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76
Approximating extent measure of points
 Journal of ACM
"... We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our tec ..."
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Cited by 98 (29 self)
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We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our technique include�approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms. 1
Nearest Neighbor Queries in a Mobile Environment
, 1999
"... Nearest neighbor queries have received much interest in recent years due to their increased importance in advanced database applications. However, past work ..."
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Cited by 54 (5 self)
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Nearest neighbor queries have received much interest in recent years due to their increased importance in advanced database applications. However, past work
Collision detection for deforming necklaces
 IN SYMP. ON COMPUTATIONAL GEOMETRY
, 2002
"... In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, rope, and other ‘linear ’ objects in the physical world. In this paper, we exploit this linearity ..."
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Cited by 36 (11 self)
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In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, rope, and other ‘linear ’ objects in the physical world. In this paper, we exploit this linearity to develop geometric structures associated with necklaces that are useful in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres built on a necklace. Such a hierarchy is easy to compute and is suitable for maintenance when the necklace deforms, as our theoretical and experimental results show. This hierarchy can be used for collision and selfcollision detection. In particular, we achieve an upper bound of O(nlog n) in two dimensions and O(n 2−2/d) in ddimensions, d ≥ 3, for collision checking. To our knowledge, this is the first subquadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be also used to detect selfcollisions of a necklace in certain ways complementary to the sphere hierarchy.
Almost tight upper bounds for vertical decompositions in four dimensions
 In Proc. 42nd IEEE Symposium on Foundations of Computer Science
, 2001
"... We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major proble ..."
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Cited by 32 (6 self)
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We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a nearlinear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n 2d−4+ε), for any ε> 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems. 1
The coverage problem in threedimensional wireless sensor networks
 In IEEE Globecom
, 2004
"... Abstract — One of the fundamental issues in sensor networks is the coverage problem, which reflects how well a sensor network is monitored or tracked by sensors. In this paper, we formulate this problem as a decision problem, whose goal is to determine whether every point in the service area of the ..."
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Cited by 26 (5 self)
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Abstract — One of the fundamental issues in sensor networks is the coverage problem, which reflects how well a sensor network is monitored or tracked by sensors. In this paper, we formulate this problem as a decision problem, whose goal is to determine whether every point in the service area of the sensor network is covered by at least α sensors, where α is a given parameter and the sensing regions of sensors are modeled by balls (not necessarily of the same radius). This problem in a 2D space is solved in [1] with an efficient polynomialtime algorithm (in terms of the number of sensors). In this paper, we show that tackling this problem in a 3D space is still feasible within polynomial time. The proposed solution can be easily translated into an efficient polynomialtime distributed protocol. I.
Projective Visual Hulls
, 2002
"... This thesis presents an imagebased method for computing the visual hull of an object bounded by a smooth surface and observed by a finite number of perspective cameras. The essential structure of the visual hull is projective: to compute an exact topological (combinatorial) description of its bound ..."
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Cited by 26 (2 self)
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This thesis presents an imagebased method for computing the visual hull of an object bounded by a smooth surface and observed by a finite number of perspective cameras. The essential structure of the visual hull is projective: to compute an exact topological (combinatorial) description of its boundary, we do not need to know the Euclidean properties of the input cameras or of the scene. Unlike most existing visual hull computation methods, ours requires only a projective reconstruction of the camera matrices, or equivalently, the epipolar geometry between each pair of cameras in the scene. Starting with a rigorous theoretical framework of oriented projective geometry and projective differential geometry, we develop a suite of algorithms to construct the visual hull and associated data structures. The thesis discusses our implementation of the algorithms, and presents experimental results on synthetic and real data sets.
Kinetic Medians and kdTrees
, 2002
"... We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor. ..."
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Cited by 22 (8 self)
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We propose algorithms for maintaining two variants of kd trees of a set of moving points in the plane. A pseudo kdtree allows the number of points stored in the two children to di#er by a constant factor.
Adhoc Topk Query Answering for Data Streams
, 2007
"... A topk query retrieves the k highest scoring tuples from a data set with respect to a scoring function defined on the attributes of a tuple. The efficient evaluation of topk queries has been an active research topic and many different instantiations of the problem, in a variety of settings, have b ..."
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Cited by 21 (1 self)
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A topk query retrieves the k highest scoring tuples from a data set with respect to a scoring function defined on the attributes of a tuple. The efficient evaluation of topk queries has been an active research topic and many different instantiations of the problem, in a variety of settings, have been studied. However, techniques developed for conventional, centralized or distributed databases are not directly applicable to highly dynamic environments and online applications, like data streams. Recently, techniques supporting topk queries on data streams have been introduced. Such techniques are restrictive however, as they can only efficiently report topk answers with respect to a prespecified (as opposed to adhoc) set of queries. In this paper we introduce a novel geometric representation for the topk query problem that allows us to raise this restriction. Utilizing notions of geometric arrangements, we design and analyze algorithms for incrementally maintaining a data set organized in an arrangement representation under streaming updates. We introduce query evaluation strategies that operate on top of an arrangement data structure that are able to guarantee efficient evaluation for adhoc queries. The performance of our core technique is augmented by incorporating tuple pruning strategies, minimizing the number of tuples that need to be stored and manipulated. This results in a main memory indexing technique supporting both efficient incremental updates and the evaluation of adhoc topk queries. A thorough experimental study evaluates the efficiency of the proposed technique.
On Levels in Arrangements of Curves
 Proc. 41st IEEE
, 2002
"... Analyzing the worstcase complexity of the klevel in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously ..."
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Cited by 20 (3 self)
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Analyzing the worstcase complexity of the klevel in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudoparabolas (curves that pairwise intersect at most twice) to O(nk k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudoparabolas into pseudosegments, as well as a new observation for cutting pseudosegments into pieces that can be extended to pseudolines. We mention applications to parametric and kinetic minimum spanning trees.
Complete, exact, and efficient computations with cubic curves
 In Proc. 20th Annu. ACM Symp. Comput. Geom
, 2004
"... The BentleyOttmann sweepline method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realiza ..."
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Cited by 17 (6 self)
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The BentleyOttmann sweepline method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realizations. As a result, we obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Conics and cubic splines are special cases of cubic curves. The algorithm is complete in that it handles all possible degeneracies including singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement.