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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 280 (41 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.
Filtering Search: A new approach to queryanswering
 SIAM J. Comput
, 1986
"... Abstract. We introduce a new technique for solving problems of the following form: preprocess a set ofobjects so that those satisfying a given property with respect to a query object canbe listed very effectively. Wellknown problems that fall into this category include range search, point enclosure ..."
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Abstract. We introduce a new technique for solving problems of the following form: preprocess a set ofobjects so that those satisfying a given property with respect to a query object canbe listed very effectively. Wellknown problems that fall into this category include range search, point enclosure, intersection, and nearneighbor problems. The approach which we take is very general and rests on a new concept called filtering search.We show on a number ofexamples how it can be used to improve the complexity ofknown algorithms and simplify their implementations as well. In particular, filtering search allows us to improve on the worstcase complexity ofthe best algorithms known so far for solving the problems mentioned above. Key words, computational geometry, database, data structures, filtering search, retrieval problems
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 57 (3 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.
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 28 (4 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
How Hard Is Halfspace Range Searching?
, 1993
"... We investigate the complexity of halfspace range searching: Given n points in d space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worstcase query time t in the Fredman/Yao arithmetic ..."
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We investigate the complexity of halfspace range searching: Given n points in d space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worstcase query time t in the Fredman/Yao arithmetic model of computation. We show that t must be at least on the order of (n= log n) 1\Gamma d\Gamma1 d(d+1) =m 1=d : Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound of O \Gamma n=m 1=d \Delta upper bound established by Matousek. We also show that it is possible to devise a sequence of n inserts and halfspace range queries that require a total time of n 2\GammaO(1=d) . Our results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension. For example they show that, with linear storage, circular range queries in the plane require\Omega \Gamma n 1=3 \Delta time (modulo a logarithmic factor).
Optimal halfspace range reporting in three dimensions
 In Proceedings of the 20 th ACMSIAM Symposium on Discrete Algorithms
, 2009
"... We give the first optimal solution to a standard problem in computational geometry: threedimensional halfspace range reporting. We show that n points in 3d can be stored in a linearspace data structure so that all k points inside a query halfspace can be reported in O(log n + k) time. The data st ..."
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We give the first optimal solution to a standard problem in computational geometry: threedimensional halfspace range reporting. We show that n points in 3d can be stored in a linearspace data structure so that all k points inside a query halfspace can be reported in O(log n + k) time. The data structure can be built in O(n log n) expected time. The previous methods with optimal query time required superlinear (O(n log log n)) space. We also mention consequences, for example, to higher dimensions and to externalmemory data structures. As an aside, we partially answer another open question concerning the crossing number in Matouˇsek’s shallow partition theorem in the 3d case (a tool used in many known halfspace range reporting methods). 1
FarthestPoint Queries with Geometric and Combinatorial Constraints
 COMPUTAT. GEOM. THEORY APPL
, 2006
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Using Simplicial Partitions to Determine a Closest Point to a Query Line
"... In this note we show that simplicial partitions can be used to answer the following queries efficiently: given a point set in the plane, determine a point that is closest to an arbitrary query line. ..."
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Cited by 3 (1 self)
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In this note we show that simplicial partitions can be used to answer the following queries efficiently: given a point set in the plane, determine a point that is closest to an arbitrary query line.
Abstract Computing the Closest Point to a Circle
"... In this paper we consider the problem of computing the closest point to the boundary of a circle among a set S of n points. We present two algorithms to solve this problem. One algorithm runs in O(n 3) preprocessing time and space and O(log 2 n) query time. The other algorithm runs in O(n 1+ɛ) prepr ..."
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In this paper we consider the problem of computing the closest point to the boundary of a circle among a set S of n points. We present two algorithms to solve this problem. One algorithm runs in O(n 3) preprocessing time and space and O(log 2 n) query time. The other algorithm runs in O(n 1+ɛ) preprocessing time and O(n log n) space and O(n 2/3+ɛ) query time. Thus we exhibit a tradeoff between preprocessing and query times For dimensions d ≥ 3 we present an algorithm with O(n⌈d/2⌉+ɛ ′ ) preprocessing time to report an approximate closest point to the boundary of ddimensional query sphere R in O(n1−1/(d+1)+ɛ) query time. 1