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The effect of corners on the complexity of approximate range searching
 DISCRETE COMPUT GEOM (2009 ) 41 : 398–443 399
, 2009
"... Given an nelement point set in R d, the range searching problem involves preprocessing these points so that the total weight, or for our purposes the semigroup sum, of the points lying within a given query range η can be determined quickly. In εapproximate range searching we assume that η is bound ..."
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Cited by 12 (4 self)
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Given an nelement point set in R d, the range searching problem involves preprocessing these points so that the total weight, or for our purposes the semigroup sum, of the points lying within a given query range η can be determined quickly. In εapproximate range searching we assume that η is bounded, and the sum is required to include all the points that lie within η and may additionally include any of the points lying within distance ε · diam(η) of η’s boundary. In this paper we contrast the complexity of approximate range searching based on properties of the semigroup and range space. A semigroup (S, +) is idempotent if x + x = x for all x ∈ S, and it is integral if for all k ≥ 2, the kfold sum x +···+x is not equal to x. Recent research has shown that the computational complexity of approximate spherical range searching is significantly lower for idempotent semigroups than it is for integral semigroups in terms of the dependencies on ε. In this
Tradeoffs in Approximate Range Searching Made Simpler
 XXI BRAZILIAN SYMPOSIUM ON COMPUTER GRAPHICS AND IMAGE PROCESSING
, 2008
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Approximation Algorithm for the Kinetic Robust KCenter Problem
, 2009
"... Two complications frequently arise in realworld applications, motion and the contamination of data by outliers. We consider a fundamental clustering problem, the kcenter problem, within the context of these two issues. We are given a finite point set S of size n and an integer k. In the standard k ..."
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Cited by 4 (3 self)
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Two complications frequently arise in realworld applications, motion and the contamination of data by outliers. We consider a fundamental clustering problem, the kcenter problem, within the context of these two issues. We are given a finite point set S of size n and an integer k. In the standard kcenter problem, the objective is to compute a set of k center points to minimize the maximum distance from any point of S to its closest center, or equivalently, the smallest radius such that S can be covered by k disks of this radius. In the discrete kcenter problem the disk centers are drawn from the points of S, and in the absolute kcenter problem the disk centers are unrestricted. We generalize this problem in two ways. First, we assume that points are in continuous motion, and the objective is to maintain a solution over time. Second, we assume that some given robustness parameter 0 < t ≤ 1 is given, and the objective is to compute the smallest radius such that there exist k disks of this radius that cover at least ⌈tn ⌉ points of S. We present a kinetic data structure (in the KDS framework) that maintains a (3 + ε)approximation for the robust discrete kcenter problem and a (4 + ε)approximation for the robust absolute kcenter problem, both under the assumption that k is a constant. We also improve on a previous 8approximation for the nonrobust discrete kinetic kcenter problem, for arbitrary k, and show that our data structure achieves a (4 + ε)approximation. All these results hold in any metric space of constant doubling dimension, which includes Euclidean space of constant dimension.
Approximate Halfspace Range Counting
, 2008
"... We present a simple scheme extending the shallow partitioning data structures of Matouˇsek, that supports efficient approximate halfspace rangecounting queries in R d with relative error ε. Specifically, the problem is, given a set P of n points in R d, to preprocess them into a data structure that ..."
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Cited by 3 (3 self)
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We present a simple scheme extending the shallow partitioning data structures of Matouˇsek, that supports efficient approximate halfspace rangecounting queries in R d with relative error ε. Specifically, the problem is, given a set P of n points in R d, to preprocess them into a data structure that returns, for a query halfspace h, a number t so that (1−ε)h∩P  ≤ t ≤ (1+ε)h∩P . One of our data structures requires linear storage and O(n 1+δ) preprocessing time, for any δ> 0, and answers a query in time O ( ε −γ n 1−1/⌊d/2 ⌋ 2 blog ∗ n) , for any γ> 2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step. The query efficiency is guaranteed with high probability, for all queries. The construction can also be fully derandomized, at the expense of increasing preprocessing time.
Tight lower bounds for halfspace range searching
 In Symposium on Computational Geometry (2010
"... We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in R d, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting m denote the space ..."
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Cited by 3 (1 self)
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We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in R d, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting m denote the space requirements, we prove a lower bound for general semigroups of ˜ Ω ( n 1−1/(d+1) /m 1/(d+1)) and for integral semigroups of ˜ Ω ( n/m 1/d). Our lower bounds are proved in the semigroup arithmetic model. Neglecting logarithmic factors, our result for integralsemigroupsmatchesthebestknownupperbounddueto Matouˇsek. Our result for general semigroups improves upon the best known lower bound due to Brönnimann, Chazelle, and Pach. Moreover, Fonseca and Mount have recently shown that, given uniformly distributed points, halfspace range queries over idempotent semigroups can be answered in O ( n 1−1/(d+1) /m 1/(d+1)) time in the semigrouparithmetic model. As our lower bounds are established for uniformly distributed point sets, it follows that they also resolve the computational complexity of halfspace range searching over idempotent semigroups in this important special case.
Approximate Range Searching In The Absolute Error Model
, 2007
"... Range searching is a well known problem in computational geometry. We consider this problem in the context of approximation, where an approximation parameter ε > 0 is provided. Most prior work on this problem has focused on the relative error model, where each range shape R is bounded, and points wi ..."
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Cited by 2 (2 self)
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Range searching is a well known problem in computational geometry. We consider this problem in the context of approximation, where an approximation parameter ε > 0 is provided. Most prior work on this problem has focused on the relative error model, where each range shape R is bounded, and points within distance ε · diam(R) of the range’s boundary may or may not be included. We introduce a different approximation model, called the absolute error model, in which points within distance ε of the range’s boundary may or may not be included, regardless of the diameter of the range. We consider sets of ranges consisting of general convex bodies, axisaligned rectangles, halfspaces, Euclidean balls, and simplices. We examine a variety of problem formulations, including range searching under general commutative semigroups, idempotent semigroups, groups, range emptiness, and range reporting. We apply our data structures to several related problems, including range sketching, approximate nearest neighbor searching, exact idempotent range searching, approximate range searching in the data stream model, and
A Unified Approach to Approximate Proximity Searching
"... Abstract. The inability to answer proximity queries efficiently for spaces of dimension d>2 has led to the study of approximation to proximity problems. Several techniques have been proposed to address different approximate proximity problems. In this paper, we present a new and unified approach to ..."
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Abstract. The inability to answer proximity queries efficiently for spaces of dimension d>2 has led to the study of approximation to proximity problems. Several techniques have been proposed to address different approximate proximity problems. In this paper, we present a new and unified approach to proximity searching, which provides efficient solutions for several problems: spherical range queries, idempotent spherical range queries, spherical emptiness queries, and nearest neighbor queries. In contrast to previous data structures, our approach is simple and easy to analyze, providing a clear picture of how to exploit the particular characteristics of each of these problems. As applications of our approach, we provide simple and practical data structures that match the best previous results up to logarithmic factors, as well as advanced data structures that improve over the best previous results for all aforementioned proximity problems. 1
GEOMETRIC ALGORITHMS FOR OBJECTS IN MOTION
"... In this thesis, the theoretical analysis of realworld motivated problems regarding objects in motion is considered. Specifically, four major results are presented addressing the issues of robustness, data collection and compression, realistic theoretical analyses of this compression, and data retri ..."
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In this thesis, the theoretical analysis of realworld motivated problems regarding objects in motion is considered. Specifically, four major results are presented addressing the issues of robustness, data collection and compression, realistic theoretical analyses of this compression, and data retrieval. Robust statistics is the study of statistical estimators that are robust to data outliers. The combination of robust statistics and data structures for moving objects has not previously been studied. In studying this intersection, we consider a problem in the context of an established kinetic data structures framework (called KDS) that relies on advance point motion information and calculates properties continuously. Using the KDS model, we present an approximation algorithm for the kinetic robust kcenter problem, a clustering problem that requires k clusters but allows some outlying points to remain unclustered. For many practical problems that inspired the exploration into robustness, the KDS model is inapplicable due to the point motion restrictions and the advanceflight plans required. We present a new framework for kinetic data that allows calculations on moving points via sensorrecorded observations. This new framework
c ○ 2010 Society for Industrial and Applied Mathematics APPROXIMATE HALFSPACE RANGE COUNTING ∗
"... Abstract. We present a simple scheme extending the shallow partitioning data structures of Matouˇsek, which supports efficient approximate halfspace rangecounting queries in Rd with relative error ε. Specifically, the problem is, given a set P of n points in Rd,topreprocessthemintoadata structure t ..."
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Abstract. We present a simple scheme extending the shallow partitioning data structures of Matouˇsek, which supports efficient approximate halfspace rangecounting queries in Rd with relative error ε. Specifically, the problem is, given a set P of n points in Rd,topreprocessthemintoadata structure that returns, for a query halfspace h, anumbertsothat (1 − ε)h ∩ P ≤t ≤ (1 + ε)h ∩ P . One of our data structures requires linear storage and O(n1+δ) preprocessing time, for any δ>0, and answers a query in time O(ε−γ n1−1/⌊d/2⌋2b log ∗ n) for any γ>2/⌊d/2⌋; the choice of γ and δ affects b and the implied constants. Several variants and extensions are also discussed. As presented, the construction of the structure is mostly deterministic, except for one critical randomized step, and so are the query, storage, and preprocessing costs. The quality of approximation, for every query, is guaranteed with high probability. The construction can also be fully derandomized, at the expense of increasing preprocessing time.