The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and ffl ? 0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance fflw of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in R d can be preprocessed in O(n log n) time and O(n) space, such that approximate queries can be answered in O(logn + (1=ffl) d ) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in const...

We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time

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...

Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearest-neighbor searching, clustering, finite-element mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since three-dimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worst-case running time \Omega (n2). However, this behavior is almost never observed in practice except for highly-contrived inputs. For all practical purposes, three-dimensional Delaunay triangulations appear to have linear complexity. This frustrating

Abstract We present space-time tradeoffs for approximate spherical range counting queries. Given a set S of n data points in Rdalong with a positive approximation factor ffl, the goal is to preprocess the points so that, given any Euclidean ball B,we can return the number of points of any subset of S that contains all the points within a (1- ffl)-factor contraction ofB, but contains no points that lie outside a (1 + ffl)-factor expansion of B.In many applications of range searching it is desirable to offer a tradeoff between space and query time. Wepresent here the first such tradeoffs for approximate range counting queries. Given 0 < ffl < = 1/2 and a parameterfl, where 2 < = fl < = 1/ffl, we show how to construct a data structure of space O(nfld log(1/ffl)) that allows us toanswer ffl-approximate spherical range counting queries in time O(log(nfl) + 1/(fflfl)d-1). The data structure can be built in time O(nfld log(n/ffl) log(1/ffl)). Here n, ffl, and fl areasymptotic quantities, and the dimension d is assumed to be a fixed constant.At one extreme (low space), this yields a data structure of space O(n log(1/ffl)) that can answer approximate range queries in time O(log n + (1/ffl)d-1) which, up to a factorof O(log 1/ffl) in space, matches the best known result

Consider a set of geometric objects, such as points, line segments, or axesparallel hyperrectangles in IR d, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum inter-point separation or minimum diameter that ever occurs. In particular, two open problems from the literature are solved: Deciding in o(n 2) time if there is a collision in a set of n moving points in IR 2, where the points move at constant but possibly different velocities, and the analogous problem for detecting a red-blue collision between sets of red and blue moving points. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search. 1

We prove a general lemma on the existence of (1/r)-cuttings of geometric objects in I = that stisfy certain properties. We use this lemma to construct (1/r)-cuttings of (azymptotically) optimal size for arrangements of line segments in the plane and arrangements of triangles in 3-space; for line segments in the plane we obtain a cutting of size O(r + Ar2/n2), and for triangles in 3-space our cutting haz size O(r2(r) + Ara/nZ). Here A is the combinatorial complexity of the arrangement. Finally, we use these results to obtain new results for several problems concerning line segments in the plane and triangles in 3-spce.

Abstract We present new algorithms for approximate range counting,where, for a specified &quot;> 0, we want to count the number of data points in a query range, up to relative error of&quot;. We first describe a general framework, adapted from Cohen [10], for this task, and then specialize it to two important instances of range counting: halfspaces in R3 anddisks in the plane. The technique reduces the approximate

We give the first nontrivial worst-case results for dynamic versions of various basic geometric optimization and measure problems under the semi-online model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1-center, largest empty circle, convex hull volume in three dimensions, volume of the union of axis-parallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1-center problems can be solved fully dynamically. Some applications are mentioned.

Answering range queries is a problem of fundamental importance in spatial information retrieval and computational geometry. The objective is to store a set of n points P in R d, each associated with a weight, so that it is possible to count, or more generally to compute some function of the weights of the points lying inside a given query range. Range searching is among the most heavily studied problems, and many search structures have been proposed and analyzed [1, 7]. There is a spectrum of space-time tradeoffs. The most relevant work to ours involves halfspace range counting queries, which Matouˇsek [6] has shown can be answered in n/m 1/d time from a data structure of space O(m). Nearly matching lower bounds were given by by Brönnimann, Chazelle and Pach [4] (or BCP). Given the relatively high complexity of range searching, it is natural to consider the problem in the context of approximation. We are given an approximation parameter ε> 0 and assume that ranges are bounded. Let η denote a range, and let diam(η) denote its diameter. All the points that lie in the range must be counted, and any of the points that lie within distance ε · diam(η) of the range’s boundary may be counted as well. Arya and Mount [3] showed that in any fixed dimension d with O(n log n) preprocessing time and O(n) space, ε-approximate range queries for any bounded convex range can be answered in time O(log n+1/ε d−1) [3]. Later, Chazelle, Liu, and Magen [5] considered approximate halfspace range and Euclidean ball searching in the high dimensional setting. Ignoring polylogarithmic factors, they showed that is possible to answer queries in O(d/ε 2) time with O(dn O(1/ε2) ) space.