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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 193 (9 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
Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (19 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
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 65 (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 35 (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...
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 20 (3 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...
Randomized Data Structures for the Dynamic ClosestPair Problem
, 1993
"... We describe a new randomized data structure, the sparse partition, for solving the dynamic closestpair problem. Using this data structure the closest pair of a set of n points in Ddimensional space, for any fixed D, can be found in constant time. If a frame containing all the points is known in adv ..."
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Cited by 10 (2 self)
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We describe a new randomized data structure, the sparse partition, for solving the dynamic closestpair problem. Using this data structure the closest pair of a set of n points in Ddimensional space, for any fixed D, can be found in constant time. If a frame containing all the points is known in advance, and if the floor function is available at unitcost, then the data structure supports insertions into and deletions from the set in expected O(log n) time and requires expected O(n) space. Here, it is assumed that the updates are chosen by an adversary who does not know the random choices made by the data structure. This method is more efficient than any deterministic algorithm for solving the problem in dimension D ? 1. The data structure can be modified to run in O(log 2 n) expected time per update in the algebraic computation tree model of computation. Even this version is more efficient than the currently best known deterministic algorithm for D ? 2. 1 Introduction We ...
On Enumerating and Selecting Distances
 Int. J. Comput. Geom. Appl
, 1999
"... Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selec ..."
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Cited by 9 (2 self)
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Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n + n 2=3 k 1=3 log 5=3 n). We also describe outputsensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search. Keywords: distance enumeration, distance selection, closest pairs, range counting, randomized algorithms. 1 Introduction Finding the closest pair of an npoint set has a long history in computational geometry (see [34] for a nice survey). In the plane, the problem can be solved in O(n log n) time using the Delaunay triangulation. In an arbitrary fixed dimension d, the first O(n log n) algorithm, based on di...