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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 256 (40 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.
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 94 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 78 (22 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
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.
Product Range Spaces, Sensitive Sampling, and Derandomization
, 1993
"... We introduce the concept of a sensitive Eapproximation, and use it to derive a more efficient algorithm for computing &nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VCdimensional case. This generalizes and simpli ..."
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Cited by 46 (6 self)
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We introduce the concept of a sensitive Eapproximation, and use it to derive a more efficient algorithm for computing &nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VCdimensional case. This generalizes and simplifies results from previous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(n log3 n) deterministic algorithm for computing the diameter of an npoint set in 3dimensional space.
Parallel Algorithms for HigherDimensional Convex Hulls
"... We give fast randomized and deterministic parallel methods for constructing convex hulls in R^d, for any fixed d. Our methods are for the weakest sharedmemory model,the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In particular, we show that the convex ..."
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Cited by 44 (14 self)
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We give fast randomized and deterministic parallel methods for constructing convex hulls in R^d, for any fixed d. Our methods are for the weakest sharedmemory model,the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In particular, we show that the convex hull of n points in R^d can be constructed in O(log n) time using O(n log n + nbd=2c) work, with high probability. We also show that it can be constructed deterministically in O(log² n) time using O(n log n) work for d = 3 and in O(log n) time using O(nbd=2c logc(dd=2e\Gamma bd=2c) n) work, for d * 4, where c? 0is a constant, which is optimal for even d * 4. We also showhow to make our 3dimensional methods outputsensitive with only a small increase in running time.These methods can be applied to other problems as well. A variation of the convex hull algorithm for even dimensions deterministically constructs a (1=r)cutting of n hyperplanes in IR d in O(log n) time using optimal O(nrd\Gamma 1) work; when r = n, we obtain their arrangement and a pointlocation data structure for it. With appropriate modifications, our deterministic 3dimensional convex hull algorithmcan be used to compute, in the same resource bounds, the intersection of n balls of equal radius in R³. This leads to asequential algorithm for computing the diameter of a point set in IR3 with running time O(n log³ n), which is arguably simpler than an algorithm with the same running time by Brönnimann et al.
Constructing Levels in Arrangements and Higher Order Voronoi Diagrams
 SIAM J. COMPUT
, 1994
"... We give simple randomized incremental algorithms for computing the klevel in an arrangement of n hyperplanes in two and threedimensional space. The expected running time of our algorithms is O(nk+nff(n) log n) for the planar case, and O(nk 2 +n log 3 n) for the threedimensional case. Both bo ..."
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Cited by 42 (10 self)
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We give simple randomized incremental algorithms for computing the klevel in an arrangement of n hyperplanes in two and threedimensional space. The expected running time of our algorithms is O(nk+nff(n) log n) for the planar case, and O(nk 2 +n log 3 n) for the threedimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the klevel in an arrangement of discs or xmonotone Jordan curves in the plane. Our approach can also be used to compute the klevel; this yields a randomized algorithm for computing the orderk Voronoi diagram of n points in the plane in expected time O(k(n \Gamma k) log n + n log 3 n).
An ExpanderBased Approach to Geometric Optimization
 IN PROC. 9TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1993
"... We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach ..."
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Cited by 40 (16 self)
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We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach
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
Efficient detection of motion patterns in spatiotemporal data sets
 In Proceedings of the 13th International Symposium of ACM Geographic Information Systems
, 2004
"... Abstract Moving point object data can be analyzed through the discovery of patterns. We consider the computational efficiency of detecting four such spatiotemporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., 2004. These patterns are large enough subgr ..."
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Cited by 27 (7 self)
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Abstract Moving point object data can be analyzed through the discovery of patterns. We consider the computational efficiency of detecting four such spatiotemporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., 2004. These patterns are large enough subgroups of the moving point objects that exhibit similar movement in the sense of direction, heading for the same location, and/or proximity. By the use of techniques from computational geometry, including approximation algorithms, we improve the running time bounds of existing algorithms to detect these patterns. 1 Introduction Moving point object data is becoming increasingly more available since the development of GPS andradio transmitters. One of the objectives of spatiotemporal data mining [16, 23] is to analyze such data sets for interesting patterns. For example, a group of caribou with radio collars gives rise to the positionsof each caribou at a sequence of time steps. Analyzing this data gives insight into entity behavior, in particular, migration patterns [22]. The analysis of moving objects also has applications in sports (e.g.,soccer players [12]) and in socioeconomic geography [9]. There is ample research on data mining of moving objects (e.g., [13, 25, 26, 28]) in particular, on thediscovery of similar trajectories or clusters. In general the input is a set of