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32
Applications of Random Sampling in Computational Geometry, II
 Discrete Comput. Geom
, 1995
"... We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms ..."
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Cited by 396 (12 self)
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We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divideandconquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...
Las Vegas algorithms for linear and integer programming when the dimension is small
 J. ACM
, 1995
"... Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algor ..."
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Cited by 103 (2 self)
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Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algorlthm N gwen for integer hnear programmmg. Let p bound the number of bits required to specify the ratmnal numbers defmmg an input constraint or the ob~ective function vector. Let n and d be as before. Then, the algorithm requires expected 0(2d dn + S~dm In n) + dc)’d) ~ in H operations on numbers with O(1~p bits d ~ ~ ~z + ~, where the constant factors do not depend on d or p. The expectations are with respect to the random choices made by the algorithms, and the bounds hold for any gwen input. The techmque can be extended to other convex programming problems. For example, m algorlthm for finding the smallest sphere enclosing a set of /z points m Ed has the same t]me bound
Sphere Packing Numbers for Subsets of the Boolean nCube with Bounded VapnikChervonenkis Dimension
, 1992
"... : Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn ..."
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Cited by 93 (4 self)
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: Let V ` f0; 1g n have VapnikChervonenkis dimension d. Let M(k=n;V ) denote the cardinality of the largest W ` V such that any two distinct vectors in W differ on at least k indices. We show that M(k=n;V ) (cn=(k + d)) d for some constant c. This improves on the previous best result of ((cn=k) log(n=k)) d . This new result has applications in the theory of empirical processes. 1 The author gratefully acknowledges the support of the Mathematical Sciences Research Institute at UC Berkeley and ONR grant N0001491J1162. 1 1 Statement of Results Let n be natural number greater than zero. Let V ` f0; 1g n . For a sequence of indices I = (i 1 ; . . . ; i k ), with 1 i j n, let V j I denote the projection of V onto I, i.e. V j I = f(v i 1 ; . . . ; v i k ) : (v 1 ; . . . ; v n ) 2 V g: If V j I = f0; 1g k then we say that V shatters the index sequence I. The VapnikChervonenkis dimension of V is the size of the longest index sequence I that is shattered by V [VC71] (t...
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.
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
On Rectangular Partitionings in Two Dimensions: Algorithms, Complexity, and Applications
 In Proceedings of the 7th International Conference on Database Theory
, 1999
"... . Partitioning a multidimensional data set into rectangular partitions subject to certain constraints is an important problem that arises in many database applications, including histogrambased selectivity estimation, loadbalancing, and construction of index structures. While provably optimal ..."
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Cited by 44 (7 self)
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. Partitioning a multidimensional data set into rectangular partitions subject to certain constraints is an important problem that arises in many database applications, including histogrambased selectivity estimation, loadbalancing, and construction of index structures. While provably optimal and efficient algorithms exist for partitioning onedimensional data, the multidimensional problem has received less attention, except for a few special cases. As a result, the heuristic partitioning techniques that are used in practice are not well understood, and come with no guarantees on the quality of the solution. In this paper, we present algorithmic and complexitytheoretic results for the fundamental problem of partitioning a twodimensional array into rectangular tiles of arbitrary size in a way that minimizes the number of tiles required to satisfy a given constraint. Our main results are approximation algorithms for several partitioning problems that provably approxima...
RAY SHOOTING AND OTHER APPLICATIONS OF SPANNING TREES WITH LOW STABBING NUMBER
, 1992
"... This paper considers the following problem: Given a set G of n (possibly intersecting) line segments in the plane, prcproccss it so that, given a query ray p emanating from a point p, one can quickly compute the intersection point &(G, p) of p with a segment of G that lies nearest to p. The paper pr ..."
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Cited by 32 (12 self)
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This paper considers the following problem: Given a set G of n (possibly intersecting) line segments in the plane, prcproccss it so that, given a query ray p emanating from a point p, one can quickly compute the intersection point &(G, p) of p with a segment of G that lies nearest to p. The paper presents an algorithm that preproccsses G, in time 0 ( 3/2 log n), into a data structure of size O(nc(n) log4 n), so that for a query ray p, /,(, p) can be computed in time O(v/nc(ni log2 n), where w is a constant < 4.33 and a(n) is a functional inverse of Ackermann’s function. If the given segments are nonintersecting, the storage goes down to O(n log3 n) and the query time becomes O(v/ log2 n). The main tool used is spanning trees (on the set of segment endpoints) with low stabbing number, i.e., with the property that no line intersects more than O(x/) edges of the tree. Such trees make it possible to obtain faster algorithms for several other problems, including implicit point location, polygon containment, and implicit hidden surface removal.
Parallel Linear Programming in Fixed Dimension Almost Surely In Constant Time
, 1992
"... For any fixed dimension d, the linear programming problem with n inequality constraints can be solved on a probabilistic CRCW PRAM with O(n) processors almost surely in constant time. The algorithm always finds the correct solution. With nd/log² d processors, the probability that the algorithm wi ..."
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Cited by 17 (1 self)
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For any fixed dimension d, the linear programming problem with n inequality constraints can be solved on a probabilistic CRCW PRAM with O(n) processors almost surely in constant time. The algorithm always finds the correct solution. With nd/log² d processors, the probability that the algorithm will not finish within O(d² log² d) time tends to zero exponentially with n.