Results 1  10
of
99
Crossing numbers and hard Erdős problems in discrete geometry
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the min ..."
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Cited by 113 (1 self)
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We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
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...
A sumproduct estimate in finite fields, and applications
"... Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a Sze ..."
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Cited by 46 (3 self)
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Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a SzemerédiTrotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the threedimensional Kakeya problem in finite fields. 1.
Geometric matching under noise: combinatorial bounds and algorithms
 ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1999
"... In geometric pattern matching, we are given two sets of points P and Q in d dimensions, and the problem is to determine the rigid transformation that brings P closest to Q, under some distance measure. More generally, each point can be modelled as a ball of small radius, and we may wish to nd a tran ..."
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Cited by 40 (9 self)
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In geometric pattern matching, we are given two sets of points P and Q in d dimensions, and the problem is to determine the rigid transformation that brings P closest to Q, under some distance measure. More generally, each point can be modelled as a ball of small radius, and we may wish to nd a transformation approximating the closest distance between P and Q. This problem has many applications in domains such as computer vision and computational chemistry In this paper we present improved algorithms for this problem, by allowing the running time of our algorithms to depend not only on n, (the number of points in the sets), but also on, the diameter of the point set. The dependence on also allows us to e ectively process point sets that occur in practice, where diameters tend to be small ([EVW94]). Our algorithms are also simple to implement, in contrast to much of the earlier work. To obtain the abovementioned results, we introduce a novel discretization technique to reduce geometric pattern matching to combinatorial pattern matching. In addition, we address various generalizations of the classical problem rst posed by Erdos: \Given a set of n points in the plane, how many pairs of points can be exactly a unit distance apart?". The combinatorial bounds we prove enable us to obtain improved results for geometric pattern matching and may have other applications.
New Lower Bounds for Hopcroft's Problem
, 1996
"... We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst cas ..."
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Cited by 33 (6 self)
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We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...
Applications of the crossing number
 In Proc. 10th Annu. ACM Sympos. Comput. Geom
, 1994
"... Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1 ..."
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Cited by 28 (6 self)
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Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1
Some Geometric Applications of Dilworth’s Theorem
, 1993
"... A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no 3 vertices are collinear. We settle an old question of Avital, Hanani, Erdos, Kupitz and Perles by showing that every geometric graph with n vertices and m> k4n edges cent ains k+ 1 pairwise disjoint ..."
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Cited by 27 (11 self)
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A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no 3 vertices are collinear. We settle an old question of Avital, Hanani, Erdos, Kupitz and Perles by showing that every geometric graph with n vertices and m> k4n edges cent ains k+ 1 pairwise disjoint edges. We also prove that, given a set of points V and a set of axisparallel rectangles in the plane, then either there are k + 1 rectangles such that no point of V belongs to more than one of them, or we can find an at most 2. 105 ks element subset of V meeting all rectangles. This improves a result of Ding, Seymour and Winkler. Both proofs are based on Dilworth’s theorem on
Distinct Distances in the Plane
, 2001
"... It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1. ..."
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Cited by 16 (0 self)
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It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1.
Fourier bases and a distance problem of Erdős
 Amer. J. Math
, 1999
"... Abstract. We prove that no ball admits a nonharmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erdős, which says that the number of distances determined by n points in Rd is at least Cdn 1 d +ǫd, ǫd> 0. Introduction and statement of results Fourier bases ..."
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Cited by 16 (2 self)
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Abstract. We prove that no ball admits a nonharmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erdős, which says that the number of distances determined by n points in Rd is at least Cdn 1 d +ǫd, ǫd> 0. Introduction and statement of results Fourier bases. Let D be a domain in R d, i.e., D is a Lebesgue measurable subset of R d with finite nonzero Lebesgue measure. We say that D is a spectral set if L 2 (D) has orthogonal basis of the form EΛ = {e 2πix·λ}
ErdösFalconer distance problem, exponential sums, and Fourier analytic approach to incidence theorems in vector spaces over finite fields,(2006
"... Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields with respect to the cubic metric. Estimates for discrete Airy sums and Adolphson/Sperber estimates for exponential sums in terms of Newton polyhedra play a crucial role. Similar techniques are used to study th ..."
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Cited by 14 (5 self)
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Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields with respect to the cubic metric. Estimates for discrete Airy sums and Adolphson/Sperber estimates for exponential sums in terms of Newton polyhedra play a crucial role. Similar techniques are used to study the incidence problem between points and cubic and quadratic curves. As a result we obtain a nontrivial range of exponents that appear to be difficult to attain using combinatorial methods.