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42
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.
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
MinimumVolume Enclosing Ellipsoids and Core Sets
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 2005
"... We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly improved ..."
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Cited by 25 (4 self)
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We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly improved complexity bound of O(nd (0, 1). As a byproduct, our algorithm returns a core set with the property that the minimum volume enclosing ellipsoid of provides a good approximation to that of S.
Private coresets
, 2009
"... A coreset of a point set P is a small weighted set of points that captures some geometric properties of P. Coresets have found use in a vast host of geometric settings. We forge a link between coresets, and differentially private sanitizations that can answer any number of queries without compromisi ..."
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Cited by 18 (2 self)
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A coreset of a point set P is a small weighted set of points that captures some geometric properties of P. Coresets have found use in a vast host of geometric settings. We forge a link between coresets, and differentially private sanitizations that can answer any number of queries without compromising privacy. We define the notion of private coresets, which are simultaneously both coresets and differentially private, and show how they may be constructed. We first show that the existence of a small coreset with low generalized sensitivity (i.e., replacing a single point in the original point set slightly affects the quality of the coreset) implies (in an inefficient manner) the existence of a private coreset for the same queries. This greatly extends the works of Blum, Ligett, and Roth [STOC 2008] and McSherry and Talwar [FOCS 2007]. We also give an efficient algorithm to compute private coresets for kmedian and kmean queries in ℜ d, immediately implying efficient differentially private sanitizations for such queries. Following McSherry and Talwar, this construction also gives efficient coalition proof (approximately dominant strategy) mechanisms for location problems. Unlike coresets which only have a multiplicative approximation factor, we prove that private coresets must exhibit additive error. We present a new technique for showing lower bounds on this error.
A SpaceOptimal DataStream Algorithm for Coresets in the Plane
"... Given a point set P ⊆ R², a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the (1+ε)expansion of W also contains P. We present a datastream algorithm for maintaining an εkernel of a stream of points in R² that uses O(1/√ε) space and takes O(log(1/ε)) amortized time to process ..."
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Cited by 15 (5 self)
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Given a point set P ⊆ R², a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the (1+ε)expansion of W also contains P. We present a datastream algorithm for maintaining an εkernel of a stream of points in R² that uses O(1/√ε) space and takes O(log(1/ε)) amortized time to process each point. This is the first spaceoptimal datastream algorithm for this problem. As a consequence, we obtain improved datastream approximation algorithms for other extent measures, such as width, robust kernels, as well as εkernels in higher dimensions.
Approximate Range Searching: The Absolute Model
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 2009
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Geometric optimization problems over sliding windows
 Internat. J. Comput. Geom. Appl
, 2004
"... Abstract. We study the problem of maintaining a (1+ffl)factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store O( 1 ffl log R) points at any time, where the parameter R denotes the "spread ..."
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Cited by 13 (0 self)
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Abstract. We study the problem of maintaining a (1+ffl)factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store O( 1 ffl log R) points at any time, where the parameter R denotes the "spread " of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang's recent solution by two logarithmic factors. We then extend our onedimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constantfactor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the twodimensional case.
On the minimum volume covering ellipsoid of ellipsoids
 SIAM Journal on Optimization
, 2006
"... We study the problem of computing a (1+ɛ)approximation to the minimum volume covering ellipsoid of a given set S of the convex hull of m fulldimensional ellipsoids in R n. We extend the firstorder algorithm of Kumar and Yıldırım that computes an approximation to the minimum volume covering ellips ..."
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Cited by 11 (2 self)
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We study the problem of computing a (1+ɛ)approximation to the minimum volume covering ellipsoid of a given set S of the convex hull of m fulldimensional ellipsoids in R n. We extend the firstorder algorithm of Kumar and Yıldırım that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in R n, which, in turn, is a modification of Khachiyan’s algorithm. For fixed ɛ> 0, we establish a polynomialtime complexity, which is linear in the number of ellipsoids m. In particular, the iteration complexity of our algorithm is identical to that for a set of m points. The main ingredient in our analysis is the extension of polynomialtime complexity of certain subroutines in the algorithm from a set of points to a set of ellipsoids. As a byproduct, our algorithm returns a finite “core ” set X ⊆ S with the property that the minimum volume covering ellipsoid of X provides a good approximation to that of S. Furthermore, the size of X depends only on the dimension n and ɛ, but not on the number of ellipsoids m. We also discuss the extent to which our algorithm can be used to compute the minimum volume covering ellipsoid of the convex hull of other sets in R n. We adopt the real number model of computation in our analysis.
Robust shape fitting via peeling and grating coresets
 In Proc. 17th ACMSIAM Sympos. Discrete Algorithms
, 2006
"... Let P be a set of n points in R d. A subset S of P is called a (k, ε)kernel if for every direction, the direction width of S εapproximates that of P, when k “outliers ” can be ignored in that direction. We show that a (k, ε)kernel of P of size O(k/ε (d−1)/2) can be computed in time O(n+k 2 /ε d−1 ..."
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Cited by 10 (3 self)
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Let P be a set of n points in R d. A subset S of P is called a (k, ε)kernel if for every direction, the direction width of S εapproximates that of P, when k “outliers ” can be ignored in that direction. We show that a (k, ε)kernel of P of size O(k/ε (d−1)/2) can be computed in time O(n+k 2 /ε d−1). The new algorithm works by repeatedly “peeling” away (0, ε)kernels from the point set. We also present a simple εapproximation algorithm for fitting various shapes through a set of points with at most k outliers. The algorithm is incremental and works by repeatedly “grating ” critical points into a working set, till the working set provides the required approximation. We prove that the size of the working set is independent of n, and thus results in a simple and practical, nearlinear εapproximation algorithm for shape fitting with outliers in low dimensions. We demonstrate the practicality of our algorithms by showing their empirical performance on various inputs and problems. 1
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
, 2008
"... Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bou ..."
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Cited by 10 (3 self)
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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a threedimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higherdimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this longstanding limitation, answering, for example, a question of Willard (SODA’92).