Results 1  10
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37
Random Sampling, Halfspace Range Reporting, and Construction of (≤k)Levels in Three Dimensions
 SIAM J. COMPUT
, 1999
"... Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the co ..."
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Cited by 38 (8 self)
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Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the orderk Voronoi diagram in O(n log n + nk log k) expected time.
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 32 (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 27 (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.
Robust Geometric Computing in Motion
 International Journal of Robotics Research
"... Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely is ..."
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Cited by 24 (2 self)
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Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely issues related to arithmetic precision and degenerate input. The paper starts with a discussion of the gap between the theory and practice of geometric algorithms, together with a brief review of existing solutions to some of the problems that this dichotomy brings about. We then turn to an overview of the CGAL project and library. The CGAL project is a joint effort by a number of research groups in Europe and Israel to produce a robust software library of geometric algorithms and data structures. The library is now available for use with significant functionality. We describe the main goals and results of the project. The central part of the paper is devoted to arrangements (i.e., space subdivisions induced by geometric objects) and motion planning. We concentrate on the maps and arrangements part of the CGAL library. Then we describe two packages developed on top of CGAL for constructing robust geometric primitives for motion algorithms. KEY WORDS—Computational geometry, robustness and precision, arrangements, algorithmic motion planning 1.
Sweeping and Maintaining Twodimensional Arrangements on Quadrics
"... We show how to compute and maintain the twodimensional arrangement on a quadric that is induced by intersection curves with other quadrics. The key idea is to parameterize the quadric by two variables, which then allows to implicitly compute the arrangement in a modified parameter space. We give ..."
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Cited by 17 (9 self)
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We show how to compute and maintain the twodimensional arrangement on a quadric that is induced by intersection curves with other quadrics. The key idea is to parameterize the quadric by two variables, which then allows to implicitly compute the arrangement in a modified parameter space. We give details of a possible parameterization and explain how to implement the needed geometric and topological predicates.
Topology and arrangement computation of semialgebraic planar curves
 CAGD
, 2008
"... We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise lin ..."
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Cited by 15 (4 self)
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We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties. The method isolates singular points from regular parts and deals with them independently. The topology near singular points is guaranteed through topological degree computation. In either case the topology inside regions is recovered from information on the boundary of a cell of the subdivision. Obtained regions are segmented to provide an efficient insertion operation while dynamically maintaining an arrangement structure. We use enveloping techniques of the polynomial represented in the Bernstein basis to achieve both efficiency and certification. It is finally shown on examples that this algorithm is able to handle curves defined by high degree polynomials with large coefficients, to identify regions of interest and use the resulting structure for either efficient rendering of implicit curves, point localization or boolean operation computation.
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
 In Proc. 44th IEEE Sympos. Found. Comput. Sci
, 2003
"... We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which ..."
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Cited by 13 (2 self)
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We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the klevel has subquadratic (O(n 2s )) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degrees polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved klevel results for most of the curve families studied earlier, including a nearO(n ) bound for parabolas.
Remarks on kLevel Algorithms in the Plane
, 1999
"... In light of recent developments, this paper reexamines the fundamental geometric problem of how to construct the klevel in an arrangement of n lines in the plane. ffl The author's recent dynamic data structure for planar convex hulls improves a decadeold sweepline algorithm by Edelsbrunner ..."
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Cited by 12 (6 self)
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In light of recent developments, this paper reexamines the fundamental geometric problem of how to construct the klevel in an arrangement of n lines in the plane. ffl The author's recent dynamic data structure for planar convex hulls improves a decadeold sweepline algorithm by Edelsbrunner and Welzl, which now runs in O(n log m+m log 1+" n) deterministic time and O(n) space, where m is the output size and " is any positive constant. We discuss simplification of the data structure in this particular application, by viewing the problem kinetically. ffl HarPeled recently announced a randomized algorithm with an expected running time of O((n + m)ff(n) log n). We observe that a version of an earlier randomized incremental algorithm by Agarwal, de Berg, Matousek, and Schwarzkopf yields almost the same result. ffl The current combinatorial bound by Dey shows that m = O(nk 1=3 ) in the worst case. We give an algorithm that guarantees O(n log n + nk 1=3 ) expected time. 1 Introd...
Generalized penetration depth computation
 In SPM ’06: Proceedings of the 2006 ACM symposium on Solid and physical modeling
, 2006
"... Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must ..."
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Cited by 11 (2 self)
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Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalized PD. When an object undergoes rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object under all possible rigid transformations to separate the overlapping objects. We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both the objects are nonconvex, we treat them as a combination of the above two cases, and present an algorithm that computes a lower and an upper bound on generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of polygonal robots undergoing translational and rotational motion in a plane. In particular, we use generalized PD computation for checking path nonexistence.
Computing the EulerPoincaré characteristics of sign conditions
 Comput. Complexity
"... Abstract. Computing various topological invariants of semialgebraic sets in single exponential time is an active area of research. Several algorithms are known for deciding emptiness, computing the number of connected components of semialgebraic sets in single exponential time etc. However, an alg ..."
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Cited by 11 (6 self)
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Abstract. Computing various topological invariants of semialgebraic sets in single exponential time is an active area of research. Several algorithms are known for deciding emptiness, computing the number of connected components of semialgebraic sets in single exponential time etc. However, an algorithm for computing all the Betti numbers of a given semialgebraic set in single exponential time is still lacking. In this paper we describe a new, improved algorithm for computing the Euler–Poincaré characteristic (which is the alternating sum of the Betti numbers) of the realization of each realizable sign condition of a family of polynomials restricted to a real variety. The complexity of the algorithm is sk ′ +1O(d) k + sk ′ ((k ′ log2(s) + k log2(d))d) O(k) , where s is the number of polynomials, k the number of variables, d a bound on the degrees, and k ′ the real dimension of the variety. A consequence of our result is that the Euler–Poincaré characteristic of any locally closed semialgebraic set can be computed with the same complexity. The best previously known single exponential time algorithm for computing the Euler–Poincaré characteristic of semialgebraic sets worked only for a more restricted class of closed semialgebraic sets and had a complexity of (ksd) O(k).