Results 1  10
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12
Applications of Random Sampling in Computational Geometry, II
 Discrete Comput. Geom
, 1995
"... We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms ..."
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Cited by 396 (12 self)
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We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divideandconquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...
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...
LowDimensional Linear Programming with Violations
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given half ..."
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Cited by 46 (3 self)
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Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3d runs in near O(n + k ) expected time. Interestingly, the idea is based on concavechain decompositions (or covers) of the ( k)level, previously used in proving combinatorial klevel bounds.
Point Sets With Many KSets
, 1999
"... For any n, k, n 2k > 0, we construct a set of n points in the plane with ne p log k ksets. This improves the bounds of Erd}os, Lovasz, et al. As a consequence, we also improve the lower bound of Edelsbrunner for the number of halving hyperplanes in higher dimensions. 1 Introduction For a ..."
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Cited by 45 (0 self)
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For any n, k, n 2k > 0, we construct a set of n points in the plane with ne p log k ksets. This improves the bounds of Erd}os, Lovasz, et al. As a consequence, we also improve the lower bound of Edelsbrunner for the number of halving hyperplanes in higher dimensions. 1 Introduction For a set P of n points in the ddimensional space, a kset is subset P 0 P such that P 0 = P \H for some halfspace H, and jP 0 j = k. The problem is to determine the maximum number of ksets of an npoint set in the ddimensional space. Even in the most studied two dimensional case, we are very far from the solution, and in higher dimensions even much less is known. The rst results in the two dimensional case are due to Erd}os, Lovasz, Simmons and Straus [L71], [ELSS73]. They established an upper bound O(n p k), and a lower bound (n log k). Despite great interest in this problem [W86], [E87], [S91], [EVW97], [AACS98], partly due to its importance in the analysis of geometric alg...
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 23 (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.
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 provided a ..."
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Cited by 9 (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.
The 2D kset Problem in Computational Geometry\Lambda
, 2002
"... 1 Introduction The kset problem is one of the most challenging open problems in combinatorial geometry. The simplestvariant of the problem is as follows: Let S be a set of n points in the plane. We assume they are in generalposition, i.e., no three of them are collinear and no parallel connecting l ..."
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1 Introduction The kset problem is one of the most challenging open problems in combinatorial geometry. The simplestvariant of the problem is as follows: Let S be a set of n points in the plane. We assume they are in generalposition, i.e., no three of them are collinear and no parallel connecting lines. A
Depth estimation via sampling
, 2008
"... In this chapter, we introduce a “trivial” but yet powerful idea. Given a set S of objects, a point p that is contained in some of the objects, and let its weight be the number of objects that contains it. We can estimate the depth/weight of p by counting the number of objects that contains it in a ..."
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In this chapter, we introduce a “trivial” but yet powerful idea. Given a set S of objects, a point p that is contained in some of the objects, and let its weight be the number of objects that contains it. We can estimate the depth/weight of p by counting the number of objects that contains it in a random sample of the objects. In fact, by considering points induced by the sample, we can bound the number of “light ” vertices induced by S. This idea can be extended to bounding the number of “light” configurations induced by a set of objects. This approach leads to a sequence of short, beautiful, elegant and correct proofs of several hallmark results in discrete geometry. While the results in this chapter are not directly related to approximation algorithms, the insights and general approach would be useful for us later, or so one hopes. 8.1 The at most klevels Let L be a set of n lines in the plane. A point p ∈ � ℓ∈L ℓ is of level k, if there are k lines of L strictly below it. The klevel is the closure of set of points of level k. Namely, the klevel is an xmonotone curve along the lines of L.
kSets and Continuous Motion in R³
, 2010
"... We prove several new results concerning ksets of point sets on the 2sphere (equivalently, for signed point sets in the plane) and ksets in 3space. Specific results include spherical generalizations of (i) Lovász’ lemma (regarding the number of spherical kedges intersecting a given great circle) ..."
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We prove several new results concerning ksets of point sets on the 2sphere (equivalently, for signed point sets in the plane) and ksets in 3space. Specific results include spherical generalizations of (i) Lovász’ lemma (regarding the number of spherical kedges intersecting a given great circle) and of (ii) the crossing identity for kedges due to Andrzejak et al. As a new ingredient compared to the planar case, the latter involves the winding number of kfacets around a given point in 3space, as introduced by Lee and by Welzl, independently. As a corollary, we obtain a crossing identity for the number of pinched crossings (crossing pairs of triangles sharing one vertex) of kfacets in 3space.