Abstract not found

z Sivan Toledo x

Let P be a set of n points in R d (where d is a small fixed positive integer), and let \Gamma be a collection of subsets of R d , each of which is defined by a constant number of bounded degree polynomials. We consider the following \Gamma-range searching problem: Given P , build a data structure for efficient answering of queries of the form `Given a fl 2 \Gamma, count (or report) the points of P lying in fl'. Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1\Gamma1=b+ffi ) query time, where d b 2d \Gamma 3 and ffi ? 0 is an arbitrarily small constant. The actual value of b is related to the problem of partitioning arrangements of algebraic surfaces into constant-complexity cells. We present some of the applications of \Gamma-range searching problem, including improved ray shooting among triangles in R³.

We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in fixed dimension into a lineartime deterministic one. The constant of proportionality is d O(d) , which is better than for previously known such algorithms. We show that the algorithm works in a fairly general abstract setting, which allows us to solve various other problems (such as finding the maximum volume ellipsoid inscribed into the intersection of n halfspaces) in linear time.

In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the k-level of the arrangement A(F) is O(k 3+" /(n=k)), for any " ? 0, where /(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F . This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings, in the sense of [51], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static...

We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d. A point c ∈ IR d is a β-center point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d + 1)-center point; our algorithm finds an Ω(1/d 2)-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log 2 d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak ɛ-nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly. 1

We introduce the concept of a sensitive E-approx-imation, and use it to derive a more efficient algorithm for computing &-nets. We define and investigate prod-uct range spaces, for which we establish sampling the-orems analogous to the standard finite VC-dimensional case. This generalizes and simplifies results from previ-ous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(n log3 n) deterministic algorithm for com-puting the diameter of an n-point set in 3-dimensional space.

We give fast randomized and deterministic parallel meth-ods for constructing convex hulls in IR d, for any fixed d. Our methods are for the weakest shared-memory model,the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In partic-ular, we show that the convex hull of n points in IRd canbe constructed in O(log n) time using O(n log n + nbd=2c)work, with high probability. We also show that it can be constructed deterministically in O(log2 n) time using O(n log n) work for d = 3 and in O(log n) time using O(nbd=2c logc(dd=2e\Gamma bd=2c) n) work, for d * 4, where c? 0is a constant, which is optimal for even d * 4. We also showhow to make our 3-dimensional methods output-sensitive with only a small increase in running time.These methods can be applied to other problems as well. A variation of the convex hull algorithm for even dimen-sions deterministically constructs a (1=r)-cutting of n hy-perplanes in IR d in O(log n) time using optimal O(nrd\Gamma 1) work; when r = n, we obtain their arrangement and a pointlocation data structure for it. With appropriate modifications, our deterministic 3-dimensional convex hull algorithmcan be used to compute, in the same resource bounds, the intersection of n balls of equal radius in R³. This leads to asequential algorithm for computing the diameter of a point set in IR3 with running time O(n log³ n), which is arguablysimpler than an algorithm with the same running time by Brönnimann et al.

We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach