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...

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 "well-behaved" curves in the plane.

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 NP-complete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1

We derive improved bounds on the number of k-dimensional simplices spanned by a set of n points in R(sup d) that are congruent to a given k-simplex, for k < or = d - 1. Let f(sup d)(sub K)(n) be the maximum number of k-simplices spanned by a set of n points in R(sup d) that are congruent to a given k-simplex. 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.

We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudo-segments, n circles, or n unit circles. The bounds are worst-case optimal for unit circles; they are also worst-case optimal for the case of pseudo-segments, except when the number of faces is very small, in which case our upper bound is a polylogarithmic factor from the best-known lower bound. For general circles, the bounds nearly coincide with the best-known bounds for the number of incidences between m points and n circles, recently obtained in [9].

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension 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.

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.

Recent papers concerned with the Point Set Pattern Matching Problem (PSPM) ( nding all congruent copies of a pattern in a sample set) in Euclidean 3-space, R³, have given algorithms with running times that have decreased as known output bounds for the problem have decreased. In this paper, a recent result of [2] is used to show that the volume of the output is O(kn

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 Davenport-Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type subgraph avoidance problems. Forbidden submatrices have found many applications in discrete geometry and the analysis of both geometric and non-geometric 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 0-1 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 0-1 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

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