Journal ArticleDOI
Optimal Randomized Parallel Algorithms for Computational Geometry I
H J Reif,Sandeep Sen +1 more
TLDR
In this paper, the authors present parallel algorithms for 3-D maxima and two-set dominance counting by an application of integer sorting, which have running time of O(logn)$ using $n$ processors, with very high probability.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.read more
Citations
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Journal ArticleDOI
Parallel techniques for computational geometry
TL;DR: A survey of techniques for solving geometric problems in parallel, both for shared memory parallel machines and for networks of processors, and the hybrid RAM/ARRAY model and its connection to I/O complexity are given.
Proceedings ArticleDOI
Optimal parallel randomized algorithms for the Voronoi diagram of line segments in the plane and related problems
TL;DR: An optimal parallel randomized algorithm for the Voronoi diagram of a set of non-intersecting line segments in the plane using efficient randomized search techniques and random sampling at “two stages” of the algorithm.
Proceedings ArticleDOI
Dynamic point location in arrangements of hyperplanes
Ketan Mulmuley,Sandeep Sen +1 more
TL;DR: Algorithms for maintaining data structures supporting fast (polylogarithmic) point-location and ray-shooting queries in arrangements of hyperplanes using random bits are presented, which has a versatile quality and is likely to have further applications to other dynamic algorithms.
Posted Content
Parallelism in Randomized Incremental Algorithms
TL;DR: It is shown that most sequential randomized incremental algorithms are in fact parallel, and three types of dependences found in the algorithms studied are identified and a framework for analyzing each type of algorithm is presented.
Book ChapterDOI
Deterministic Parallel Computational Geometry.
Mikhail J. Atallah,Danny Z. Chen +1 more
TL;DR: This work describes general methods for designing deterministic parallel algorithms in computational geometry and focuses on techniques for shared-memory parallel machines.
References
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Computational geometry. an introduction
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Journal ArticleDOI
A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations
TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.
Proceedings ArticleDOI
Applications of random sampling in computational geometry, II
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Journal ArticleDOI
Parallel merge sort
TL;DR: A parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small.
Journal ArticleDOI
Optimal Search in Planar Subdivisions
TL;DR: This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage.