H J Reif

Bio: H J Reif is an academic researcher from Duke University. The author has contributed to research in topics: Parallel algorithm & Integer sorting. The author has an hindex of 1, co-authored 1 publications receiving 45 citations.

Optimal Randomized Parallel Algorithms for Computational Geometry I

Abstract: We present parallel algorithms for some fundamental problems in computational geometry which have running time of $O(logn)$ using $n$ processors, with very high probability (approaching 1 as $n~ \rightarrow~ \infty$). These include planar point location, triangulation and trapezoidal decomposition. We also present optimal algorithms for 3-D maxima and two-set dominance counting by an application of integer sorting. Most of these algorithms run on CREW PRAM model and have optimal processor-time product which improve on the previously best known algorithms of Atallah and Goodrich [3] for these problems. The crux of these algorithms is a useful data structure which emulates the plane sweeping paradigm used for sequential algorithms. We extend some of the techniques used by Reischuk [22] Reif and Valiant [21] for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.

Applications of random sampling in computational geometry, II

Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

Cascading Divide-and-conquer: A Technique for Designing Parallel Algorithms

Abstract: We present techniques for parallel divide-and-conquer, resulting in improved parallel algorithms for a number of problems. The problems for which we give improved algorithms include intersection detection, trapezoidal decomposition (hence, polygon triangulation), and planar point location (hence, Voronoi diagram construction). We also give efficient parallel algorithms for fractional cascading, 3-dimensional maxima, 2-set dominance counting, and visibility from a point. All of our algorithms run in O(log n) time with either a linear or sub-linear number of processors in the CREW PRAM model.

A deterministic view of random sampling and its use in geometry

Abstract: A number of efficient probabilistic algorithms based on the combination of divide-and-conquer and random sampling have been recently discovered. It is shown that all those algorithms can be derandomized with only polynomial overhead. In the process. results of independent interest concerning the covering of hypergraphs are established, and various probabilistic bounds in geometry complexity are improved. For example, given n hyperplanes in d-space and any large enough integer r, it is shown how to compute, in polynomial time, a simplicial packing of size O(r/sup d/) that covers d-space, each of whose simplices intersects O(n/r) hyperplanes. It is also shown how to locate a point among n hyperplanes in d-space in O(log n) query time, using O(n/sup d/) storage and polynomial preprocessing. >

The probabilistic method yields deterministic parallel algorithms

Abstract: A method is provided for converting randomized parallel algorithms into deterministic parallel algorithms. The approach is based on a parallel implementation of the method of conditional probabilities. Results obtained by applying the method to the set balancing problem, lattice approximation, edge-coloring graphs, random sampling, and combinatorial constructions are presented. The general form in which the method of conditional probabilities is applied sequentially is described. The reason why this form does not lend itself to parallelization are discussed. The general form of the case for which the method of conditional probabilities can be applied in the parallel context is given. >