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39
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
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.
Geometric approximation via coresets
 Combinatorial and Computational Geometry, MSRI
, 2005
"... Abstract. The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem o ..."
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Cited by 60 (7 self)
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Abstract. The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem on Q using a relatively inefficient algorithm. The solution for Q is then translated to an approximate solution to the original point set P. This paper describes the ways in which this paradigm has been successfully applied to various optimization and extent measure problems. 1.
Hellytype theorems and generalized linear programming
 DISCRETE COMPUT. GEOM
, 1994
"... This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use the ..."
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Cited by 60 (0 self)
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This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems which we call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use these results to explore the class GLP and show new applications to geometric optimization, and also to prove Helly theorems. In general, a GLP is a set...
A discrete subexponential algorithm for parity games
 STACS’03
, 2003
"... We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly ..."
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Cited by 34 (8 self)
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We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly
Maintaining Approximate Extent Measures of Moving Points
 In Proc. 12th ACMSIAM Sympos. Discrete Algorithms
, 2001
"... We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smal ..."
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Cited by 31 (4 self)
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We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smallest enclosing disk, width, and smallest area or perimeter bounding rectangle of a set of moving points in R 2 so that the number of events is only a constant. This contrasts with\Omega\Gamma n 2 ) events that data structures for the maintenance of those exact properties have to handle. 1 Introduction With the rapid advances in positioning systems, e.g., GPS, adhoc networks, and wireless communication, it is becoming increasingly feasible to track and record the changing position of continuously moving objects. These developments have raised a wide range of challenging geometric problems involving moving objects, including efficient data structures for answering proximity queries, fo...
Practical Methods for Shape Fitting and Kinetic Data Structures using Core Sets
 In Proc. 20th Annu. ACM Sympos. Comput. Geom
, 2004
"... The notion of εkernel was introduced by Agarwal et al. [5] to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the expanded slab (1 + ε)W contains P. They illustrat ..."
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Cited by 27 (8 self)
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The notion of εkernel was introduced by Agarwal et al. [5] to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the expanded slab (1 + ε)W contains P. They illustrated the significance of εkernel by showing that it yields approximation algorithms for a wide range of geometric optimization problems. We present a simpler and more practical algorithm for computing the εkernel of a set P of points in R d. We demonstrate the practicality of our algorithm by showing its empirical performance on various inputs. We then describe an incremental algorithm for fitting various shapes and use the ideas of our algorithm for computing εkernels to analyze the performance of this algorithm. We illustrate the versatility and practicality of this technique by implementing approximation algorithms for minimum enclosing cylinder, minimumvolume bounding box, and minimumwidth annulus. Finally, we show that εkernels can be effectively used to expedite the algorithms for maintaining extents of moving points. 1
Linear programming  randomization and abstract frameworks
 In Proc. 13th annu. Symp. on Theoretical Aspects of Computer Science (STACS
, 1996
"... Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this point is a randomized ‘combinatorial ’ algorithm which solves a linear program over d variables and n constraints with expected O(d 2 n + e O( d ..."
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Cited by 24 (9 self)
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Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this point is a randomized ‘combinatorial ’ algorithm which solves a linear program over d variables and n constraints with expected O(d 2 n + e O( d log d)) arithmetic operations. The bound relies on two algorithms by Clarkson, and the subexponential algorithms due to Kalai, and to Matouˇsek, Sharir & Welzl. frameworks like LPtype problems and abstract optimization problems (due to Gärtner) which allow the application of these algorithms to a number of nonlinear optimization problems (like polytope distance and smallest enclosing ball of points).
The smallest enclosing ball of balls: combinatorial structure and algorithms
 International Journal of Computational Geometry & Application
, 2004
"... We develop algorithms for computing the smallest enclosing ball of a set of n balls in ddimensional space. Unlike previous methods, we explicitly address small cases (n ≤ d +1), derive the necessary primitive operations and show that they can efficiently be realized with rational arithmetic. An exa ..."
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Cited by 24 (3 self)
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We develop algorithms for computing the smallest enclosing ball of a set of n balls in ddimensional space. Unlike previous methods, we explicitly address small cases (n ≤ d +1), derive the necessary primitive operations and show that they can efficiently be realized with rational arithmetic. An exact implementation (along with a fast 1 and robust floatingpoint version) is available as part of the CGAL library. 2 Our algorithms are based on novel insights into the combinatorial structure of the problem. As it turns out, results for smallest enclosing balls of points do not extend as one might expect. For example, we show that Welzl’s randomized lineartime algorithm for computing the ball spanned by a set of points fails to work for balls. Consequently, David White’s adaptation of the method to the ball case—as the only available implementation so far it is mentioned in many link collections—is incorrect and may crash or, in the better case, produce wrong balls. In solving the small cases we may assume that the ball centers are affinely independent; in this case, the problem is surprisingly wellbehaved: via a geometric transformation and suitable generalization, it fits into the combinatorial model of unique sink orientations whose rich structure has recently received considerable attention. One consequence is that Welzl’s algorithm does work for small instances; moreover, there is a wide variety of pivoting methods for unique sink orientations which have the potential of being fast in practice even for high dimensions. Partly supported by the IST Programme of the EU and