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Approximating extent measure of points
 Journal of ACM
"... We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our tec ..."
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Cited by 96 (28 self)
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We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our technique include�approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms. 1
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...
Coordination And Geometric Optimization Via Distributed Dynamical Systems
, 2003
"... This paper discusses dynamical systems for diskcovering and spherepacking problems. We present facility location functions from geometric optimization and characterize their differentiable properties. We design and analyze a collection of distributed control laws that are related to nonsmooth grad ..."
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Cited by 63 (26 self)
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This paper discusses dynamical systems for diskcovering and spherepacking problems. We present facility location functions from geometric optimization and characterize their differentiable properties. We design and analyze a collection of distributed control laws that are related to nonsmooth gradient systems. The resulting dynamical systems promise to be of use in coordination problems for networked robots; in this setting the distributed control laws correspond to local interactions between the robots. The technical approach relies on concepts from computational geometry, nonsmooth analysis, and the dynamical system approach to algorithms.
Exact and Approximation Algorithms for Clustering
, 1997
"... In this paper we present a n O(k 1\Gamma1=d ) time algorithm for solving the kcenter problem in R d , under L1 and L 2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete kcenter problem, as well. We also describe a simple (1 + ffl)approximation algorith ..."
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Cited by 57 (5 self)
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In this paper we present a n O(k 1\Gamma1=d ) time algorithm for solving the kcenter problem in R d , under L1 and L 2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete kcenter problem, as well. We also describe a simple (1 + ffl)approximation algorithm for the kcenter problem, with running time O(n log k) + (k=ffl) O(k 1\Gamma1=d ) . Finally, we present a n O(k 1\Gamma1=d ) time algorithm for solving the Lcapacitated kcenter problem, provided that L = \Omega\Gamma n=k 1\Gamma1=d ) or L = O(1). We conclude with a simple approximation algorithm for the Lcapacitated kcenter problem. The work on this paper was partially supported by a National Science Foundation Grant CCR9301259, by an Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award and matching funds from Xerox Corporation, and by a grant from the U.S.Israeli Binational Science Foundation. y Department of Computer Science, Box ...
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...
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.
LowDimensional Linear Programming with Violations
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given half ..."
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Cited by 46 (3 self)
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Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3d runs in near O(n + k ) expected time. Interestingly, the idea is based on concavechain decompositions (or covers) of the ( k)level, previously used in proving combinatorial klevel bounds.
Approximation Algorithms for kLine Center
, 2002
"... Given a set P of n points in Rd and an integer k> = 1, let w * denote the minimumvalue so that P can be covered by k cylinders of width at most w*. We describe analgorithm that, given P and an "> 0, computes k cylinders of width at most (1 + ")w*that cover P. The running time of the algori ..."
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Cited by 34 (5 self)
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Given a set P of n points in Rd and an integer k> = 1, let w * denote the minimumvalue so that P can be covered by k cylinders of width at most w*. We describe analgorithm that, given P and an "> 0, computes k cylinders of width at most (1 + ")w*that cover P. The running time of the algorithm is O(n log n), with the constant ofproportionality depending on k, d, and ". The running times of the fastest algorithmsthat compute w * exactly are of the order of nO(dk). An approximation algorithm withnearlinear dependence on n for k> 1 was only known for the planar 2line centerproblem, i.e., the case k = 2, d = 2.We believe that the techniques used in showing this result are quite useful in themselves. We first show that there exists a small "certificate " Q ` P, whose size doesnot depend on n, such that for any kcylinders that cover Q, an enlargement of thesecylinders by a factor of (1 + ") covers P. We only establish the existence of a small certificate and our proof does not give us an efficient way of constructing one. We then observe that a wellknown scheme based on sampling and iterated reweighting gives usan efficient algorithm for solving the problem. Only the existence of a small certificate is used to establish the correctness of the algorithm. This technique is quite generaland can be used in other contexts as well.
Polynomial Time Approximation Schemes for Geometric kClustering
 J. OF THE ACM
, 2001
"... The JohnsonLindenstrauss lemma states that n points in a high dimensional Hilbert space can be embedded with small distortion of the distances into an O(log n) dimensional space by applying a random linear transformation. We show that similar (though weaker) properties hold for certain random linea ..."
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Cited by 30 (5 self)
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The JohnsonLindenstrauss lemma states that n points in a high dimensional Hilbert space can be embedded with small distortion of the distances into an O(log n) dimensional space by applying a random linear transformation. We show that similar (though weaker) properties hold for certain random linear transformations over the Hamming cube. We use these transformations to solve NPhard clustering problems in the cube as well as in geometric settings. More specifically, we address the following clustering problem. Given n points in a larger set (for example, R^d) endowed with a distance function (for example, L² distance), we would like to partition the data set into k disjoint clusters, each with a "cluster center", so as to minimize the sum over all data points of the distance between the point and the center of the cluster containing the point. The problem is provably NPhard in some high dimensional geometric settings, even for k = 2. We give polynomial time approximation schemes for this problem in several settings, including the binary cube {0, 1}^d with Hamming distance, and R^d either with L¹ distance, or with L² distance, or with the square of L² distance. In all these settings, the best previous results were constant factor approximation guarantees. We note that our problem is similar in flavor to the kmedian problem (and the related facility location problem), which has been considered in graphtheoretic and fixed dimensional geometric settings, where it becomes hard when k is part of the input. In contrast, we study the problem when k is fixed, but the dimension is part of the input.