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Programming Parallel Algorithms
, 1996
"... In the past 20 years there has been treftlendous progress in developing and analyzing parallel algorithftls. Researchers have developed efficient parallel algorithms to solve most problems for which efficient sequential solutions are known. Although some ofthese algorithms are efficient only in a th ..."
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Cited by 237 (11 self)
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In the past 20 years there has been treftlendous progress in developing and analyzing parallel algorithftls. Researchers have developed efficient parallel algorithms to solve most problems for which efficient sequential solutions are known. Although some ofthese algorithms are efficient only in a theoretical framework, many are quite efficient in practice or have key ideas that have been used in efficient implementations. This research on parallel algorithms has not only improved our general understanding ofparallelism but in several cases has led to improvements in sequential algorithms. Unf:ortunately there has been less success in developing good languages f:or prograftlftling parallel algorithftls, particularly languages that are well suited for teaching and prototyping algorithms. There has been a large gap between languages
On constructing minimum spanning trees in kdimensional space and related problems
 SIAM JOURNAL ON COMPUTING
, 1982
"... . The problem of finding a minimum spanning tree connecting n points in a kdimensional space is discussed under three common distance metrics: Euclidean, rectilinear, and L. By employing a subroutine that solves the post office problem, we show that, for fixed k _> 3, such a minimum spanning t ..."
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Cited by 222 (1 self)
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. The problem of finding a minimum spanning tree connecting n points in a kdimensional space is discussed under three common distance metrics: Euclidean, rectilinear, and L. By employing a subroutine that solves the post office problem, we show that, for fixed k _> 3, such a minimum spanning tree can be found in time O(n2a<k)(1og n)la<k)), where a(k) = 2+1). The bound can be improved to O((n log n) 1"8) for points in 3dimensional Euclidean space. We also obtain o(n 2) algorithms for finding a farthest pair in a set of n points and for other related problems.
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
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ClosestPoint Problems in Computational Geometry
, 1997
"... This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, th ..."
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Cited by 73 (14 self)
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This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate postoffice problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divideandconquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...
An Optimal Algorithm for Closest Pair Maintenance
 Discrete Comput. Geom
, 1995
"... Given a set S of n points in kdimensional space, and an L t metric, the dynamic closest pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension k and fixed metric L t , we give a data structure of size O(n) ..."
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Cited by 34 (0 self)
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Given a set S of n points in kdimensional space, and an L t metric, the dynamic closest pair problem is defined as follows: find a closest pair of S after each update of S (the insertion or the deletion of a point). For fixed dimension k and fixed metric L t , we give a data structure of size O(n) that maintains a closest pair of S in O(logn) time per insertion and deletion. The running time of algorithm is optimal up to constant factor because \Omega\Gammaaus n) is a lower bound, in algebraic decisiontree model of computation, on the time complexity of any algorithm that maintains the closest pair (for k = 1). The algorithm is based on the fairsplit tree. The constant factor in the update time is exponential in the dimension. We modify the fairsplit tree to reduce it. 1 Introduction The dynamic closest pair problem is one of the very wellstudied proximity problem in computational geometry [6, 1720, 22, 2426, 2831]. We are given a set S of n points in kdimensional space...
SpaceEfficient Geometric DivideandConquer Algorithms
, 2006
"... We develop a number of spaceefficient tools including an approach to simulate divideandconquer spaceefficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multidimensional set that is sorted in another dimension. ..."
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Cited by 22 (5 self)
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We develop a number of spaceefficient tools including an approach to simulate divideandconquer spaceefficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multidimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divideandconquer. Specifically, we present a deterministic algorithm running in O(n log n) time using O(1) extra memory given inputs of size n for the closest pair problem and a randomized solution running in O(n log n) expected time and using O(1) extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in O(n log n + k) time using O(1) extra space where n is the number of horizontal and vertical line segments and k is the number of intersections.
An Optimal Algorithm for the onLine Closest Pair Problem
 Algorithmica
, 1994
"... We give an algorithm that computes the closest pair in a set of n points in k dimensional space online, in O(n log n) time. The algorithm only uses algebraic functions and, therefore, is optimal. The algorithm maintains a hierarchical subdivision of kspace into hyperrectangles, which is stored i ..."
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Cited by 19 (4 self)
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We give an algorithm that computes the closest pair in a set of n points in k dimensional space online, in O(n log n) time. The algorithm only uses algebraic functions and, therefore, is optimal. The algorithm maintains a hierarchical subdivision of kspace into hyperrectangles, which is stored in a binary tree. Centroids are used to maintain a balanced decomposition of this tree. 1 Introduction The closest pair problem is one of the classical problems in computational geometry. In this problem, we have to compute the closest pairor its distancein a set of n points in kdimensional space. Distances are measured in an arbitrary, but fixed, L t metric. Let p = (p 1 ; : : : ; p k ) and q = (q 1 ; : : : ; q k ) be two points in kdimensional space. Then the L t distance d t (p; q) between p and q is defined by d t (p; q) := / k X i=1 jp i \Gamma q i j t !1=t ; if 1 t ! 1, and for t = 1, it is defined by d1 (p; q) := max 1ik jp i \Gamma q i j: We observe, as many o...
New Techniques For Exact And Approximate Dynamic ClosestPoint Problems
, 1994
"... . Let S be a set of n points in IR D . It is shown that a range tree can be used to find an L1nearest neighbor in S of any query point, in O((logn) D\Gamma1 log log n) time. This data structure has size O(n(log n) D\Gamma1 ) and an amortized update time of O((logn) D\Gamma1 log log n). This result ..."
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Cited by 11 (2 self)
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. Let S be a set of n points in IR D . It is shown that a range tree can be used to find an L1nearest neighbor in S of any query point, in O((logn) D\Gamma1 log log n) time. This data structure has size O(n(log n) D\Gamma1 ) and an amortized update time of O((logn) D\Gamma1 log log n). This result is used to solve the (1 + ffl)approximate L 2 nearest neighbor problem within the same bounds (up to a constant factor that depends on ffl and D). In this o problem, for any query point p, a point q 2 S is computed such that the euclidean distance between p and q is at most (1 + ffl) times the euclidean distance between p and its true nearest neighbor. This is the first dynamic data structure for this problem having close to linear size and polylogarithmic query and update times. New dynamic data structures are given that maintain a closest pair of S. For D 3, a structure of size O(n) is presented with amortized update time O((logn) D\Gamma1 log log n). The constant factor in t...