ^ \Gamma n2 \Delta.
Kinetic Medians and kdTrees
, 2002
"... We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor. ..."
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Cited by 22 (8 self)
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We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor.
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.