Abstract not found

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.

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.

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 give simple randomized incremental algorithms for computing the k-level in an arrangement of n hyperplanes in two- and three-dimensional space. The expected running time of our algorithms is O(nk+nff(n) log n) for the planar case, and O(nk 2 +n log 3 n) for the three-dimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the k-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also be used to compute the k-level; this yields a randomized algorithm for computing the order-k Voronoi diagram of n points in the plane in expected time O(k(n \Gamma k) log n + n log 3 n).

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f . By a standard lifting map, we obtain outputsensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E 2 and also leads to improved output-sensitive results on constructing convex hulls in E d for any even constant d ? 4. 1 Introduction Geometric structures induced by n points in Euclidean d-dimensional space, such as the convex hull, Voronoi diagram, or Delaunay triangulation, can be of larger size than the point set that defines them. In many practical situat...

We show that the well-known random incremental construction of Clarkson and Shor [14] can be adapted via gradations to provide efficient external-memory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line segments in the plane with K pairwise intersections, that requires \Theta( log M=B N ) expected disk accesses, where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to obtain algorithms also for the problems of 3-d half-space intersections, 2-d and 3-d convex hulls, 2-d abstract Voronoi diagrams and batched planar point location, which require an optimal expected number of disk accesses and are simpler than the ones previously known. The results extend to an external-memory model with multiple disks. Additionally, under reasonable conditions on the parameters N;M;B, these results can be notably simplified originating practical algorithms which still achieve optimal expected bounds.

We describe the following data structures. For halfspace range reporting, in 3-space using expected preprocessing time O(n log n), worst-case storage O(n log log n) and worst-case reporting time O(log n + k), where n is the number of data points and k the number of points reported; in d-space, with d even, using worst-case preprocessing time O(n log n), storage O(n) and reporting time O(n 1 1=bd=2c log c n + k), where c is a constant. For ray shooting in a convex polytope in d-space determined by n facets, using deterministic preprocessing time O((n= log n) bd=2c log c n) and storage O((n= log n) bd=2c 2 c log n ) and with query time O(log n). For ray shooting in arbitrary direction among n hyperplanes using preprocessing O(n d = log bd=2c n) and query time O(log n). We also describe a randomized algorithm for constructing the k-level of n planes in 3-space. In the case of planes dual to points in convex position, in which the size of the k-level is O(nk), the a...

Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linear-space data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a re-examination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a re-examination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are non-uniform spatially or temporally (in which case one desires data structures that adapt to s...

We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, non-random constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...