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Geometric Range Searching and Its Relatives
 CONTEMPORARY MATHEMATICS
"... ... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems. ..."
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Cited by 256 (40 self)
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... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.
On approximating the depth and related problems
 SIAM J. Comput
"... 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 63 (11 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. 1
FixedDimensional Linear Programming Queries Made Easy
 Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... We derive two results from Clarkson's randomized algorithm for linear programming in a fixed dimension d. The first is a simple general method that reduces the problem of answering linear programming queries to the problem of answering halfspace range queries. For example, this yields a randomized ..."
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Cited by 35 (8 self)
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We derive two results from Clarkson's randomized algorithm for linear programming in a fixed dimension d. The first is a simple general method that reduces the problem of answering linear programming queries to the problem of answering halfspace range queries. For example, this yields a randomized data structure with O(n) space and O(n 1\Gamma1=bd=2c 2 O(log n) ) query time for linear programming on n halfspaces (d ? 3). The second result is a simpler proof of the following: a sequence of q linear programming queries on n halfspaces can be answered in O(n log q) time, if q n ff d for a certain constant ff d ? 0. Unlike previous methods, our algorithms do not require parametric searching. 1 Introduction One of the major discoveries in computational geometry is that fixeddimensional linear programming can be solved in linear time [Meg84]. It was observed that the introduction of randomization leads to considerably simpler solutions [Sei91, Cla95]. The goal of this paper is...
Random Sampling, Halfspace Range Reporting, and Construction of (≤k)Levels in Three Dimensions
 SIAM J. COMPUT
, 1999
"... Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the co ..."
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Cited by 33 (7 self)
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Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the orderk Voronoi diagram in O(n log n + nk log k) expected time.
Almost tight upper bounds for vertical decompositions in four dimensions
 In Proc. 42nd IEEE Symposium on Foundations of Computer Science
, 2001
"... We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major problem i ..."
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Cited by 31 (6 self)
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We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n 2d−4+ε), for any ε> 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems. 1
Arrangements
, 1997
"... INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes ..."
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Cited by 28 (13 self)
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INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes have served as a unifying structure for many problems in discrete and computational geometry. With the recent advances in the study of arrangements of curved (algebraic) surfaces, arrangements have emerged as the underlying structure of geometric problems in a variety of `physical world' application domains such as robot motion planning and computer vision. This chapter is devoted to arrangements of hyperplanes and of curved surfaces in lowdimensional Euclidean space, with an emphasis on combinatorics and algorithms. In the first section we in
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 klevel of the a ..."
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Cited by 28 (11 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.
Dynamic data structures for fat objects and their applications
, 2000
"... We present several efficient dynamic data structures for pointenclosure queries, involving convex fat objects in R² or R³. Our planar structures are actually fitted for a more general class of objects – (β, δ)covered objects – which are not necessarily convex, see definition below. These structure ..."
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Cited by 26 (12 self)
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We present several efficient dynamic data structures for pointenclosure queries, involving convex fat objects in R² or R³. Our planar structures are actually fitted for a more general class of objects – (β, δ)covered objects – which are not necessarily convex, see definition below. These structures are more efficient than alternative known structures, because they exploit the fatness of the objects. We then apply these structures to obtain efficient solutions to two problems: (i) finding a perfect containment matching between a set of points and a set of convex fat objects, and (ii) finding a piercing set for a collection of convex fat objects, whose size is optimal up to some constant
Boundary labeling: Models and efficient algorithms for rectangular maps
, 2004
"... Abstract. In this paper, we present boundary labeling, a new approach for labeling point sets with large labels. We first place disjoint labels around an axisparallel rectangle that contains the points. Then we connect each label to its point such that no two connections intersect. Such an approach ..."
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Cited by 24 (7 self)
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Abstract. In this paper, we present boundary labeling, a new approach for labeling point sets with large labels. We first place disjoint labels around an axisparallel rectangle that contains the points. Then we connect each label to its point such that no two connections intersect. Such an approach is common e.g. in technical drawings and medical atlases, but so far the problem has not been studied in the literature. The new problem is interesting in that it is a mixture of a labelplacement and a graphdrawing problem. 1