Results 1  10
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25
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...
On the Number of Incidences Between Points and Curves
 Combinatorics, Probability and Computing 7
"... We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane. ..."
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Cited by 30 (14 self)
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We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane.
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
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
On the Number of Congruent Simplices in a Point Set
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2002
"... We derive improved bounds on the number of kdimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex, for k < or = d  1. Let f(sup d)(sub K)(n) be the maximum number of ksimplices spanned by a set of n points in R(sup d) that are congruent to a given ..."
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Cited by 12 (2 self)
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We derive improved bounds on the number of kdimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex, for k < or = d  1. Let f(sup d)(sub K)(n) be the maximum number of ksimplices spanned by a set of n points in R(sup d) that are congruent to a given ksimplex. We prove that f(sup 3)(sub 2) (n) = O(n sup 5/3) times 2 sup O(alpha (sup 2)(n)), f(sup 4)(sub 2)(n) = O(n (sup 2) + epsilon), f(sup 5)(sub 2) (n)) = theta (n (sup 7/3), and f(sup 4)(sub 3)(n) = O(n (sup 9/4) plus epsilon). We also derive a recurrence to bound f(sup d)(sub k) (n) for arbitrary values of k and d, and use it to derive the bound f(sup d)(sub k(n) = O(n(sup d/2)) for d < or = 7 and k < or = d  2. Following Erdo's and Purdy, we conjecture that this bound holds for larger values of d as well, and for k < or = d  2.
Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract)
, 2010
"... We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but ..."
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Cited by 10 (5 self)
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We first describe a reduction from the problem of lowerbounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3. Applying these bounds, we obtain, among several other results, the upper bound O(s 3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s 3 /k 12/7). One of our unresolved conjectures is that this number is O(s 3 /k 2), for k ≥ 2. If true, it would imply the lower bound Ω(s / log s) on the number of distinct distances in the plane.
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 9 (5 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]. 1
Some scalable parallel algorithms for geometric problems
 Journal of Parallel and Distributed Computing
, 1999
"... This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary ..."
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Cited by 6 (2 self)
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This paper considers a variety of geometric pattern recognition problems on input sets of size n using a coarse grained multicomputer model consisting of p processors with 0(n p) local memory each (i.e., 0(n p) memory cells of 3(log n) bits apiece), where the processors are connected to an arbitrary interconnection network. It introduces efficient scalable parallel algorithms for a number of geometric problems including the rectangle finding problem, the maximal equally spaced collinear points problem, and the point set pattern matching problem. All of the algorithms presented are scalable in that they are applicable and efficient over a very wide range of ratios of problem size to number of processors. In addition to the practicality imparted by scalability, these algorithms are easy to implement in that all required communications can be achieved by a small number of calls to standard global routing operations.
On Nonlinear Forbidden 01 Matrices: A Refutation of a FürediHajnal Conjecture
"... A matrix A P t0, 1u m n is said to avoid a forbidden pattern P P t0, 1u k l if no k l submatrix of A matches P, where a 0 in P can match either a 0 or 1 in A. Let ExpP, nq be the maximum weight (i.e., number of 1s) of an n n matrix avoiding the pattern P or all patterns in the set P. The theory of f ..."
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Cited by 5 (4 self)
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A matrix A P t0, 1u m n is said to avoid a forbidden pattern P P t0, 1u k l if no k l submatrix of A matches P, where a 0 in P can match either a 0 or 1 in A. Let ExpP, nq be the maximum weight (i.e., number of 1s) of an n n matrix avoiding the pattern P or all patterns in the set P. The theory of forbidden submatrices subsumes many extremal problems in combinatorics and graph theory, including DavenportSchinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turántype subgraph avoidance problems. Forbidden submatrices have found many applications in discrete geometry and the analysis of both geometric and nongeometric algorithms. In general terms, to bound the complexity of an arrangement of objections or the running time of an algorithm, one transcribes the objects or operations as a 01 matrix that provably avoids some forbidden pattern or collection of patterns P. This method is useful only to the extent that ExpP, nq can be tightly bounded, for specific P s or whole classes of P s. A 01 matrix can be interpreted as the incidence matrix of a bipartite graph where vertices on either side of the bipartition are ordered. In 1992, Füredi and Hajnal conjectured that imposing