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42
On approximating the depth and related problems
 SIAM J. COMPUT
, 2008
"... We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear pr ..."
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Cited by 70 (14 self)
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We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries.
Shape Fitting with Outliers
 SIAM J. Comput
, 2003
"... we present an algorithm that "approximates the extent between the top and bottom k levels of the arrangement of H in time O(n+(k=") ), where c is a constant depending on d. The algorithm relies on computing a subset of H of size O(k=" ), in near linear time, such that the k ..."
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Cited by 34 (10 self)
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we present an algorithm that "approximates the extent between the top and bottom k levels of the arrangement of H in time O(n+(k=") ), where c is a constant depending on d. The algorithm relies on computing a subset of H of size O(k=" ), in near linear time, such that the klevel of the arrangement of the subset approximates that of the original arrangement. Using this algorithm, we propose ecient approximation algorithms for shape tting with outliers for various shapes. This is the rst algorithms to handle outliers eciently for the shape tting problems considered.
A dynamic data structure for 3d convex hull and 2d nearest neighbor queries
 In: Proceedings of the seventeenth ACMSIAM symposium on Discrete algorithm
, 2006
"... We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log 3 n) expected amortized time, deletions take O(log 6 n) expected amortized time, and extremepoint queries take O(log 2 n) worstca ..."
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Cited by 27 (4 self)
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We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log 3 n) expected amortized time, deletions take O(log 6 n) expected amortized time, and extremepoint queries take O(log 2 n) worstcase time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(n ε)time method by Agarwal and Matouˇsek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane). 1
On approximate range counting and depth
 In Proc. 23rd Annu. ACM Sympos. Comput. Geom
, 2007
"... ABSTRACT We improve the previous results by Aronov and HarPeled (SODA'05) and Kaplan and Sharir (SODA'06) and present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in O(log n k) expected time, where k is the actual value ..."
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Cited by 23 (1 self)
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ABSTRACT We improve the previous results by Aronov and HarPeled (SODA'05) and Kaplan and Sharir (SODA'06) and present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in O(log n k) expected time, where k is the actual value of the count. This is the first optimal method for the problem in the standard decision tree model; moreover, unlike previous methods, the new method is Las Vegas instead of Monte Carlo. In addition, we describe new results for several related problems, including approximate Tukey depth queries in 3D, approximate regression depth queries in 2D, and approximate linear programming with violations in low dimensions. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problemsgeometrical problems and computations
Fast Algorithms for Computing the Smallest kEnclosing Disc
 In Proc. 11th Annu. European Sympos. Algorithms, volume 2832 of Lect. Notes in Comp. Sci
, 2003
"... We consider the problem of nding, for a given n point set P in the plane and an integer k n, the smallest circle enclosing at least k points of P . We present a randomized algorithm that computes in O(nk) expected time such a circle, improving over previously known algorithms. ..."
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Cited by 22 (4 self)
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We consider the problem of nding, for a given n point set P in the plane and an integer k n, the smallest circle enclosing at least k points of P . We present a randomized algorithm that computes in O(nk) expected time such a circle, improving over previously known algorithms.
Coloring Geometric Range Spaces
"... Given a set of points in R 2 or R 3, we aim to color them such that every region of a certain family (for instance disks) containing at least a certain number of points contains points of many different colors. Using k colors, it is not always possible to ensure that every region containing k points ..."
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Cited by 19 (8 self)
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Given a set of points in R 2 or R 3, we aim to color them such that every region of a certain family (for instance disks) containing at least a certain number of points contains points of many different colors. Using k colors, it is not always possible to ensure that every region containing k points contains all k colors. Thus, we introduce two relaxations: either we allow the number of colors to increase to c(k), or we require that the number of points in each region increases to p(k). We give upper bounds on c(k) and p(k) for halfspaces, disks, and pseudodisks. We also consider the dual question, where we want to color regions instead of points. This is related to previous results of Pach, Tardos and Tóth on decompositions of coverings.
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 16 (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
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
 In Proc. 44th IEEE Sympos. Found. Comput. Sci
, 2003
"... We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which ..."
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Cited by 13 (2 self)
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We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degrees polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved klevel results for most of the curve families studied earlier, including a nearO(n ) bound for parabolas.