Results 1  10
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20
Cutting Circles into Pseudosegments and Improved Bounds for Incidences
 Geom
, 2000
"... We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m poi ..."
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Cited by 21 (11 self)
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We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree. 1 Introduction Let P be a finite set of points in the plane and C a finite set of circles. Let I = I(P, C) denote the number of incidences between the points and the circles. Let I(m, n) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles, and let I # (m, n, X) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles with at most X intersecting pairs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 ...
The ClarksonShor Technique Revisited and Extended
 Comb., Prob. & Comput
, 2001
"... We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds. ..."
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Cited by 18 (3 self)
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We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds.
Improved Bounds on Planar ksets and klevels
 Discrete Comput. Geom
, 1997
"... We prove an O(nk 1=3 ) upper bound for planar ksets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in arrangement ..."
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Cited by 16 (0 self)
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We prove an O(nk 1=3 ) upper bound for planar ksets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in arrangements of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees and parametric matroids in general. 1 Introduction The problem of determining the optimum asymptotic bound on the number of ksets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer k n, a kset is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains elusive even in ! ...
ThreeDimensional Delaunay Mesh Generation
 Discrete and Computational Geometry
, 2004
"... We propose an algorithm to compute a conforming Delaunay mesh of a bounded domain specified by a piecewise linear complex. Arbitrarily small input angles are allowed, and the input complex is not required to be a manifold. ..."
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Cited by 15 (5 self)
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We propose an algorithm to compute a conforming Delaunay mesh of a bounded domain specified by a piecewise linear complex. Arbitrarily small input angles are allowed, and the input complex is not required to be a manifold.
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
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Cited by 12 (0 self)
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
Partitioning Colored Point Sets Into Monochromatic Parts
, 2002
"... We show that any twocolored set of n points in general position in the plane can be partitioned into at most d n+1 2 e monochromatic subsets, whose convex hulls are pairwise disjoint. This bound cannot be improved in general. We present an O(n log n) time algorithm for constructing a partition into ..."
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Cited by 9 (1 self)
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We show that any twocolored set of n points in general position in the plane can be partitioned into at most d n+1 2 e monochromatic subsets, whose convex hulls are pairwise disjoint. This bound cannot be improved in general. We present an O(n log n) time algorithm for constructing a partition into fewer parts, if the coloring is unbalanced, i.e., the sizes of the two color classes differ by more than one. The analogous question for kcolored point sets (k > 2) and its higher dimensional variant are also considered.
On the complexity of many faces in arrangements of pseudosegments and of circles
 IN DISCRETE AND COMPUTATIONAL GEOMETRY: THE GOODMANPOLLACK FESTSCHRIFT
"... We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudosegments, n circles, or n unit circles. The bounds are worstcase optimal for unit circles; they are also worstcase optimal for the case of pseudosegments, except when the number of faces is very small, i ..."
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Cited by 8 (4 self)
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We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudosegments, n circles, or n unit circles. The bounds are worstcase optimal for unit circles; they are also worstcase optimal for the case of pseudosegments, except when the number of faces is very small, in which case our upper bound is a polylogarithmic factor from the bestknown lower bound. For general circles, the bounds nearly coincide with the bestknown bounds for the number of incidences between m points and n circles, recently obtained in [9].
On ksets in four dimension
, 2004
"... We show, with an elementary proof, that the number of halving simplices in a set of n points in R 4 in general position is O(n 4−2/45). This improves the previous bound of O(n4−1/134). Our main new ingredient is a bound on the maximum number of halving simplices intersecting a fixed 2plane. 1 ..."
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Cited by 5 (0 self)
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We show, with an elementary proof, that the number of halving simplices in a set of n points in R 4 in general position is O(n 4−2/45). This improves the previous bound of O(n4−1/134). Our main new ingredient is a bound on the maximum number of halving simplices intersecting a fixed 2plane. 1
Extremal Configurations and Levels in Pseudoline Arrangements
"... This paper studies a variety of problems involving certain types of extreme con gurations in arrangements of (xmonotone) pseudolines. For example, we obtain a very simple proof of the bound O(nk ) on the maximum complexity of the kth level in an arrangement of n pseudolines, which becomes ev ..."
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Cited by 4 (0 self)
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This paper studies a variety of problems involving certain types of extreme con gurations in arrangements of (xmonotone) pseudolines. For example, we obtain a very simple proof of the bound O(nk ) on the maximum complexity of the kth level in an arrangement of n pseudolines, which becomes even simpler in the case of lines. We thus simplify considerably previous proofs by Dey and by Tamaki and Tokuyama.