Results 1  10
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41
Approximate Range Searching
 in Proc. 11th Annu. ACM Sympos. Comput. Geom
, 1995
"... The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the ..."
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Cited by 86 (20 self)
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The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and ffl ? 0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance fflw of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in R d can be preprocessed in O(n log n) time and O(n) space, such that approximate queries can be answered in O(logn + (1=ffl) d ) time. The only assumption we make about ranges is that the intersection of a range and a ddimensional cube can be answered in const...
Range Searching
, 1996
"... Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , an ..."
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Cited by 70 (1 self)
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Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a rangesearching problem. A typical rangesearching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.
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
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
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Cited by 33 (6 self)
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We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...
Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Cited by 30 (2 self)
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating
Spacetime tradeoffs for approximate spherical range counting
 In Proc. 16th ACMSIAM Sympos. Discrete Algorithms
, 2005
"... Abstract We present spacetime tradeoffs for approximate spherical range counting queries. Given a set S of n data points in Rdalong with a positive approximation factor ffl, the goal is to preprocess the points so that, given any Euclidean ball B,we can return the number of points of any subset of ..."
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Cited by 24 (10 self)
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Abstract We present spacetime tradeoffs for approximate spherical range counting queries. Given a set S of n data points in Rdalong with a positive approximation factor ffl, the goal is to preprocess the points so that, given any Euclidean ball B,we can return the number of points of any subset of S that contains all the points within a (1 ffl)factor contraction ofB, but contains no points that lie outside a (1 + ffl)factor expansion of B.In many applications of range searching it is desirable to offer a tradeoff between space and query time. Wepresent here the first such tradeoffs for approximate range counting queries. Given 0 < ffl < = 1/2 and a parameterfl, where 2 < = fl < = 1/ffl, we show how to construct a data structure of space O(nfld log(1/ffl)) that allows us toanswer fflapproximate spherical range counting queries in time O(log(nfl) + 1/(fflfl)d1). The data structure can be built in time O(nfld log(n/ffl) log(1/ffl)). Here n, ffl, and fl areasymptotic quantities, and the dimension d is assumed to be a fixed constant.At one extreme (low space), this yields a data structure of space O(n log(1/ffl)) that can answer approximate range queries in time O(log n + (1/ffl)d1) which, up to a factorof O(log 1/ffl) in space, matches the best known result
SemiOnline Maintenance of Geometric Optima and Measures
, 2003
"... We give the first nontrivial worstcase results for dynamic versions of various basic geometric optimization and measure problems under the semionline model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time i ..."
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Cited by 19 (6 self)
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We give the first nontrivial worstcase results for dynamic versions of various basic geometric optimization and measure problems under the semionline model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1center, largest empty circle, convex hull volume in three dimensions, volume of the union of axisparallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1center problems can be solved fully dynamically. Some applications are mentioned.
Randomized incremental construction of threedimensional convex hulls and planar Voronoi diagrams, and approximate range counting
 in Proceedings of the Seventeenth Annual ACMSIAM Symposium on Discrete Algorithms
, 2006
"... We present new algorithms for approximate range counting, where, for a specified ε> 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [12], for this task, and then specialize it to two important instan ..."
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Cited by 19 (7 self)
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We present new algorithms for approximate range counting, where, for a specified ε> 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [12], for this task, and then specialize it to two important instances of range counting: halfspaces in R 3 and disks in the plane. The technique reduces the approximate range counting problem to that of finding the minimum rank of a data object in the range, with respect to a random permutation of the input. A major technical step in our analysis, which we believe to be of independent interest, is a bound of O(n log n) on the expected complexity of the overlay of all the Voronoi faces that are generated during a randomized incremental construction of the Voronoi diagram of n points in the plane. The same bound holds for the expected complexity of the overlay of all the faces of the minimization diagram of the lower envelope of n planes in R 3, or for the expected complexity of the overlay of all the normal (or Gaussian) diagram faces of the convex hull of n points in R 3, that are generated during a randomized incremental construction of the lower envelope or of the hull, respectively. All these bounds are tight in the worst case.
Fast algorithms for collision and proximity problems involving moving geometric objects
 Comput. Geom. Theory Appl
, 1996
"... Consider a set of geometric objects, such as points, line segments, or axesparallel hyperrectangles in IR d, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining wheth ..."
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Cited by 17 (1 self)
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Consider a set of geometric objects, such as points, line segments, or axesparallel hyperrectangles in IR d, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum interpoint separation or minimum diameter that ever occurs. In particular, two open problems from the literature are solved: Deciding in o(n 2) time if there is a collision in a set of n moving points in IR 2, where the points move at constant but possibly different velocities, and the analogous problem for detecting a redblue collision between sets of red and blue moving points. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search. 1
Optimal Partition Trees
, 2010
"... We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG’92, Matouˇsek gave a partition tree method for ddimensional simplex range searching achieving O(n) space and O(n 1−1/d) query time. Although this method is generally be ..."
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Cited by 15 (2 self)
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We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG’92, Matouˇsek gave a partition tree method for ddimensional simplex range searching achieving O(n) space and O(n 1−1/d) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n 1+ε) preprocessing time for any fixed ε> 0. An earlier method by Matouˇsek (SoCG’91) requires O(n log n) preprocessing time but O(n1−1/d log O(1) n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n1−1/d) query time with high probability. Our method has several advantages: • It is conceptually simpler than Matouˇsek’s SoCG’92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children’s cells at each node). • It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods). • A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n 1−1/⌊d/2 ⌋ ) query time for halfspace range emptiness in even dimensions d ≥ 4. Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points,...). 1